Research ArticleMathematical Modelling of Human African TrypanosomiasisUsing Control Measures

Hamenyimana Emanuel Gervas 123 Nicholas Kwasi-Do Ohene Opoku 14

and Shamsuddeen Ibrahim 1

1African Institute for Mathematical Sciences Biriwa Cape Coast Ghana2University of Dar es Salaam Dar es Salaam Tanzania3University of Dodoma Dodoma Tanzania4University of Cape Coast Cape Coast Ghana

Correspondence should be addressed to Hamenyimana Emanuel Gervas hamenyimanaaimsedugh

Received 10 July 2018 Revised 12 September 2018 Accepted 22 October 2018 Published 22 November 2018

Academic Editor Konstantin Blyuss

Copyrightcopy 2018HamenyimanaEmanuelGervas et al(is is anopenaccess article distributedunder theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Human African trypanosomiasis (HAT) commonly known as sleeping sickness is a neglected tropical vector-borne diseasecaused by trypanosome protozoa It is transmitted by bites of infected tsetse fly In this paper we first present the vector-hostmodel which describes the general transmission dynamics of HAT In the tsetse fly population the HAT is modelled by threecompartments while in the human population the HATis modelled by four compartments(e next-generationmatrix approachis used to derive the basic reproduction number R0 and it is also proved that if R0 le 1 the disease-free equilibrium is globallyasymptotically stable which means the disease dies out (e disease persists in the population if the value of R0 gt 1 Furthermorethe optimal control model is determined by using the Pontryaginrsquos maximum principle with control measures such as educationtreatment and insecticides used to optimize the objective function (e model simulations confirm that the use of the threecontrol measures is very efficient and effective to eliminate HAT in Africa

1 Introduction

Human African trypanosomiasis (HAT) commonly known assleeping sickness is a vector-borne tropical disease which iscaused by Trypanosoma brucei protozoa species It is one of theneglected tropical diseases which affect people in sub-SaharanAfrica specifically those living in rural areas HAT is caused bytwo species of protozoa which are Trypanosoma brucei gam-biense (TBG) which causes the chronic form of HATin centraland western Africa and Trypanosoma brucei rhodesiense(TBR) which causes the acute form of the disease in easternand southern Africa [1](e HATdisease has killed millions ofpeople since the beginning of 20th century and it is transmittedfrom one individual to another by tsetse flies (genus Glossina)TBG is transmitted by riverine tsetse species while TBR istransmitted by savanna tsetse species [1] Rhodesiense HAT isan acute disease that can lead to death if not treated within 6months while gambiense HAT is a slow chronic progressivedisease which causes death with an average duration of 3 years

[2] (e signs and symptoms for both forms of HAT are notspecific and their appearances vary from one person to anotherat the first stage of HAT the disease is not severe and the signsand symptoms such as intermittent fever headache prurituslymphadenopathies asthenia anemia cardiac disorders en-docrine disturbances musculoskeletal pains and hep-atosplenomegaly may be observed while in the second stage ofHAT sleep disorders and neuropsychiatric disorders are likelyto dominate (e HAT disease can be treated by using drugssuch as suramin eflornithine melarsoprol and pentamidine

(edisease is reported to affect about 37 sub-SaharanAfricancountries it affects much rural areas where there are suitableenvironments for the tsetse flies to live and reproduce and theperiurban areas can also be affected (e transmission of HATcan occur during human activities such as hunting farming aswell as fishing [3] (e transmission of HATneeds the reservoirreservoir is a species that can permanentlymaintain the pathogenand from which the pathogen can be transmitted to the targetpopulation [4] Rhodesiense HAT is zoonotic which requires

HindawiComputational and Mathematical Methods in MedicineVolume 2018 Article ID 5293568 13 pageshttpsdoiorg10115520185293568

a nonhuman reservoir (animals) for maintaining its populationwhile in gambiense HAT humans act as key reservoir [4]

Mathematical models have been used to study the trans-mission and effective control of diseases simply and cheaply withno need of expensive and complicated experiments [5] So fardifferent models have been developed and formulated by dif-ferent researchers One of the important modelling work onHAT was done by Rogers [6] the model explained the math-ematical framework on transmission of HAT in multiple hostpopulations [6] Rogersrsquo model was generalized byHargrove et al[7] and a new parameter which allows the tsetse flies to feed offmultiple hosts was introduced (e model compared the ef-fectiveness of two methods used to control HAT insecticide-treated cattle and the use of trypanocide drugs to treat cattle(eyfound out that treating cattle with insecticides is more effectiveand a cheaper approach to control HAT than using trypanocidedrugs Kajunguri [8] developed a model which was based ona constant population with a fixed number of domestic animalshuman and tsetse flies in one of the villages in West Africa (emajor findings of theirmodel estimated that the cattle populationcontributes to about 92 of the total TBR transmission while therest 8 is the contribution of human population in transmissionof the disease(e study by Kajunguri [8] which also formulateda multihost model was used to study the control of tsetse fliesand TBR in southern Uganda (ey found out that the effectiveapplication of insecticides brings about a cost-effectivemethod ofcontrol and eliminating the disease (ey realized that usinginsecticides for controlling HATis more effective and efficient inthe area where there are few wild hosts

Due to low mortality rate of the disease and poverty of itssufferers the efforts toward the control of HAT has reducedMost attention is given to popular diseases such asHIVAIDStuberculosis malaria and ebola although the disease is stilla threat to the lives of sub-Saharan African people Moreoververy few studies have been carried out on applying optimalcontrol theory to HAT transmission models In this paper weuse optimal control theory to study the transmission dy-namics of HAT diseases by using education treatment andinsecticides as the control measures

(e rest of this paper is outlined as follows Section 2represents the vector-host model and the underlying as-sumptions In section 3 the model equilibria and stabilities aredetermined whereas in Section 4 the optimal control model isanalyzed by modifying the previous one to control the HAT byusing control measures (education insecticides and treatment)In addition the numerical simulations for the optimal controlmodel are done in this section andwe use the results obtained tocompare the efforts of each control measure to control the HATin Africa Finally we provide the conclusion in Section 5

2 Model Formulation

In this section the vector-host model as well as the necessarydifferential equations to describe the transmission of HAT fromtsetse fly to human and vice versa are developed (e trans-mission of HATin the human population is modelled using foursubclasses Susceptible SH Exposed EH Infectious IH and Re-covered RH(e total human population NH is thus defined by

NH SH + EH + IH + RH (1)

(e transmission of HAT in the vector (tsetse flies)population is also divided into Susceptible (SV) Exposed(EV) and Infectious (IV) (e total population of the tsetseflies NV is also defined by

NV SV + EV + IV (2)

We assume a constant population for both host andvector It is also assumed that the tsetse fly cannot recoverfrom the disease and the infected tsetse fly remains infectiousthroughout the rest of its life there is no disease-induceddeath rate for tsetse flies and the recruitment rates are as-sumed to be constant due to birth and immigration

In ourmodel the recruitment rate of hosts and vectors arerepresented by πH and πV respectively (e susceptible hostgets the disease when bitten by infectious tsetse fly andsusceptible tsetse fly gets the disease when it bites an infectioushuman at the rate a (e natural mortality rate for humansand vectors are represented by μH and μV respectively (eparameter ω represents the disease-induced death rate forhumans while ξH and ξV are the force of infection for humansand vectors respectively (e parameter σ represents percapita rate of a vector becoming infectious and the rest of theparameters are explained in Table 1 Assuming that thetransmission per bite from infectious tsetse fly to human is athen the rate of infection per susceptible human is given by

ξH apHIV

NV (3)

and also if we further assume that a is the tsetse-fly bitingrate that is the average number of bites per tsetse fly perunit then the rate of infection per susceptible tsetse fly canbe represented by

ξV apVIH

NH (4)

From the model diagram in Figure 1 the followingdifferential equations are derived

dSH

dt πHNH + ρRH minus

apHIV

NVSH minus μHSH

dEH

dt

apHIV

NVSH minus εEH minus μHEH

dIH

dt εEH minus μHIH minusωIH minus τIH

dRH

dt τIH minus ρRH minus μERH

dSV

dt πVNV minus μVSV minus

apVIH

NHSV

dEV

dt

apVIHNH

SV minus μVEV minus σEV

dIV

dt σEV minus μVIV

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

2 Computational and Mathematical Methods in Medicine

From system (5) the dimensionless technique is used toderive another equivalent differential equation we denotesh (SHNH) eh (EHNH) ih (IHNH) rh (RHNH)sv (SVNV) ev (EVNV) and iv (IVNV) and sub-stitute in system (5) to obtain the following new equivalentequations

dsh

dt πh + ρrh minus aphivsh minus μhsh

deh

dt aphivsh minus εeh minus μheh

dih

dt εeh minus μhih minusωih minus τih

drh

dt τih minus ρrh minus μhrh

dsvdt

πv minus μvsv minus apvihsv

dev

dt apvihsv minus μvev minus σev

div

dt σev minus μviv

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

Table 1 shows the description of the model parametersand variables

21 Positivity and Boundedness of the Solutions In thissubsection we show that system (6) is epidemiologically andmathematically well defined in the positive invariant region

D sh eh ih rh sv ev iv( 1113857 isin R7+ nh le

πh

μh nv le

πv

μv1113896 1113897 (7)

Theorem 1 2ere exists a domain D in which the solution(sh eh ih rh sv ev iv) is contained and bounded

Proof We provide the proof following the idea by Olaniyiand Obabiyi [9] Given the solution set (sh eh ih rh sv ev iv)

with the positive initial conditions (sh0 eh0 ih0 rh0 sv0

ev0 iv0) we define

nh sh eh ih rh( 1113857 sh(t) + eh(t) + ih(t) + rh(t) and

nv sv ev iv( 1113857 sv(t) + ev(t) + iv(t)(8)

(e derivatives of nh and nv with respect to time alongthe solution of system (6) for human and tsetse flies re-spectively are obtained by

nhprime dsh

dt+

deh

dt+

dih

dt+

drh

dt

πh minus sh + eh + ih + rh( 1113857μh minusωih

πh minus nhμh

nvprime dsv

dt+

dev

dt+

div

dt

πv minus sv + ev + iv( 1113857μv

πv minus nvμv

(9)

From these differential equations it follows that nhprime leπh minus μhnh and nvprime le πv minus μvnv We obtain the solutions asfollows

nh leπhμh

1minus exp minusμht( 1113857( 1113857 + nh sh0 eh0 ih0 rh01113872 1113873exp minusμht( 1113857

nv leπvμv

1minus exp minusμvt( 1113857( 1113857 + nv sv0 ev0 iv01113872 1113873exp minusμvt( 1113857

(10)

By taking the limits of both nh and nv above as t⟶infinwe obtain nh le (πhμh) and nv le (πvμv) hence the solutionsare contained in the region D (is implies that all solutionsof the human and tsetse fly population are contained in theregion D and are nonnegative this guarantees that thepositive invariant region for system (6) exists and is given by

D sh eh ih rh sv ev iv( 1113857 isin R7+ nh le

πhμh

nv leπv

μv1113896 1113897

(11)

3 Model Equilibria and Stability Analysis

In this section we give the model equilibria the basic re-production number R0 and the stabilities at both disease-free and endemic equilibrium

31 Disease-Free Equilibrium (DFE) (e DFE in system (6)is when there are no HAT infections within the human andtsetse fly population (us the existence of the DFE is givenby E0 ((πhμh) 0 0 0 (πvμv) 0 0)

Table 1 (e description of model variables and parameters

Variable Descriptionsh Susceptible human populationsv Susceptible tsetse fly populationeh and ev Exposed human and tsetse fly population respectivelyih and iv Infectious human and tsetse population respectivelyrh Recovered human populationParameter Descriptionπh Recruitment rate for human populationπv Recruitment rate for tsetse fly population

phProportion of bites by the infectious vector on

susceptible human population

pvProportion of bites by susceptible vector on an

infectious human populationa (e biting rate of the tsetse fliesσ Per capita rate of a vector becoming infectiousε Per capita rate of human becoming infectiousω Disease induced death rate

ρ (e rate at which the recovered human can becomesusceptible again

τ Recovery rateμ Natural death rateξh Force of infection for human populationξv Force of infection for tsetse flies

Computational and Mathematical Methods in Medicine 3

32 Endemic Equilibrium (EE) e EE is the nontrivialequilibrium point at which the HATdisease persists in bothhuman and tsetse y population us the EE is obtained asfollows Elowast (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) where

slowasth πh ρ + μh( ) + ρτilowasth[ ] μv σ + μv( ) apvilowasth + μv( )[ ]

a2σphpvπvilowasth + μvμh σ + μv( ) apvilowasth + μv( )[ ] μh + ρ( )

elowasth ω + τ + μh( )ilowasth

ε

rlowasth τilowasth

μh + ρ

slowastv πv

apvilowasth + μv

elowastv apvπvi

lowasth

σ + μv( ) apvilowasth + μv( )

ilowastv aσpvπvilowasth

σ + μv( ) apvilowasth + μv( )μv

ilowasth ρ + μh( ) a2εphpvπhπvσ minus μ2vμh ε + μh( ) σ + μv( ) μh + τ + ω( )[ ]

B

(12)

and the term

B ( apv( aσphπv( ερω + μh( ρ(τ + ω) + ε(ρ + τ + ω)+ μh ε + ρ + τ + ω + μh( ))) + μh ε + μh( ) ρ + μh( )middot τ + ω + μh( )μv σ + μv( )))

(13)

33 Basic ReproductionNumberR0 e basic reproductionnumber R0 is dened as the number of secondary in-fections caused by one infected host or vector in a com-pletely susceptible population [10] e next-generationmatrix approach as done by Van den Driessche andWatmough in [5 11] is applied to derive

F aphivsh

0

apvihsv

V

μheh + εehminusεeh + μhih + ωih + τih

μvev + δevminusδev + μviv

(14)

By denoting matrix F (zFzxi) and V (zVzxi)where xi eh ih ev iv the spectral radius of the next-generation matrix FVminus1 gives the value of R0

F

0 0 0 aphsh

0 0 0 0

0 apvsv 0 0

0 0 0 0

V

ε + μh 0 0 0

minusε μh + ω + τ 0 0

0 0 μv + δ 0

0 0 minusδ μv

FVminus1

0 0aδph

μv minusδ + μv( )minusaphμv

0apv

μh + τ + ω0 0

0 0 0 0

0 0 0 0

(15)

лH

лV

SH

SV

μHSH μHEH

ξHSH

ξVSV

EH

EV

εEH

ρRH

ωIH

IHτIH

μHIH

μVIVμVEVμVSV

μHRH

RH

IVσEV

Figure 1 Compartmental model for the transmission of human African trypanosomiasis

4 Computational and Mathematical Methods in Medicine

(e spectral radius σ(FVminus1) gives

R0 σ FVminus1

1113872 1113873

a2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

1113971

(16)

One infected human in a population of susceptible vectorswill cause Rv infected vectors likewise one infected vector ina population will cause Rh infected humans [5](erefore thebasic reproduction number can be rewritten as R0

RhRv

1113968

where Rh (aεphπhμh(μh + ε)(μh + τ + ω)) and Rv

(σapvπvμ2v(σ + μv)) (us R0 can also be defined as thesquare root of the product of the number of infected humansin the susceptible population caused by one infected tsetsefly in its infectious lifetime and the number of infected tsetse

flies caused by one infected human during the infectiousperiod [12]

34 Local Stability of Disease-Free Equilibrium (DFE)

Theorem 2 If R0 le 1 the DFE given by E0 is locally as-ymptotically stable in the region defined by (7) and it isunstable when R0 gt 1

Proof (e DFE is locally stable if all eigenvalues of Jacobianmatrix JE0

are negative (e matrix has all eigenvalues neg-ative only if the trace of JE0

lt 0 and determinant of JE0gt 0 By

linearizing system (6) around E0 we obtain the followingJacobian matrix

JE0

minusμh 0 0 ρ 0 0 minusaphsh

0 minus ε + μh( 1113857 0 0 0 0 aphsh

0 ε minus ω + τ + μh( 1113857 0 0 0 00 0 τ minus ρ + μh( 1113857 0 0 00 0 minusapvsv 0 minusμv 0 00 0 apvsv 0 0 minus σ + μv( 1113857 00 0 0 0 0 σ minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(17)

(e trace of matrix JE0is such that

tr JE01113872 1113873 minus( μh + ε + μh + μh + ω + τ + ρ + μh

+ μv + σ + μv + μv1113857

minus 4μh + 3μv + ε + ω + τ + ρ + σ( 1113857lt 0

(18)

Using the basic properties of matrix algebra as in [13] itis clear that the eigenvalues λ1 minusμh and λ2 minusμv of thematrix JE0

have negative real parts (e reduced matrix is

JE1

minus ε + μh( 1113857 0 0 0 aphsh

ε minus ω + τ + μh( 1113857 0 0 0

0 τ minus ρ + μh( 1113857 0 0

0 apvsv 0 minus σ + μv( 1113857 0

0 0 0 σ minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(19)

From matrix JE1 the eigenvalue λ3 minus(ρ + μh) has

negative real part (e remaining matrix is further reducedby using the reduction techniques and we obtain

JE2

minus ε + μh( 1113857 0 0 aphsh

0 minus ω + τ + μh( 1113857 0aphshεε + μh

0 apvsv minus σ + μv( 1113857 0

0 0 σ minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(20)

Using the properties of matrix algebra the matrix JE2has

eigenvalue minus(ε + μh) which has negative real part Wefurther reduce to a 2 times 2 matrix by using the same reductiontechniques (e matrix is

JE3

minus ω + τ + μh( 1113857aphshεε + μh

apvsvσσ + μv

minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (21)

From the reduced 2 times 2 matrix the trace is negative andthe determinant is

Det JE31113872 1113873 ω + τ + μh( 1113857μv minus

aphshεε + μh

timesapvsvσσ + μv

ω + τ + μh( 1113857μv 1minusa2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 11138571113890 1113891

(22)

Since

R0

a2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

1113971

(23)

then by letting RT (a2εphpvπhπvσμ2vμh(ε + μh)(σ + μv)(μh + τ + ω)) we find our determinant as

Det JE31113872 1113873 ω + τ + μh( 1113857μv 1minusRT1113858 1113859 (24)

(e value of RT can be seen to be positive because all theparameters are positive As a result the determinant in (24)

Computational and Mathematical Methods in Medicine 5

is positive if and only if RT lt 1 (erefore the DFE is locallystable if RT le 1

35 Global Stability of Disease-Free Equilibrium (DFE)To show that the DFE is globally stable we apply Lyapunovrsquostheorem in [5]

Theorem 3 2eDFE defined by E0 is globally asymptoticallystable in the region defined by (7) if R0 le 1 Otherwise unstableif R0 gt 1

Proof We define Lyapunovrsquos function as

V k1 sh minus sh0 minus sh0 lnsh

sh01113888 1113889 + k2eh + k3ih

+ k4 sv minus sv0 minus sv0 lnsv

sv01113888 1113889 + k5ev + k6iv

(25)

satisfying system (6) where k1 k2 k3 k4 k5 k6 gt 0 are to bedetermined and sh0 (πhμh) and sv0 (πvμv) We firstshow that Vgt 0 for all Ene ((πhμh) 0 0 0 (πvμv) 0 0) It isenough to check that

k1sh0sh

sh0minus 1minus ln

sh

sh01113888 1113889gt 0

k4sv0svsv0minus 1minus ln

svsv0

1113888 1113889gt 0

(26)

(e function g(m) mminus 1minus lnm such thatm (shsh0) (svsv0) has minimum value equal to zerowhen m 1 hence g(m)gt 0 for all mgt 0 (us Lyapunovrsquosfunction Vgt 0 (e function V is radially unbounded be-cause as |m|⟶infin the function g(m)⟶infin We nowtake the derivative of V with respect to time and use system(6) to replace the derivatives in the right hand side such that

Vprime k1 1minussh0sh

1113888 1113889dsh

dt+ k2

deh

dt+ k3

dih

dt+ k4 1minus

sv0sv

1113888 1113889dsv

dt

+ k5dev

dt+ k6

div

dt

k1 1minussh0sh

1113888 1113889 πh + ρrh minus aphivsh minus μhsh1113858 1113859

+ k2 aphivsh minus εeh minus μheh1113858 1113859

+ k3 εeh minus μhih minusωih minus τih1113858 1113859

+ k4 1minussv0sv

1113888 1113889 πv minus μvsv minus apvihsv1113858 1113859

+ k5 apvihsv minus μvev minus σev1113858 1113859 + k6 σev minus μviv1113858 1113859

2k1πh minus aphshk1iv + ρrhk1 minus k1μhsh

minus k1π2h

μhsh+

aphπhiv

μhk1 minus

πhρrh

μhshk1 + k2aphshiv

minus k2 ε + μh( 1113857eh + k3εeh minus k3 τ + μh + ω( 1113857ih

+ 2k4πv minus k4apvsvih minus μvsvk4 minusπ2v

μvsvk4

+apvihπv

μvk4 minus k5 μv + σ( 1113857ev

+ k5apvsvih + k6σev minus k6μviv(27)

(e terms with rh are ignored because if sh eh ih areglobally stable then rh⟶ 0 at any time t and the DFE forsystem (6) is globally stable Taking k1 k2 (1μh + ε)k4 k5 (1μv + σ) k3 (1ε) and k6 (1σ) the de-rivative of V with respect to time becomes

Vprime minusπh

μh + επhμhsh

+μhsh

πhminus 21113888 1113889minus

τ + μh + ω( 1113857

εih

+aphπhμh + ε( 1113857μh

iv minusπv

μv + σπv

μvsv+μvsvπvminus 21113888 1113889

minusμvσ

iv +apvπv

μv μv + σ( 1113857ih

minusπh

μh + επh

μhsh+μhsh

πhminus 21113888 1113889 +

πv

μv + σπv

μvsv+μvsvπvminus 21113888 11138891113890 1113891

+τ + μh + ω( 1113857

εεapvπv

μv σ + μv( 1113857 μh + τ + ω( 1113857minus 11113888 1113889ih

+μvσ

aσphπhμvμh μh + ε( 1113857

minus 11113888 1113889iv

minusπh

μh + επh

μhsh+μhsh

πhminus 21113888 1113889 +

πv

μv + σπv

μvsv+μvsvπvminus 21113888 11138891113890 1113891

+τ + μh + ω( 1113857

εRv minus 1( 1113857ih +

μv

σRh minus 1( 1113857iv

(28)

(e terms ((πhμhsh) + (μhshπh)minus 2) and ((πvμvsv) +

(μvsvπv)minus 2) are positive because if we suppose m

(πhμhsh) (πvμvsv) we have m + (1m)minus 2 (m2 minus 2m +

1m) ((mminus 1)2m)gt 0 for all mgt 1 and since Rv le 1 andRh le 1 then Vprime is negative (us we have Vprime lt 0 for allE0 ne ((πhμh) 0 0 0 (πvμv) 0 0)

(us the largest compact invariant set in D is the sin-gleton set E0 Hence system (6) is globally asymptoticallystable

6 Computational and Mathematical Methods in Medicine

36 Local Stability of Endemic Equilibrium (EE)

Theorem 4 2e unique endemic equilibrium defined by Elowast

is locally asymptotically stable in the region defined by (7) ifR0 gt 1 but is unstable if R0 le 1

Proof We give the proof of this theorem based on the ap-proach used by Olaniyi and Obabiyi [9 14] From the EE

points defined in (12) since all values are positive we expressthe value of ilowasth in terms of R0 to obtain

ilowasth

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

B

middota2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857minus 11113890 1113891

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

BR20 minus 11113960 1113961

(29)

where

B ( apv( aσphπv( ερω + μh( ρ(τ + ω) + ε(ρ + τ + ω)

+ μh ε + ρ + τ + ω + μh( 111385711138571113857 + μh ε + μh( 1113857 ρ + μh( 1113857

middot τ + ω + μh( 1113857μv σ + μv( 111385711138571113857

(30)

Since the basic reproduction number R0 (a2εphpvπhπvσμ2vμh(ε + μh)(σ + μv)(μh + τ + ω))

1113968 if we

let

RT a2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857 (31)

then we can find ilowasth as

ilowasth

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

BRT minus 11113858 1113859

(32)

(e value of B is clearly positive because all parametersare positive Hence ilowasth gt 0 if and only if RT gt 1 implying thatthe EE is locally asymptotically stable if RT gt 1

37 Global Stability of Endemic Equilibrium (EE) To showthe global stability of the EE we use Lyapunovrsquos theoremtogether with the following lemma

Lemma 1 Suppose that y1 y2 middot middot middot yn are n positive num-bers 2en their arithmetic mean is greater than or equal tothe geometric mean that is (y1 + y2 + middot middot middot + ynn)ge(y1y2 middot middot middot yn)1n

Theorem 5 2e EE defined by Elowast is globally asymptoticallystable if R0 gt 1 otherwise unstable

Proof (e proof is based on the idea as explained byMartcheva [5] We define Lyapunovrsquos function as

V k1 sh minus slowasth minus slowasth ln

shslowasth

1113888 1113889 + k2 eh minus elowasth minus elowasth ln

ehelowasth

1113888 1113889

+ k3 ih minus ilowasth minus ilowasth ln

ih

ilowasth1113888 1113889 + k4 sv minus s

lowastv minus slowastv ln

sv

slowastv1113888 1113889

+ k5 ev minus elowastv minus elowastv ln

ev

elowastv1113888 1113889 + k6 iv minus i

lowastv minus ilowastv ln

iv

ilowastv1113888 1113889

(33)

satisfying system (6) with k1 k2 k3 k4 k5 k6 gt 0 to bedetermined (e function V is nonnegative for all(sh eh ih rh sv ev iv)ne (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) and radiallyunbounded

We need to prove that Vprime lt 0 for all(sh eh ih rh sv ev iv)ne (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) We find thederivative of V with respect to time and replace the de-rivatives sh

prime ehprime ihprime rhprime svprime evprime ivprime with system (6) We also ignore

the rh terms because if sh eh ih are globally stable thenrh⟶ 0 at any time t and EE is globally stable

Vprime k1 1minusslowasthsh

1113888 1113889 πh minus aphivsh minus μhsh1113858 1113859 + k2 1minuselowastheh

1113888 1113889

middot aphivsh minus μh + ε( 1113857eh1113858 1113859 + k3 1minusilowasthih

1113888 1113889

middot εeh minus ω + τ + uh( 1113857ih1113858 1113859 + k4 1minusslowastvsv

1113888 1113889

middot πv minus apvihsv minus μvsv1113858 1113859 + k5 1minuselowastvev

1113888 1113889

middot apvihsv minus μv + σ( 1113857ev1113858 1113859 + k6 1minusilowastviv

1113888 1113889 σev minus uviv1113858 1113859

(34)

We now substitute πh aphilowastv slowasth + μhslowasth and πv

apvilowasthslowastv + μvslowastv at the endemic equilibrium and then simplifyand put similar terms together to obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth minus k1aphivsh minus k1aph

slowast2h ilowastvsh

+ k1aphivslowasth + k2aphshiv minus k2 ε + μh( 1113857eh minus k2aphshiv

elowastheh

+ k2 ε + μh( 1113857elowasth + k3εeh minus k3 ω + τ + μh( 1113857i

lowasth minus k3εeh

ilowasthih

+ k3 ω + τ + μh( 1113857ilowasth minus k4

sv minus slowastv( 11138572μv

sv+ k4apvi

lowasthslowastv

minus k4apvihsv minus k4apvslowast2v ilowasth

sv+ k4apvihs

lowastv + k5apvsvih

minus k5 σ + μv( 1113857ev minus k5apvsvihelowastvev

+ k5 σ + μv( 1113857elowastv + k6σev

minus k6μvilowastv minus k6σev

ilowastviv

+ k6μvilowastv

(35)

Computational and Mathematical Methods in Medicine 7

We suppose k1 k2 and k4 k5 and multiply and dividethe same equilibrium value to some of the fractions to obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth minus k1aph

slowast2h ilowastvsh

+ k1aphivslowasth

minus k2 ε + μh( 1113857eh minus k2aphshivelowasthslowasth ilowastvslowasth ilowastv eh

+ k2 ε + μh( 1113857elowasth

+ k3εeh minus k3 ω + τ + μh( 1113857ih minus k3εehilowasthelowasthihelowasth

+ k3 ω + τ + μh( 1113857ilowasth minus k4

sv minus slowastv( 11138572μv

sv+ k4apvi

lowasthslowastv

minus k4apvslowast2v ilowasth

sv+ k4apvihs

lowastv minus k5 σ + μv( 1113857ev

minus k5apvsvihelowastv slowastv ilowasthslowastv ilowasthev

+ k5 σ + μv( 1113857elowastv + k6σev minus k6μvi

lowastv

minus k6σevelowastv ilowastvivelowastv

+ k6μvilowastv

(36)

We choose k3 k2(μh + εε) such that k3(ω + τ+ μh)ilowasth

k2(ε + μh)elowasth and choose k6 k5(μh + σσ) such thatk6μvilowastv k5(σ + μv)elowastv Now aphslowasth ilowastv (μh + ε)elowasth andapvslowastv ilowastv (μv + σ)elowastv because k1 k2 and k4 k5 re-

spectively We now obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

+ k1aphivslowasth minus k3 ω + τ + μh( 1113857ih + k3εminus k2 ε + μh( 11138571113858 1113859eh

minus k4sv minus slowastv( 1113857

2μvsv

+ k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsv

slowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891

+ k4apvihslowastv minus k6μviv + k6σ minus k5 σ + μv( 11138571113858 1113859ev

(37)

Suppose k6 (k1aphslowasthμv) and k3 (k4apvslowastv ω + τ+

μh) We then substitute and simplify to get

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

minus k4sv minus slowastv( 1113857

2μvsv

+ k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsvslowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891

(38)

From Lemma 1 the terms

k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsv

slowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891le 0

(39)

(erefore Vprime lt 0 for all (sh eh ih rh sv ev iv)ne (slowasth

elowasth ilowasth rlowasth slowastv elowastv ilowastv ) implying that the endemic equilibrium isglobally asymptotically stable if R0 gt 1

4 Analysis of Optimal Control Model

In this section we formulate the optimal control model bymodifying system (6) to an optimal control problem Wethus define some linear functions ci(t) 1 for i 1 2 3 It isimportant to note that controls are fully effective whenci(t) 1 and not effective when ci(t) 0 (e forces ofinfection ξh and xiv which correspond to the human andvector population respectively are reduced by the factor(1minus c1) where c1 measures the level of success obtained dueto the effort of educating people on the dangers of exposingtheir skin and encouraging them to wear long sleeves andlong pants during the day to minimize tsetse fly-humancontacts (e factor c2 represents the effort of treatment tocontrol the disease and the factor c3 also represents theeffort of using insecticides to ensure that the breeding sites ofthe tsetse fly are minimized Hence taking into account theassumptions and extensions made we try to find the mosteffective strategy that reduces the HAT infection in thepopulation at a veryminimum costWith the use of boundedLebesgue measurable control we define the objectivefunction to be minimized as

J c1 c2 c3( 1113857 1113946tF

0( M1eh + M2ih + M3ev + M4iv

+12k1c

21 +

12k2c

22 +

12k3c

231113857 dt

(40)

(us the dynamics of the controls that minimizes theobjective function is given by

dshdt

πh + ρrh minus 1minus c1( 1113857aphivsh minus μhsh

deh

dt 1minus c1( 1113857aphivsh minus εeh minus μheh

dih

dt εeh minus μhih minusωih minus c2τih

drh

dt c2τih minus ρrh minus μhrh

dsv

dt πv minus c3μvsv minus 1minus c1( 1113857apvihsv

dev

dt 1minus c1( 1113857apvihsv minus c3μvev minus σev

div

dt σev minus c3μviv

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

subject to the initial conditions sh ge 0 eh ge 0 ih ge 0 rh ge 0

sv ge 0 ev ge 0 and iv ge 0 (e associated effective reproduc-tion number for (41) denoted by RE is obtained as

RE

a2ε 1minus c1( 11138572phπhπvσ

c3μh ε + μh( 1113857μ2v σ + c3μv( 1113857 μh + c2τ + ω( 1113857

1113971

Rc

1113968

(42)

8 Computational and Mathematical Methods in Medicine

(e goal is to minimize the exposed and infectious humanpopulations (eh ih) the exposed and infectious vector pop-ulations (ev iv) and the cost of implementing the control bythe use of possible ci i 1 2 3 (e functional objectiveincludes the social cost which relates to the resources that areneeded for educating people on personal protection (12)k1c

21

the application of treatment (12)k2c22 and spraying of tsetse

fly operations (12)k3c23 (e quantities M1 and M2 re-

spectively represent the associated cost with minimizing theexposed and infected human population while M3 and M4also represent the cost associated withminimizing the exposedand infected vectors respectively (e quantity tF is the timeperiod of intervention As explained in [15] the costs cor-responding to M1eh M2ih M3ev and M4iv are linear whilethe cost control functions (12)k1c

21 (12)k2c

22 and (12)k3c

23

should be nonlinear and take a quadratic form (erefore weseek to minimize the objective function over the given timeinterval [0 tF] Pontryaginrsquos maximum principle is used tosolve this optimal control problem and the derivation of thenecessary conditions (e Lagrangian of the optimal controlproblem is given by

L M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22 +

12k3c

231113874 1113875

(43)

To determine the Lagrangian minimum value we definethe Hamiltonian H for the control problem as

H M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22

+12k3c

23 + λsh

dsh

dt+ λeh

deh

dt+ λih

dih

dt+ λrh

drh

dt

+ λsv

dsv

dt+ λev

dev

dt+ λiv

div

dt

(44)

where λsh λeh

λih λrh

λsv λev

and λivare adjoint variables or

costate variables (e differential equations of adjoint var-iables are obtained by taking the partial derivatives of theHamiltonian equation with respect to the state variableswhich gives

dλsh

dt λsh

μh + λshminus λeh

1113872 1113873 1minus c1( 1113857aphiv

dλeh

dt minusM1 + λeh

minus λih1113872 1113873ε + λeh

μh

dλih

dt minusM2 + λih

minus λrh1113872 1113873c2τ + λih

ω + μh( 1113857 + λsvminus λev

1113872 1113873 1minus c1( 1113857apvsv

dλrh

dt λrhminus λsh

1113872 1113873ρ + λrhμh

dλsv

dt λsvminus λev

1113872 1113873 1minus c1( 1113857apvih + λsvμvc3

dλev

dt minusM3 + λev

minus λiv1113872 1113873σ + λev

μvc3

dλiv

dt minusM4 + λsh

minus λeh1113872 1113873 1minus c1( 1113857aphsh + λiv

μvc3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Theorem 6 Given the optimal controls clowast1 clowast2 clowast3 and thesolutions sh eh ih rh sv ev iv of the corresponding stateequations (41) and (40) which minimize J(c1 c2 c3) over theregion Ω then there exist adjoint variablesλsh

λeh λih

λrh λsv

λev λiv

satisfying

minusdλi

dt

zH

zi i isin sh eh ih rh sv ev iv1113864 1113865 (46)

and the optimal solution clowast1 clowast2 clowast3 is given by

clowast1 min 1 max 0 1113954c1( 11138571113864 1113865

clowast2 min 1 max 0 1113954c2( 11138571113864 1113865

clowast3 min 1 max 0 1113954c3( 11138571113864 1113865

(47)

Proof (e Pontryaginrsquos maximum principle described in[16 17] is applied Corollary 41 in [17] shows the existenceof an optimal control due to the convexity of the integrand Jwith respect to c1 c2 c3 and Lipschitz property of the statesystem with respect to the state variables By using theoptimal conditions

zH

zc1 0

zH

zc2 0

zH

zc3 0

(48)

we obtain

zH

zc1 k1c1 + λsh

minus λeh1113872 1113873aphivsh + λsv

minus λev1113872 1113873apvihsv 0

zH

zc2 k2c2 + λrh

minus λih1113872 1113873τih 0

zH

zc3 k3c3 minus λsv

μvsv + λevμvev + λiv

μviv1113872 1113873 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)

Table 2 Parameters values used for simulations

Parameter Value Referenceπh 0000215day [14]πv 007day [14]ph 062 [6]pv 0065 [6 15]a Varying Assumedσ 0001 Assumedε 0083 [18]ω 0004 [3]ρ 002 [6]τ 0125 [3]μh 000044 Assumedμv 0034 [15]

Computational and Mathematical Methods in Medicine 9

Solving (49) we have

c1 λeh minus λsh( )aphivsh + λev minus λsv( )apvihsv

k1

c2 λih minus λrh( )τih

k2

c3 λsvμvsv + λevμvev + λivμviv

k3

(50)

As stated earlier the lower and upper boundaries for thecontrol parameters are 0 and 1 respectively If c1 c2 c3 lt 1then c1 c2 c3 0 and if c1 c2 c3 gt 1 then c1 c2 c3 1 otherwise c1 c1 c2 c2 c3 c3 erefore for thecontrol parameters clowast1 c

lowast2 clowast3 we obtain the optimum value of

the function J(c1 c2 c3)

41 Optimal Control Simulations e Octave programminglanguage is used to simulate the optimal control model usingthe set of parameters obtained from previously reportedstudies and datasets which have been cited Some of theseparameters are assumed for the sake of illustrations Table 2represents the values of the model parameters used forsimulationse following initial conditions were considered

sh(0) 30

eh(0) 7

ih(0) 2

rh(0) 0

sv(0) 40

ev(0) 10

iv(0) 3

(51)

and the weight constants were assumed to be

M1 1

M2 2

M3 2

M4 2

k1 2

k2 10

k3 5

(52)

Figures 2 and 3 represent the control proles at dierentvalues of c1 c2 and c3 while the rest of the plots are thegraphs of infectious human and vector population plottedagainst time in days and they represent the eect of optimalcontrols c1 c2 and c3 in reducing the number of individualsinfected From Figure 4 we observe that the use of treatmentand insecticides only has a signicant impact in reducing thenumber of infectious individuals and they show that thisstrategy is eective to control tsetse ies and infected humanpopulations In Figure 5 we observe that the use of edu-cation and insecticides reduce the number of infectiousindividuals but the results depicted in Figure 5(a) shows thatthis strategy is not eective and ecient to control theinfectious human population From Figure 6 we observethat the use of education and treatment reduces the numberof infectious individuals but the results from Figure 6(b)shows that this strategy is not eective and ecient tocontrol the population of infectious tsetse ies Lastly theresults depicted from Figure 7 shows that the strategy ofusing education treatment and insecticides is very ecientand eective to reduce the number of infected individualserefore the use of education treatment and insecticidessimultaneously is very ecient and eective to eliminateHAT in Africa

In epidemiology a reproduction number less than unityimplies that the disease can be eradicated in the long run

06

04

02

0

Con

trol p

rofil

e

c1 = 0c2 ne 0c3 ne 0

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 = 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 2 (a) Control prole when c1 0 c2 ne 0 and c3 ne 0 at Rc 00075 (b) Control prole when c1 ne 0 c2 0 and c3 ne 0 at Rc 000013

10 Computational and Mathematical Methods in Medicine

Hence choosing suitable parameters for the controls c1 c2and c3 it was observed that the eective reproductionnumbers obtained for Figures 2ndash7were 00075 000013 00440000075 00075 000013 0044 and 0000075 respectivelyis shows that incorporating all the control measures that iseducating individuals giving treatment and applying in-secticides is an eective method to help reduce the numbersecondary infections in the population which correspondswith eradicating the disease in the long run

5 Conclusion

In this paper we studied and analyzed the model fortransmission of HAT and determined the basic reproduction

number e local and global stabilities of disease-freeequilibrium and endemic equilibrium were also provedFor the optimal control model education treatment andinsecticides as control measures were used to optimize theobjective function dened by Equation (40) e numericalsimulations of the optimal control model show that the beststrategy to reduce the number of infected individuals isthrough the use of education treatment and insecticidesis is the eective and ecient method to eliminate thedisease Furthermore the national authorities non-governmental organizations (NGOs) and stakeholdersmust not lose their interest in controlling the disease be-cause neglecting this disease may cause rapid reoccurrenceand much eect to the people who are at risk

06

04

02

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(a)

4

35

3

25

15

1

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(b)

Figure 4 Simulations of the model showing the eorts of treatment and insecticides only on infectious individuals at Rc 00075

c1 ne 0c2 ne 0c3 = 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 ne 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 3 (a) Control prole when c1 ne 0 c2 ne 0 and c3 0 at Rc 0044 (b) Control prole when c1 ne 0 c2 ne 0 and c3 ne 0 at Rc 0000075

Computational and Mathematical Methods in Medicine 11

20

15

10

5Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(a)

12

10

8

6

4

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(b)

Figure 7 Simulations of the model showing the eorts of education treatment and insecticides on infectious individuals at Rc 0000075

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(a)

44

42

38

4

34

36

32

3

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(b)

Figure 6 Simulations of the model showing the eorts of education and treatment only on infectious individuals at Rc 0044

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

20

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

(a)

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

Infe

ctio

us v

ecto

r pop

ulat

ion

12

8

10

6

4

20 20 40 60 80 100 120

Time (days)

(b)

Figure 5 Simulations of the model showing the eorts of education and insecticides only on infectious individuals at Rc 000013

12 Computational and Mathematical Methods in Medicine

Data Availability

(e secondary data supporting this research are from pre-viously reported studies and datasets which have been cited(e processed data are available in cited references

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(e authors thank the support from Next Einstein Initiativeof the African Institute for Mathematical Sciences Ghana(Annual Grant for Graduate Studies) and MasterCardFoundation

References

[1] K S Rock S J Torr C Lumbala and M J KeelingldquoQuantitative evaluation of the strategy to eliminate humanafrican trypanosomiasis in the democratic republic of CongordquoParasites and Vectors vol 8 no 1 p 532 2015

[2] J R Franco P P Simarro A Diarra and J G JanninldquoEpidemiology of human african trypanosomiasisrdquo ClinicalEpidemiology vol 2014 no 6 pp 257ndash275 2014

[3] R Brun J Blum F Chappuis and C Burri ldquoHuman africantrypanosomiasisrdquo Lancet vol 375 no 9709 pp 148ndash1592010

[4] World Health Organization et al Control and Surveillance ofHuman African Trypanosomiasis Report of a WHO ExpertCommittee World Health Organization Geneva Switzerland2013

[5] M Martcheva An Introduction to Mathematical Epidemiol-ogy Springer Vol 61 Springer Berlin Germany 2015

[6] D J Rogers ldquoA general model for the african trypanoso-miasesrdquo Parasitology vol 97 no 1 pp 193ndash212 1988

[7] J W Hargrove R Ouifki D Kajunguri G A Vale andS J Torr ldquoModeling the control of trypanosomiasis usingtrypanocides or insecticide-treated livestockrdquo PLoS NeglectedTropical Diseases vol 6 no 5 p 1615 2012

[8] D Kajunguri ldquoModelling the control of tsetse and africantrypanosomiasis through application of insecticides on cattlein Southeastern Ugandardquo PhD thesis Stellenbosch Uni-versity Stellenbosch South Africa 2013

[9] S Olaniyi and O S Obabiyi ldquoMathematical model for malariatransmission dynamics in human and mosquito populationswith nonlinear forces of infectionrdquo International Journal ofPure and Applied Mathematics vol 88 no 1 pp 125ndash1562013

[10] R H Chisholm P T Campbell Y Wu S Y C TongJ McVernon and N Geard ldquoImplications of asymptomaticcarriers for infectious disease transmission and controlrdquoRoyal Society Open Science vol 5 no 2 p 172341 2018

[11] P Yan and S Liu ldquoSeir epidemic model with delayrdquo ANZIAMJournal vol 48 no 1 pp 119ndash134 2006

[12] G T Azu-Tungmah ldquoA mathematical model to control thespread of malaria in Ghanardquo PhD thesis Kwame NkrumahUniversity of Science and Technology Kumasi Ghana 2012

[13] S A Pedro S Abelman F T Ndjomatchoua R Sang andH E Z Tonnang ldquoStability bifurcation and chaos analysis ofvector-borne disease model with application to rift valleyfeverrdquo PloS One vol 9 no 10 Article ID 108172 2014

[14] S Olaniyi and O S Obabiyi ldquoQualitative analysis of malariadynamics with nonlinear incidence functionrdquo AppliedMathematical Sciences vol 8 no 78 pp 3889ndash3904 2014

[15] L S Pontryagin V G Boltyanskii and R V Gamkrelidze EfMishchenko the Mathematical 2eory of Optimal ProcessesInterscience New York NY USA 1962

[16] B Heimann ldquoFleming WHrishel RW deterministic andstochastic optimal control New York-heidelbergberlinSpringer-verlag 1975 xiii 222 s dm 60 60rdquo ZAMM-Journal of Applied Mathematics and Mechanics Zeitschriftfur Angewandte Mathematik und Mechanik vol 59 no 9p 494 1979

[17] S Davis S Aksoy and A Galvani ldquoA global sensitivityanalysis for african sleeping sicknessrdquo Parasitology vol 138no 4 pp 516ndash526 2011

[18] M Artzrouni and J P Gouteux ldquoA compartmental model ofsleeping sickness in central africardquo Journal of BiologicalSystems vol 4 no 4 pp 459ndash477 1996

Computational and Mathematical Methods in Medicine 13

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Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom

From system (5) the dimensionless technique is used toderive another equivalent differential equation we denotesh (SHNH) eh (EHNH) ih (IHNH) rh (RHNH)sv (SVNV) ev (EVNV) and iv (IVNV) and sub-stitute in system (5) to obtain the following new equivalentequations

dsh

dt πh + ρrh minus aphivsh minus μhsh

deh

dt aphivsh minus εeh minus μheh

dih

dt εeh minus μhih minusωih minus τih

drh

dt τih minus ρrh minus μhrh

dsvdt

πv minus μvsv minus apvihsv

dev

dt apvihsv minus μvev minus σev

div

dt σev minus μviv

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

Table 1 shows the description of the model parametersand variables

21 Positivity and Boundedness of the Solutions In thissubsection we show that system (6) is epidemiologically andmathematically well defined in the positive invariant region

D sh eh ih rh sv ev iv( 1113857 isin R7+ nh le

πh

μh nv le

πv

μv1113896 1113897 (7)

Theorem 1 2ere exists a domain D in which the solution(sh eh ih rh sv ev iv) is contained and bounded

Proof We provide the proof following the idea by Olaniyiand Obabiyi [9] Given the solution set (sh eh ih rh sv ev iv)

with the positive initial conditions (sh0 eh0 ih0 rh0 sv0

ev0 iv0) we define

nh sh eh ih rh( 1113857 sh(t) + eh(t) + ih(t) + rh(t) and

nv sv ev iv( 1113857 sv(t) + ev(t) + iv(t)(8)

(e derivatives of nh and nv with respect to time alongthe solution of system (6) for human and tsetse flies re-spectively are obtained by

nhprime dsh

dt+

deh

dt+

dih

dt+

drh

dt

πh minus sh + eh + ih + rh( 1113857μh minusωih

πh minus nhμh

nvprime dsv

dt+

dev

dt+

div

dt

πv minus sv + ev + iv( 1113857μv

πv minus nvμv

(9)

From these differential equations it follows that nhprime leπh minus μhnh and nvprime le πv minus μvnv We obtain the solutions asfollows

nh leπhμh

1minus exp minusμht( 1113857( 1113857 + nh sh0 eh0 ih0 rh01113872 1113873exp minusμht( 1113857

nv leπvμv

1minus exp minusμvt( 1113857( 1113857 + nv sv0 ev0 iv01113872 1113873exp minusμvt( 1113857

(10)

By taking the limits of both nh and nv above as t⟶infinwe obtain nh le (πhμh) and nv le (πvμv) hence the solutionsare contained in the region D (is implies that all solutionsof the human and tsetse fly population are contained in theregion D and are nonnegative this guarantees that thepositive invariant region for system (6) exists and is given by

D sh eh ih rh sv ev iv( 1113857 isin R7+ nh le

πhμh

nv leπv

μv1113896 1113897

(11)

3 Model Equilibria and Stability Analysis

In this section we give the model equilibria the basic re-production number R0 and the stabilities at both disease-free and endemic equilibrium

31 Disease-Free Equilibrium (DFE) (e DFE in system (6)is when there are no HAT infections within the human andtsetse fly population (us the existence of the DFE is givenby E0 ((πhμh) 0 0 0 (πvμv) 0 0)

Table 1 (e description of model variables and parameters

Variable Descriptionsh Susceptible human populationsv Susceptible tsetse fly populationeh and ev Exposed human and tsetse fly population respectivelyih and iv Infectious human and tsetse population respectivelyrh Recovered human populationParameter Descriptionπh Recruitment rate for human populationπv Recruitment rate for tsetse fly population

phProportion of bites by the infectious vector on

susceptible human population

pvProportion of bites by susceptible vector on an

infectious human populationa (e biting rate of the tsetse fliesσ Per capita rate of a vector becoming infectiousε Per capita rate of human becoming infectiousω Disease induced death rate

ρ (e rate at which the recovered human can becomesusceptible again

τ Recovery rateμ Natural death rateξh Force of infection for human populationξv Force of infection for tsetse flies

Computational and Mathematical Methods in Medicine 3

32 Endemic Equilibrium (EE) e EE is the nontrivialequilibrium point at which the HATdisease persists in bothhuman and tsetse y population us the EE is obtained asfollows Elowast (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) where

slowasth πh ρ + μh( ) + ρτilowasth[ ] μv σ + μv( ) apvilowasth + μv( )[ ]

a2σphpvπvilowasth + μvμh σ + μv( ) apvilowasth + μv( )[ ] μh + ρ( )

elowasth ω + τ + μh( )ilowasth

ε

rlowasth τilowasth

μh + ρ

slowastv πv

apvilowasth + μv

elowastv apvπvi

lowasth

σ + μv( ) apvilowasth + μv( )

ilowastv aσpvπvilowasth

σ + μv( ) apvilowasth + μv( )μv

ilowasth ρ + μh( ) a2εphpvπhπvσ minus μ2vμh ε + μh( ) σ + μv( ) μh + τ + ω( )[ ]

B

(12)

and the term

B ( apv( aσphπv( ερω + μh( ρ(τ + ω) + ε(ρ + τ + ω)+ μh ε + ρ + τ + ω + μh( ))) + μh ε + μh( ) ρ + μh( )middot τ + ω + μh( )μv σ + μv( )))

(13)

33 Basic ReproductionNumberR0 e basic reproductionnumber R0 is dened as the number of secondary in-fections caused by one infected host or vector in a com-pletely susceptible population [10] e next-generationmatrix approach as done by Van den Driessche andWatmough in [5 11] is applied to derive

F aphivsh

0

apvihsv

V

μheh + εehminusεeh + μhih + ωih + τih

μvev + δevminusδev + μviv

(14)

By denoting matrix F (zFzxi) and V (zVzxi)where xi eh ih ev iv the spectral radius of the next-generation matrix FVminus1 gives the value of R0

F

0 0 0 aphsh

0 0 0 0

0 apvsv 0 0

0 0 0 0

V

ε + μh 0 0 0

minusε μh + ω + τ 0 0

0 0 μv + δ 0

0 0 minusδ μv

FVminus1

0 0aδph

μv minusδ + μv( )minusaphμv

0apv

μh + τ + ω0 0

0 0 0 0

0 0 0 0

(15)

лH

лV

SH

SV

μHSH μHEH

ξHSH

ξVSV

EH

EV

εEH

ρRH

ωIH

IHτIH

μHIH

μVIVμVEVμVSV

μHRH

RH

IVσEV

Figure 1 Compartmental model for the transmission of human African trypanosomiasis

4 Computational and Mathematical Methods in Medicine

(e spectral radius σ(FVminus1) gives

R0 σ FVminus1

1113872 1113873

a2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

1113971

(16)

One infected human in a population of susceptible vectorswill cause Rv infected vectors likewise one infected vector ina population will cause Rh infected humans [5](erefore thebasic reproduction number can be rewritten as R0

RhRv

1113968

where Rh (aεphπhμh(μh + ε)(μh + τ + ω)) and Rv

(σapvπvμ2v(σ + μv)) (us R0 can also be defined as thesquare root of the product of the number of infected humansin the susceptible population caused by one infected tsetsefly in its infectious lifetime and the number of infected tsetse

flies caused by one infected human during the infectiousperiod [12]

34 Local Stability of Disease-Free Equilibrium (DFE)

Theorem 2 If R0 le 1 the DFE given by E0 is locally as-ymptotically stable in the region defined by (7) and it isunstable when R0 gt 1

Proof (e DFE is locally stable if all eigenvalues of Jacobianmatrix JE0

are negative (e matrix has all eigenvalues neg-ative only if the trace of JE0

lt 0 and determinant of JE0gt 0 By

linearizing system (6) around E0 we obtain the followingJacobian matrix

JE0

minusμh 0 0 ρ 0 0 minusaphsh

0 minus ε + μh( 1113857 0 0 0 0 aphsh

0 ε minus ω + τ + μh( 1113857 0 0 0 00 0 τ minus ρ + μh( 1113857 0 0 00 0 minusapvsv 0 minusμv 0 00 0 apvsv 0 0 minus σ + μv( 1113857 00 0 0 0 0 σ minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(17)

(e trace of matrix JE0is such that

tr JE01113872 1113873 minus( μh + ε + μh + μh + ω + τ + ρ + μh

+ μv + σ + μv + μv1113857

minus 4μh + 3μv + ε + ω + τ + ρ + σ( 1113857lt 0

(18)

Using the basic properties of matrix algebra as in [13] itis clear that the eigenvalues λ1 minusμh and λ2 minusμv of thematrix JE0

have negative real parts (e reduced matrix is

JE1

minus ε + μh( 1113857 0 0 0 aphsh

ε minus ω + τ + μh( 1113857 0 0 0

0 τ minus ρ + μh( 1113857 0 0

0 apvsv 0 minus σ + μv( 1113857 0

0 0 0 σ minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(19)

From matrix JE1 the eigenvalue λ3 minus(ρ + μh) has

negative real part (e remaining matrix is further reducedby using the reduction techniques and we obtain

JE2

minus ε + μh( 1113857 0 0 aphsh

0 minus ω + τ + μh( 1113857 0aphshεε + μh

0 apvsv minus σ + μv( 1113857 0

0 0 σ minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(20)

Using the properties of matrix algebra the matrix JE2has

eigenvalue minus(ε + μh) which has negative real part Wefurther reduce to a 2 times 2 matrix by using the same reductiontechniques (e matrix is

JE3

minus ω + τ + μh( 1113857aphshεε + μh

apvsvσσ + μv

minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (21)

From the reduced 2 times 2 matrix the trace is negative andthe determinant is

Det JE31113872 1113873 ω + τ + μh( 1113857μv minus

aphshεε + μh

timesapvsvσσ + μv

ω + τ + μh( 1113857μv 1minusa2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 11138571113890 1113891

(22)

Since

R0

a2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

1113971

(23)

then by letting RT (a2εphpvπhπvσμ2vμh(ε + μh)(σ + μv)(μh + τ + ω)) we find our determinant as

Det JE31113872 1113873 ω + τ + μh( 1113857μv 1minusRT1113858 1113859 (24)

(e value of RT can be seen to be positive because all theparameters are positive As a result the determinant in (24)

Computational and Mathematical Methods in Medicine 5

is positive if and only if RT lt 1 (erefore the DFE is locallystable if RT le 1

35 Global Stability of Disease-Free Equilibrium (DFE)To show that the DFE is globally stable we apply Lyapunovrsquostheorem in [5]

Theorem 3 2eDFE defined by E0 is globally asymptoticallystable in the region defined by (7) if R0 le 1 Otherwise unstableif R0 gt 1

Proof We define Lyapunovrsquos function as

V k1 sh minus sh0 minus sh0 lnsh

sh01113888 1113889 + k2eh + k3ih

+ k4 sv minus sv0 minus sv0 lnsv

sv01113888 1113889 + k5ev + k6iv

(25)

satisfying system (6) where k1 k2 k3 k4 k5 k6 gt 0 are to bedetermined and sh0 (πhμh) and sv0 (πvμv) We firstshow that Vgt 0 for all Ene ((πhμh) 0 0 0 (πvμv) 0 0) It isenough to check that

k1sh0sh

sh0minus 1minus ln

sh

sh01113888 1113889gt 0

k4sv0svsv0minus 1minus ln

svsv0

1113888 1113889gt 0

(26)

(e function g(m) mminus 1minus lnm such thatm (shsh0) (svsv0) has minimum value equal to zerowhen m 1 hence g(m)gt 0 for all mgt 0 (us Lyapunovrsquosfunction Vgt 0 (e function V is radially unbounded be-cause as |m|⟶infin the function g(m)⟶infin We nowtake the derivative of V with respect to time and use system(6) to replace the derivatives in the right hand side such that

Vprime k1 1minussh0sh

1113888 1113889dsh

dt+ k2

deh

dt+ k3

dih

dt+ k4 1minus

sv0sv

1113888 1113889dsv

dt

+ k5dev

dt+ k6

div

dt

k1 1minussh0sh

1113888 1113889 πh + ρrh minus aphivsh minus μhsh1113858 1113859

+ k2 aphivsh minus εeh minus μheh1113858 1113859

+ k3 εeh minus μhih minusωih minus τih1113858 1113859

+ k4 1minussv0sv

1113888 1113889 πv minus μvsv minus apvihsv1113858 1113859

+ k5 apvihsv minus μvev minus σev1113858 1113859 + k6 σev minus μviv1113858 1113859

2k1πh minus aphshk1iv + ρrhk1 minus k1μhsh

minus k1π2h

μhsh+

aphπhiv

μhk1 minus

πhρrh

μhshk1 + k2aphshiv

minus k2 ε + μh( 1113857eh + k3εeh minus k3 τ + μh + ω( 1113857ih

+ 2k4πv minus k4apvsvih minus μvsvk4 minusπ2v

μvsvk4

+apvihπv

μvk4 minus k5 μv + σ( 1113857ev

+ k5apvsvih + k6σev minus k6μviv(27)

(e terms with rh are ignored because if sh eh ih areglobally stable then rh⟶ 0 at any time t and the DFE forsystem (6) is globally stable Taking k1 k2 (1μh + ε)k4 k5 (1μv + σ) k3 (1ε) and k6 (1σ) the de-rivative of V with respect to time becomes

Vprime minusπh

μh + επhμhsh

+μhsh

πhminus 21113888 1113889minus

τ + μh + ω( 1113857

εih

+aphπhμh + ε( 1113857μh

iv minusπv

μv + σπv

μvsv+μvsvπvminus 21113888 1113889

minusμvσ

iv +apvπv

μv μv + σ( 1113857ih

minusπh

μh + επh

μhsh+μhsh

πhminus 21113888 1113889 +

πv

μv + σπv

μvsv+μvsvπvminus 21113888 11138891113890 1113891

+τ + μh + ω( 1113857

εεapvπv

μv σ + μv( 1113857 μh + τ + ω( 1113857minus 11113888 1113889ih

+μvσ

aσphπhμvμh μh + ε( 1113857

minus 11113888 1113889iv

minusπh

μh + επh

μhsh+μhsh

πhminus 21113888 1113889 +

πv

μv + σπv

μvsv+μvsvπvminus 21113888 11138891113890 1113891

+τ + μh + ω( 1113857

εRv minus 1( 1113857ih +

μv

σRh minus 1( 1113857iv

(28)

(e terms ((πhμhsh) + (μhshπh)minus 2) and ((πvμvsv) +

(μvsvπv)minus 2) are positive because if we suppose m

(πhμhsh) (πvμvsv) we have m + (1m)minus 2 (m2 minus 2m +

1m) ((mminus 1)2m)gt 0 for all mgt 1 and since Rv le 1 andRh le 1 then Vprime is negative (us we have Vprime lt 0 for allE0 ne ((πhμh) 0 0 0 (πvμv) 0 0)

(us the largest compact invariant set in D is the sin-gleton set E0 Hence system (6) is globally asymptoticallystable

6 Computational and Mathematical Methods in Medicine

36 Local Stability of Endemic Equilibrium (EE)

Theorem 4 2e unique endemic equilibrium defined by Elowast

is locally asymptotically stable in the region defined by (7) ifR0 gt 1 but is unstable if R0 le 1

Proof We give the proof of this theorem based on the ap-proach used by Olaniyi and Obabiyi [9 14] From the EE

points defined in (12) since all values are positive we expressthe value of ilowasth in terms of R0 to obtain

ilowasth

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

B

middota2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857minus 11113890 1113891

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

BR20 minus 11113960 1113961

(29)

where

B ( apv( aσphπv( ερω + μh( ρ(τ + ω) + ε(ρ + τ + ω)

+ μh ε + ρ + τ + ω + μh( 111385711138571113857 + μh ε + μh( 1113857 ρ + μh( 1113857

middot τ + ω + μh( 1113857μv σ + μv( 111385711138571113857

(30)

Since the basic reproduction number R0 (a2εphpvπhπvσμ2vμh(ε + μh)(σ + μv)(μh + τ + ω))

1113968 if we

let

RT a2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857 (31)

then we can find ilowasth as

ilowasth

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

BRT minus 11113858 1113859

(32)

(e value of B is clearly positive because all parametersare positive Hence ilowasth gt 0 if and only if RT gt 1 implying thatthe EE is locally asymptotically stable if RT gt 1

37 Global Stability of Endemic Equilibrium (EE) To showthe global stability of the EE we use Lyapunovrsquos theoremtogether with the following lemma

Lemma 1 Suppose that y1 y2 middot middot middot yn are n positive num-bers 2en their arithmetic mean is greater than or equal tothe geometric mean that is (y1 + y2 + middot middot middot + ynn)ge(y1y2 middot middot middot yn)1n

Theorem 5 2e EE defined by Elowast is globally asymptoticallystable if R0 gt 1 otherwise unstable

Proof (e proof is based on the idea as explained byMartcheva [5] We define Lyapunovrsquos function as

V k1 sh minus slowasth minus slowasth ln

shslowasth

1113888 1113889 + k2 eh minus elowasth minus elowasth ln

ehelowasth

1113888 1113889

+ k3 ih minus ilowasth minus ilowasth ln

ih

ilowasth1113888 1113889 + k4 sv minus s

lowastv minus slowastv ln

sv

slowastv1113888 1113889

+ k5 ev minus elowastv minus elowastv ln

ev

elowastv1113888 1113889 + k6 iv minus i

lowastv minus ilowastv ln

iv

ilowastv1113888 1113889

(33)

satisfying system (6) with k1 k2 k3 k4 k5 k6 gt 0 to bedetermined (e function V is nonnegative for all(sh eh ih rh sv ev iv)ne (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) and radiallyunbounded

We need to prove that Vprime lt 0 for all(sh eh ih rh sv ev iv)ne (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) We find thederivative of V with respect to time and replace the de-rivatives sh

prime ehprime ihprime rhprime svprime evprime ivprime with system (6) We also ignore

the rh terms because if sh eh ih are globally stable thenrh⟶ 0 at any time t and EE is globally stable

Vprime k1 1minusslowasthsh

1113888 1113889 πh minus aphivsh minus μhsh1113858 1113859 + k2 1minuselowastheh

1113888 1113889

middot aphivsh minus μh + ε( 1113857eh1113858 1113859 + k3 1minusilowasthih

1113888 1113889

middot εeh minus ω + τ + uh( 1113857ih1113858 1113859 + k4 1minusslowastvsv

1113888 1113889

middot πv minus apvihsv minus μvsv1113858 1113859 + k5 1minuselowastvev

1113888 1113889

middot apvihsv minus μv + σ( 1113857ev1113858 1113859 + k6 1minusilowastviv

1113888 1113889 σev minus uviv1113858 1113859

(34)

We now substitute πh aphilowastv slowasth + μhslowasth and πv

apvilowasthslowastv + μvslowastv at the endemic equilibrium and then simplifyand put similar terms together to obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth minus k1aphivsh minus k1aph

slowast2h ilowastvsh

+ k1aphivslowasth + k2aphshiv minus k2 ε + μh( 1113857eh minus k2aphshiv

elowastheh

+ k2 ε + μh( 1113857elowasth + k3εeh minus k3 ω + τ + μh( 1113857i

lowasth minus k3εeh

ilowasthih

+ k3 ω + τ + μh( 1113857ilowasth minus k4

sv minus slowastv( 11138572μv

sv+ k4apvi

lowasthslowastv

minus k4apvihsv minus k4apvslowast2v ilowasth

sv+ k4apvihs

lowastv + k5apvsvih

minus k5 σ + μv( 1113857ev minus k5apvsvihelowastvev

+ k5 σ + μv( 1113857elowastv + k6σev

minus k6μvilowastv minus k6σev

ilowastviv

+ k6μvilowastv

(35)

Computational and Mathematical Methods in Medicine 7

We suppose k1 k2 and k4 k5 and multiply and dividethe same equilibrium value to some of the fractions to obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth minus k1aph

slowast2h ilowastvsh

+ k1aphivslowasth

minus k2 ε + μh( 1113857eh minus k2aphshivelowasthslowasth ilowastvslowasth ilowastv eh

+ k2 ε + μh( 1113857elowasth

+ k3εeh minus k3 ω + τ + μh( 1113857ih minus k3εehilowasthelowasthihelowasth

+ k3 ω + τ + μh( 1113857ilowasth minus k4

sv minus slowastv( 11138572μv

sv+ k4apvi

lowasthslowastv

minus k4apvslowast2v ilowasth

sv+ k4apvihs

lowastv minus k5 σ + μv( 1113857ev

minus k5apvsvihelowastv slowastv ilowasthslowastv ilowasthev

+ k5 σ + μv( 1113857elowastv + k6σev minus k6μvi

lowastv

minus k6σevelowastv ilowastvivelowastv

+ k6μvilowastv

(36)

We choose k3 k2(μh + εε) such that k3(ω + τ+ μh)ilowasth

k2(ε + μh)elowasth and choose k6 k5(μh + σσ) such thatk6μvilowastv k5(σ + μv)elowastv Now aphslowasth ilowastv (μh + ε)elowasth andapvslowastv ilowastv (μv + σ)elowastv because k1 k2 and k4 k5 re-

spectively We now obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

+ k1aphivslowasth minus k3 ω + τ + μh( 1113857ih + k3εminus k2 ε + μh( 11138571113858 1113859eh

minus k4sv minus slowastv( 1113857

2μvsv

+ k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsv

slowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891

+ k4apvihslowastv minus k6μviv + k6σ minus k5 σ + μv( 11138571113858 1113859ev

(37)

Suppose k6 (k1aphslowasthμv) and k3 (k4apvslowastv ω + τ+

μh) We then substitute and simplify to get

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

minus k4sv minus slowastv( 1113857

2μvsv

+ k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsvslowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891

(38)

From Lemma 1 the terms

k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsv

slowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891le 0

(39)

(erefore Vprime lt 0 for all (sh eh ih rh sv ev iv)ne (slowasth

elowasth ilowasth rlowasth slowastv elowastv ilowastv ) implying that the endemic equilibrium isglobally asymptotically stable if R0 gt 1

4 Analysis of Optimal Control Model

In this section we formulate the optimal control model bymodifying system (6) to an optimal control problem Wethus define some linear functions ci(t) 1 for i 1 2 3 It isimportant to note that controls are fully effective whenci(t) 1 and not effective when ci(t) 0 (e forces ofinfection ξh and xiv which correspond to the human andvector population respectively are reduced by the factor(1minus c1) where c1 measures the level of success obtained dueto the effort of educating people on the dangers of exposingtheir skin and encouraging them to wear long sleeves andlong pants during the day to minimize tsetse fly-humancontacts (e factor c2 represents the effort of treatment tocontrol the disease and the factor c3 also represents theeffort of using insecticides to ensure that the breeding sites ofthe tsetse fly are minimized Hence taking into account theassumptions and extensions made we try to find the mosteffective strategy that reduces the HAT infection in thepopulation at a veryminimum costWith the use of boundedLebesgue measurable control we define the objectivefunction to be minimized as

J c1 c2 c3( 1113857 1113946tF

0( M1eh + M2ih + M3ev + M4iv

+12k1c

21 +

12k2c

22 +

12k3c

231113857 dt

(40)

(us the dynamics of the controls that minimizes theobjective function is given by

dshdt

πh + ρrh minus 1minus c1( 1113857aphivsh minus μhsh

deh

dt 1minus c1( 1113857aphivsh minus εeh minus μheh

dih

dt εeh minus μhih minusωih minus c2τih

drh

dt c2τih minus ρrh minus μhrh

dsv

dt πv minus c3μvsv minus 1minus c1( 1113857apvihsv

dev

dt 1minus c1( 1113857apvihsv minus c3μvev minus σev

div

dt σev minus c3μviv

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

subject to the initial conditions sh ge 0 eh ge 0 ih ge 0 rh ge 0

sv ge 0 ev ge 0 and iv ge 0 (e associated effective reproduc-tion number for (41) denoted by RE is obtained as

RE

a2ε 1minus c1( 11138572phπhπvσ

c3μh ε + μh( 1113857μ2v σ + c3μv( 1113857 μh + c2τ + ω( 1113857

1113971

Rc

1113968

(42)

8 Computational and Mathematical Methods in Medicine

(e goal is to minimize the exposed and infectious humanpopulations (eh ih) the exposed and infectious vector pop-ulations (ev iv) and the cost of implementing the control bythe use of possible ci i 1 2 3 (e functional objectiveincludes the social cost which relates to the resources that areneeded for educating people on personal protection (12)k1c

21

the application of treatment (12)k2c22 and spraying of tsetse

fly operations (12)k3c23 (e quantities M1 and M2 re-

spectively represent the associated cost with minimizing theexposed and infected human population while M3 and M4also represent the cost associated withminimizing the exposedand infected vectors respectively (e quantity tF is the timeperiod of intervention As explained in [15] the costs cor-responding to M1eh M2ih M3ev and M4iv are linear whilethe cost control functions (12)k1c

21 (12)k2c

22 and (12)k3c

23

should be nonlinear and take a quadratic form (erefore weseek to minimize the objective function over the given timeinterval [0 tF] Pontryaginrsquos maximum principle is used tosolve this optimal control problem and the derivation of thenecessary conditions (e Lagrangian of the optimal controlproblem is given by

L M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22 +

12k3c

231113874 1113875

(43)

To determine the Lagrangian minimum value we definethe Hamiltonian H for the control problem as

H M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22

+12k3c

23 + λsh

dsh

dt+ λeh

deh

dt+ λih

dih

dt+ λrh

drh

dt

+ λsv

dsv

dt+ λev

dev

dt+ λiv

div

dt

(44)

where λsh λeh

λih λrh

λsv λev

and λivare adjoint variables or

costate variables (e differential equations of adjoint var-iables are obtained by taking the partial derivatives of theHamiltonian equation with respect to the state variableswhich gives

dλsh

dt λsh

μh + λshminus λeh

1113872 1113873 1minus c1( 1113857aphiv

dλeh

dt minusM1 + λeh

minus λih1113872 1113873ε + λeh

μh

dλih

dt minusM2 + λih

minus λrh1113872 1113873c2τ + λih

ω + μh( 1113857 + λsvminus λev

1113872 1113873 1minus c1( 1113857apvsv

dλrh

dt λrhminus λsh

1113872 1113873ρ + λrhμh

dλsv

dt λsvminus λev

1113872 1113873 1minus c1( 1113857apvih + λsvμvc3

dλev

dt minusM3 + λev

minus λiv1113872 1113873σ + λev

μvc3

dλiv

dt minusM4 + λsh

minus λeh1113872 1113873 1minus c1( 1113857aphsh + λiv

μvc3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Theorem 6 Given the optimal controls clowast1 clowast2 clowast3 and thesolutions sh eh ih rh sv ev iv of the corresponding stateequations (41) and (40) which minimize J(c1 c2 c3) over theregion Ω then there exist adjoint variablesλsh

λeh λih

λrh λsv

λev λiv

satisfying

minusdλi

dt

zH

zi i isin sh eh ih rh sv ev iv1113864 1113865 (46)

and the optimal solution clowast1 clowast2 clowast3 is given by

clowast1 min 1 max 0 1113954c1( 11138571113864 1113865

clowast2 min 1 max 0 1113954c2( 11138571113864 1113865

clowast3 min 1 max 0 1113954c3( 11138571113864 1113865

(47)

Proof (e Pontryaginrsquos maximum principle described in[16 17] is applied Corollary 41 in [17] shows the existenceof an optimal control due to the convexity of the integrand Jwith respect to c1 c2 c3 and Lipschitz property of the statesystem with respect to the state variables By using theoptimal conditions

zH

zc1 0

zH

zc2 0

zH

zc3 0

(48)

we obtain

zH

zc1 k1c1 + λsh

minus λeh1113872 1113873aphivsh + λsv

minus λev1113872 1113873apvihsv 0

zH

zc2 k2c2 + λrh

minus λih1113872 1113873τih 0

zH

zc3 k3c3 minus λsv

μvsv + λevμvev + λiv

μviv1113872 1113873 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)

Table 2 Parameters values used for simulations

Parameter Value Referenceπh 0000215day [14]πv 007day [14]ph 062 [6]pv 0065 [6 15]a Varying Assumedσ 0001 Assumedε 0083 [18]ω 0004 [3]ρ 002 [6]τ 0125 [3]μh 000044 Assumedμv 0034 [15]

Computational and Mathematical Methods in Medicine 9

Solving (49) we have

c1 λeh minus λsh( )aphivsh + λev minus λsv( )apvihsv

k1

c2 λih minus λrh( )τih

k2

c3 λsvμvsv + λevμvev + λivμviv

k3

(50)

As stated earlier the lower and upper boundaries for thecontrol parameters are 0 and 1 respectively If c1 c2 c3 lt 1then c1 c2 c3 0 and if c1 c2 c3 gt 1 then c1 c2 c3 1 otherwise c1 c1 c2 c2 c3 c3 erefore for thecontrol parameters clowast1 c

lowast2 clowast3 we obtain the optimum value of

the function J(c1 c2 c3)

41 Optimal Control Simulations e Octave programminglanguage is used to simulate the optimal control model usingthe set of parameters obtained from previously reportedstudies and datasets which have been cited Some of theseparameters are assumed for the sake of illustrations Table 2represents the values of the model parameters used forsimulationse following initial conditions were considered

sh(0) 30

eh(0) 7

ih(0) 2

rh(0) 0

sv(0) 40

ev(0) 10

iv(0) 3

(51)

and the weight constants were assumed to be

M1 1

M2 2

M3 2

M4 2

k1 2

k2 10

k3 5

(52)

Figures 2 and 3 represent the control proles at dierentvalues of c1 c2 and c3 while the rest of the plots are thegraphs of infectious human and vector population plottedagainst time in days and they represent the eect of optimalcontrols c1 c2 and c3 in reducing the number of individualsinfected From Figure 4 we observe that the use of treatmentand insecticides only has a signicant impact in reducing thenumber of infectious individuals and they show that thisstrategy is eective to control tsetse ies and infected humanpopulations In Figure 5 we observe that the use of edu-cation and insecticides reduce the number of infectiousindividuals but the results depicted in Figure 5(a) shows thatthis strategy is not eective and ecient to control theinfectious human population From Figure 6 we observethat the use of education and treatment reduces the numberof infectious individuals but the results from Figure 6(b)shows that this strategy is not eective and ecient tocontrol the population of infectious tsetse ies Lastly theresults depicted from Figure 7 shows that the strategy ofusing education treatment and insecticides is very ecientand eective to reduce the number of infected individualserefore the use of education treatment and insecticidessimultaneously is very ecient and eective to eliminateHAT in Africa

In epidemiology a reproduction number less than unityimplies that the disease can be eradicated in the long run

06

04

02

0

Con

trol p

rofil

e

c1 = 0c2 ne 0c3 ne 0

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 = 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 2 (a) Control prole when c1 0 c2 ne 0 and c3 ne 0 at Rc 00075 (b) Control prole when c1 ne 0 c2 0 and c3 ne 0 at Rc 000013

10 Computational and Mathematical Methods in Medicine

Hence choosing suitable parameters for the controls c1 c2and c3 it was observed that the eective reproductionnumbers obtained for Figures 2ndash7were 00075 000013 00440000075 00075 000013 0044 and 0000075 respectivelyis shows that incorporating all the control measures that iseducating individuals giving treatment and applying in-secticides is an eective method to help reduce the numbersecondary infections in the population which correspondswith eradicating the disease in the long run

5 Conclusion

In this paper we studied and analyzed the model fortransmission of HAT and determined the basic reproduction

number e local and global stabilities of disease-freeequilibrium and endemic equilibrium were also provedFor the optimal control model education treatment andinsecticides as control measures were used to optimize theobjective function dened by Equation (40) e numericalsimulations of the optimal control model show that the beststrategy to reduce the number of infected individuals isthrough the use of education treatment and insecticidesis is the eective and ecient method to eliminate thedisease Furthermore the national authorities non-governmental organizations (NGOs) and stakeholdersmust not lose their interest in controlling the disease be-cause neglecting this disease may cause rapid reoccurrenceand much eect to the people who are at risk

06

04

02

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(a)

4

35

3

25

15

1

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(b)

Figure 4 Simulations of the model showing the eorts of treatment and insecticides only on infectious individuals at Rc 00075

c1 ne 0c2 ne 0c3 = 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 ne 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 3 (a) Control prole when c1 ne 0 c2 ne 0 and c3 0 at Rc 0044 (b) Control prole when c1 ne 0 c2 ne 0 and c3 ne 0 at Rc 0000075

Computational and Mathematical Methods in Medicine 11

20

15

10

5Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(a)

12

10

8

6

4

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(b)

Figure 7 Simulations of the model showing the eorts of education treatment and insecticides on infectious individuals at Rc 0000075

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(a)

44

42

38

4

34

36

32

3

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(b)

Figure 6 Simulations of the model showing the eorts of education and treatment only on infectious individuals at Rc 0044

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

20

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

(a)

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

Infe

ctio

us v

ecto

r pop

ulat

ion

12

8

10

6

4

20 20 40 60 80 100 120

Time (days)

(b)

Figure 5 Simulations of the model showing the eorts of education and insecticides only on infectious individuals at Rc 000013

12 Computational and Mathematical Methods in Medicine

Data Availability

(e secondary data supporting this research are from pre-viously reported studies and datasets which have been cited(e processed data are available in cited references

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(e authors thank the support from Next Einstein Initiativeof the African Institute for Mathematical Sciences Ghana(Annual Grant for Graduate Studies) and MasterCardFoundation

References

[1] K S Rock S J Torr C Lumbala and M J KeelingldquoQuantitative evaluation of the strategy to eliminate humanafrican trypanosomiasis in the democratic republic of CongordquoParasites and Vectors vol 8 no 1 p 532 2015

[2] J R Franco P P Simarro A Diarra and J G JanninldquoEpidemiology of human african trypanosomiasisrdquo ClinicalEpidemiology vol 2014 no 6 pp 257ndash275 2014

[3] R Brun J Blum F Chappuis and C Burri ldquoHuman africantrypanosomiasisrdquo Lancet vol 375 no 9709 pp 148ndash1592010

[4] World Health Organization et al Control and Surveillance ofHuman African Trypanosomiasis Report of a WHO ExpertCommittee World Health Organization Geneva Switzerland2013

[5] M Martcheva An Introduction to Mathematical Epidemiol-ogy Springer Vol 61 Springer Berlin Germany 2015

[6] D J Rogers ldquoA general model for the african trypanoso-miasesrdquo Parasitology vol 97 no 1 pp 193ndash212 1988

[7] J W Hargrove R Ouifki D Kajunguri G A Vale andS J Torr ldquoModeling the control of trypanosomiasis usingtrypanocides or insecticide-treated livestockrdquo PLoS NeglectedTropical Diseases vol 6 no 5 p 1615 2012

[8] D Kajunguri ldquoModelling the control of tsetse and africantrypanosomiasis through application of insecticides on cattlein Southeastern Ugandardquo PhD thesis Stellenbosch Uni-versity Stellenbosch South Africa 2013

[9] S Olaniyi and O S Obabiyi ldquoMathematical model for malariatransmission dynamics in human and mosquito populationswith nonlinear forces of infectionrdquo International Journal ofPure and Applied Mathematics vol 88 no 1 pp 125ndash1562013

[10] R H Chisholm P T Campbell Y Wu S Y C TongJ McVernon and N Geard ldquoImplications of asymptomaticcarriers for infectious disease transmission and controlrdquoRoyal Society Open Science vol 5 no 2 p 172341 2018

[11] P Yan and S Liu ldquoSeir epidemic model with delayrdquo ANZIAMJournal vol 48 no 1 pp 119ndash134 2006

[12] G T Azu-Tungmah ldquoA mathematical model to control thespread of malaria in Ghanardquo PhD thesis Kwame NkrumahUniversity of Science and Technology Kumasi Ghana 2012

[13] S A Pedro S Abelman F T Ndjomatchoua R Sang andH E Z Tonnang ldquoStability bifurcation and chaos analysis ofvector-borne disease model with application to rift valleyfeverrdquo PloS One vol 9 no 10 Article ID 108172 2014

[14] S Olaniyi and O S Obabiyi ldquoQualitative analysis of malariadynamics with nonlinear incidence functionrdquo AppliedMathematical Sciences vol 8 no 78 pp 3889ndash3904 2014

[15] L S Pontryagin V G Boltyanskii and R V Gamkrelidze EfMishchenko the Mathematical 2eory of Optimal ProcessesInterscience New York NY USA 1962

[16] B Heimann ldquoFleming WHrishel RW deterministic andstochastic optimal control New York-heidelbergberlinSpringer-verlag 1975 xiii 222 s dm 60 60rdquo ZAMM-Journal of Applied Mathematics and Mechanics Zeitschriftfur Angewandte Mathematik und Mechanik vol 59 no 9p 494 1979

[17] S Davis S Aksoy and A Galvani ldquoA global sensitivityanalysis for african sleeping sicknessrdquo Parasitology vol 138no 4 pp 516ndash526 2011

[18] M Artzrouni and J P Gouteux ldquoA compartmental model ofsleeping sickness in central africardquo Journal of BiologicalSystems vol 4 no 4 pp 459ndash477 1996

Computational and Mathematical Methods in Medicine 13

Stem Cells International

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

MEDIATORSINFLAMMATION

of

EndocrinologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Disease Markers

Hindawiwwwhindawicom Volume 2018

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32 Endemic Equilibrium (EE) e EE is the nontrivialequilibrium point at which the HATdisease persists in bothhuman and tsetse y population us the EE is obtained asfollows Elowast (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) where

slowasth πh ρ + μh( ) + ρτilowasth[ ] μv σ + μv( ) apvilowasth + μv( )[ ]

a2σphpvπvilowasth + μvμh σ + μv( ) apvilowasth + μv( )[ ] μh + ρ( )

elowasth ω + τ + μh( )ilowasth

ε

rlowasth τilowasth

μh + ρ

slowastv πv

apvilowasth + μv

elowastv apvπvi

lowasth

σ + μv( ) apvilowasth + μv( )

ilowastv aσpvπvilowasth

σ + μv( ) apvilowasth + μv( )μv

ilowasth ρ + μh( ) a2εphpvπhπvσ minus μ2vμh ε + μh( ) σ + μv( ) μh + τ + ω( )[ ]

B

(12)

and the term

B ( apv( aσphπv( ερω + μh( ρ(τ + ω) + ε(ρ + τ + ω)+ μh ε + ρ + τ + ω + μh( ))) + μh ε + μh( ) ρ + μh( )middot τ + ω + μh( )μv σ + μv( )))

(13)

33 Basic ReproductionNumberR0 e basic reproductionnumber R0 is dened as the number of secondary in-fections caused by one infected host or vector in a com-pletely susceptible population [10] e next-generationmatrix approach as done by Van den Driessche andWatmough in [5 11] is applied to derive

F aphivsh

0

apvihsv

V

μheh + εehminusεeh + μhih + ωih + τih

μvev + δevminusδev + μviv

(14)

By denoting matrix F (zFzxi) and V (zVzxi)where xi eh ih ev iv the spectral radius of the next-generation matrix FVminus1 gives the value of R0

F

0 0 0 aphsh

0 0 0 0

0 apvsv 0 0

0 0 0 0

V

ε + μh 0 0 0

minusε μh + ω + τ 0 0

0 0 μv + δ 0

0 0 minusδ μv

FVminus1

0 0aδph

μv minusδ + μv( )minusaphμv

0apv

μh + τ + ω0 0

0 0 0 0

0 0 0 0

(15)

лH

лV

SH

SV

μHSH μHEH

ξHSH

ξVSV

EH

EV

εEH

ρRH

ωIH

IHτIH

μHIH

μVIVμVEVμVSV

μHRH

RH

IVσEV

Figure 1 Compartmental model for the transmission of human African trypanosomiasis

4 Computational and Mathematical Methods in Medicine

(e spectral radius σ(FVminus1) gives

R0 σ FVminus1

1113872 1113873

a2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

1113971

(16)

One infected human in a population of susceptible vectorswill cause Rv infected vectors likewise one infected vector ina population will cause Rh infected humans [5](erefore thebasic reproduction number can be rewritten as R0

RhRv

1113968

where Rh (aεphπhμh(μh + ε)(μh + τ + ω)) and Rv

(σapvπvμ2v(σ + μv)) (us R0 can also be defined as thesquare root of the product of the number of infected humansin the susceptible population caused by one infected tsetsefly in its infectious lifetime and the number of infected tsetse

flies caused by one infected human during the infectiousperiod [12]

34 Local Stability of Disease-Free Equilibrium (DFE)

Theorem 2 If R0 le 1 the DFE given by E0 is locally as-ymptotically stable in the region defined by (7) and it isunstable when R0 gt 1

Proof (e DFE is locally stable if all eigenvalues of Jacobianmatrix JE0

are negative (e matrix has all eigenvalues neg-ative only if the trace of JE0

lt 0 and determinant of JE0gt 0 By

linearizing system (6) around E0 we obtain the followingJacobian matrix

JE0

minusμh 0 0 ρ 0 0 minusaphsh

0 minus ε + μh( 1113857 0 0 0 0 aphsh

0 ε minus ω + τ + μh( 1113857 0 0 0 00 0 τ minus ρ + μh( 1113857 0 0 00 0 minusapvsv 0 minusμv 0 00 0 apvsv 0 0 minus σ + μv( 1113857 00 0 0 0 0 σ minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(17)

(e trace of matrix JE0is such that

tr JE01113872 1113873 minus( μh + ε + μh + μh + ω + τ + ρ + μh

+ μv + σ + μv + μv1113857

minus 4μh + 3μv + ε + ω + τ + ρ + σ( 1113857lt 0

(18)

Using the basic properties of matrix algebra as in [13] itis clear that the eigenvalues λ1 minusμh and λ2 minusμv of thematrix JE0

have negative real parts (e reduced matrix is

JE1

minus ε + μh( 1113857 0 0 0 aphsh

ε minus ω + τ + μh( 1113857 0 0 0

0 τ minus ρ + μh( 1113857 0 0

0 apvsv 0 minus σ + μv( 1113857 0

0 0 0 σ minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(19)

From matrix JE1 the eigenvalue λ3 minus(ρ + μh) has

negative real part (e remaining matrix is further reducedby using the reduction techniques and we obtain

JE2

minus ε + μh( 1113857 0 0 aphsh

0 minus ω + τ + μh( 1113857 0aphshεε + μh

0 apvsv minus σ + μv( 1113857 0

0 0 σ minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(20)

Using the properties of matrix algebra the matrix JE2has

eigenvalue minus(ε + μh) which has negative real part Wefurther reduce to a 2 times 2 matrix by using the same reductiontechniques (e matrix is

JE3

minus ω + τ + μh( 1113857aphshεε + μh

apvsvσσ + μv

minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (21)

From the reduced 2 times 2 matrix the trace is negative andthe determinant is

Det JE31113872 1113873 ω + τ + μh( 1113857μv minus

aphshεε + μh

timesapvsvσσ + μv

ω + τ + μh( 1113857μv 1minusa2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 11138571113890 1113891

(22)

Since

R0

a2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

1113971

(23)

then by letting RT (a2εphpvπhπvσμ2vμh(ε + μh)(σ + μv)(μh + τ + ω)) we find our determinant as

Det JE31113872 1113873 ω + τ + μh( 1113857μv 1minusRT1113858 1113859 (24)

(e value of RT can be seen to be positive because all theparameters are positive As a result the determinant in (24)

Computational and Mathematical Methods in Medicine 5

is positive if and only if RT lt 1 (erefore the DFE is locallystable if RT le 1

35 Global Stability of Disease-Free Equilibrium (DFE)To show that the DFE is globally stable we apply Lyapunovrsquostheorem in [5]

Theorem 3 2eDFE defined by E0 is globally asymptoticallystable in the region defined by (7) if R0 le 1 Otherwise unstableif R0 gt 1

Proof We define Lyapunovrsquos function as

V k1 sh minus sh0 minus sh0 lnsh

sh01113888 1113889 + k2eh + k3ih

+ k4 sv minus sv0 minus sv0 lnsv

sv01113888 1113889 + k5ev + k6iv

(25)

satisfying system (6) where k1 k2 k3 k4 k5 k6 gt 0 are to bedetermined and sh0 (πhμh) and sv0 (πvμv) We firstshow that Vgt 0 for all Ene ((πhμh) 0 0 0 (πvμv) 0 0) It isenough to check that

k1sh0sh

sh0minus 1minus ln

sh

sh01113888 1113889gt 0

k4sv0svsv0minus 1minus ln

svsv0

1113888 1113889gt 0

(26)

(e function g(m) mminus 1minus lnm such thatm (shsh0) (svsv0) has minimum value equal to zerowhen m 1 hence g(m)gt 0 for all mgt 0 (us Lyapunovrsquosfunction Vgt 0 (e function V is radially unbounded be-cause as |m|⟶infin the function g(m)⟶infin We nowtake the derivative of V with respect to time and use system(6) to replace the derivatives in the right hand side such that

Vprime k1 1minussh0sh

1113888 1113889dsh

dt+ k2

deh

dt+ k3

dih

dt+ k4 1minus

sv0sv

1113888 1113889dsv

dt

+ k5dev

dt+ k6

div

dt

k1 1minussh0sh

1113888 1113889 πh + ρrh minus aphivsh minus μhsh1113858 1113859

+ k2 aphivsh minus εeh minus μheh1113858 1113859

+ k3 εeh minus μhih minusωih minus τih1113858 1113859

+ k4 1minussv0sv

1113888 1113889 πv minus μvsv minus apvihsv1113858 1113859

+ k5 apvihsv minus μvev minus σev1113858 1113859 + k6 σev minus μviv1113858 1113859

2k1πh minus aphshk1iv + ρrhk1 minus k1μhsh

minus k1π2h

μhsh+

aphπhiv

μhk1 minus

πhρrh

μhshk1 + k2aphshiv

minus k2 ε + μh( 1113857eh + k3εeh minus k3 τ + μh + ω( 1113857ih

+ 2k4πv minus k4apvsvih minus μvsvk4 minusπ2v

μvsvk4

+apvihπv

μvk4 minus k5 μv + σ( 1113857ev

+ k5apvsvih + k6σev minus k6μviv(27)

(e terms with rh are ignored because if sh eh ih areglobally stable then rh⟶ 0 at any time t and the DFE forsystem (6) is globally stable Taking k1 k2 (1μh + ε)k4 k5 (1μv + σ) k3 (1ε) and k6 (1σ) the de-rivative of V with respect to time becomes

Vprime minusπh

μh + επhμhsh

+μhsh

πhminus 21113888 1113889minus

τ + μh + ω( 1113857

εih

+aphπhμh + ε( 1113857μh

iv minusπv

μv + σπv

μvsv+μvsvπvminus 21113888 1113889

minusμvσ

iv +apvπv

μv μv + σ( 1113857ih

minusπh

μh + επh

μhsh+μhsh

πhminus 21113888 1113889 +

πv

μv + σπv

μvsv+μvsvπvminus 21113888 11138891113890 1113891

+τ + μh + ω( 1113857

εεapvπv

μv σ + μv( 1113857 μh + τ + ω( 1113857minus 11113888 1113889ih

+μvσ

aσphπhμvμh μh + ε( 1113857

minus 11113888 1113889iv

minusπh

μh + επh

μhsh+μhsh

πhminus 21113888 1113889 +

πv

μv + σπv

μvsv+μvsvπvminus 21113888 11138891113890 1113891

+τ + μh + ω( 1113857

εRv minus 1( 1113857ih +

μv

σRh minus 1( 1113857iv

(28)

(e terms ((πhμhsh) + (μhshπh)minus 2) and ((πvμvsv) +

(μvsvπv)minus 2) are positive because if we suppose m

(πhμhsh) (πvμvsv) we have m + (1m)minus 2 (m2 minus 2m +

1m) ((mminus 1)2m)gt 0 for all mgt 1 and since Rv le 1 andRh le 1 then Vprime is negative (us we have Vprime lt 0 for allE0 ne ((πhμh) 0 0 0 (πvμv) 0 0)

(us the largest compact invariant set in D is the sin-gleton set E0 Hence system (6) is globally asymptoticallystable

6 Computational and Mathematical Methods in Medicine

36 Local Stability of Endemic Equilibrium (EE)

Theorem 4 2e unique endemic equilibrium defined by Elowast

is locally asymptotically stable in the region defined by (7) ifR0 gt 1 but is unstable if R0 le 1

Proof We give the proof of this theorem based on the ap-proach used by Olaniyi and Obabiyi [9 14] From the EE

points defined in (12) since all values are positive we expressthe value of ilowasth in terms of R0 to obtain

ilowasth

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

B

middota2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857minus 11113890 1113891

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

BR20 minus 11113960 1113961

(29)

where

B ( apv( aσphπv( ερω + μh( ρ(τ + ω) + ε(ρ + τ + ω)

+ μh ε + ρ + τ + ω + μh( 111385711138571113857 + μh ε + μh( 1113857 ρ + μh( 1113857

middot τ + ω + μh( 1113857μv σ + μv( 111385711138571113857

(30)

Since the basic reproduction number R0 (a2εphpvπhπvσμ2vμh(ε + μh)(σ + μv)(μh + τ + ω))

1113968 if we

let

RT a2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857 (31)

then we can find ilowasth as

ilowasth

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

BRT minus 11113858 1113859

(32)

(e value of B is clearly positive because all parametersare positive Hence ilowasth gt 0 if and only if RT gt 1 implying thatthe EE is locally asymptotically stable if RT gt 1

37 Global Stability of Endemic Equilibrium (EE) To showthe global stability of the EE we use Lyapunovrsquos theoremtogether with the following lemma

Lemma 1 Suppose that y1 y2 middot middot middot yn are n positive num-bers 2en their arithmetic mean is greater than or equal tothe geometric mean that is (y1 + y2 + middot middot middot + ynn)ge(y1y2 middot middot middot yn)1n

Theorem 5 2e EE defined by Elowast is globally asymptoticallystable if R0 gt 1 otherwise unstable

Proof (e proof is based on the idea as explained byMartcheva [5] We define Lyapunovrsquos function as

V k1 sh minus slowasth minus slowasth ln

shslowasth

1113888 1113889 + k2 eh minus elowasth minus elowasth ln

ehelowasth

1113888 1113889

+ k3 ih minus ilowasth minus ilowasth ln

ih

ilowasth1113888 1113889 + k4 sv minus s

lowastv minus slowastv ln

sv

slowastv1113888 1113889

+ k5 ev minus elowastv minus elowastv ln

ev

elowastv1113888 1113889 + k6 iv minus i

lowastv minus ilowastv ln

iv

ilowastv1113888 1113889

(33)

satisfying system (6) with k1 k2 k3 k4 k5 k6 gt 0 to bedetermined (e function V is nonnegative for all(sh eh ih rh sv ev iv)ne (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) and radiallyunbounded

We need to prove that Vprime lt 0 for all(sh eh ih rh sv ev iv)ne (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) We find thederivative of V with respect to time and replace the de-rivatives sh

prime ehprime ihprime rhprime svprime evprime ivprime with system (6) We also ignore

the rh terms because if sh eh ih are globally stable thenrh⟶ 0 at any time t and EE is globally stable

Vprime k1 1minusslowasthsh

1113888 1113889 πh minus aphivsh minus μhsh1113858 1113859 + k2 1minuselowastheh

1113888 1113889

middot aphivsh minus μh + ε( 1113857eh1113858 1113859 + k3 1minusilowasthih

1113888 1113889

middot εeh minus ω + τ + uh( 1113857ih1113858 1113859 + k4 1minusslowastvsv

1113888 1113889

middot πv minus apvihsv minus μvsv1113858 1113859 + k5 1minuselowastvev

1113888 1113889

middot apvihsv minus μv + σ( 1113857ev1113858 1113859 + k6 1minusilowastviv

1113888 1113889 σev minus uviv1113858 1113859

(34)

We now substitute πh aphilowastv slowasth + μhslowasth and πv

apvilowasthslowastv + μvslowastv at the endemic equilibrium and then simplifyand put similar terms together to obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth minus k1aphivsh minus k1aph

slowast2h ilowastvsh

+ k1aphivslowasth + k2aphshiv minus k2 ε + μh( 1113857eh minus k2aphshiv

elowastheh

+ k2 ε + μh( 1113857elowasth + k3εeh minus k3 ω + τ + μh( 1113857i

lowasth minus k3εeh

ilowasthih

+ k3 ω + τ + μh( 1113857ilowasth minus k4

sv minus slowastv( 11138572μv

sv+ k4apvi

lowasthslowastv

minus k4apvihsv minus k4apvslowast2v ilowasth

sv+ k4apvihs

lowastv + k5apvsvih

minus k5 σ + μv( 1113857ev minus k5apvsvihelowastvev

+ k5 σ + μv( 1113857elowastv + k6σev

minus k6μvilowastv minus k6σev

ilowastviv

+ k6μvilowastv

(35)

Computational and Mathematical Methods in Medicine 7

We suppose k1 k2 and k4 k5 and multiply and dividethe same equilibrium value to some of the fractions to obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth minus k1aph

slowast2h ilowastvsh

+ k1aphivslowasth

minus k2 ε + μh( 1113857eh minus k2aphshivelowasthslowasth ilowastvslowasth ilowastv eh

+ k2 ε + μh( 1113857elowasth

+ k3εeh minus k3 ω + τ + μh( 1113857ih minus k3εehilowasthelowasthihelowasth

+ k3 ω + τ + μh( 1113857ilowasth minus k4

sv minus slowastv( 11138572μv

sv+ k4apvi

lowasthslowastv

minus k4apvslowast2v ilowasth

sv+ k4apvihs

lowastv minus k5 σ + μv( 1113857ev

minus k5apvsvihelowastv slowastv ilowasthslowastv ilowasthev

+ k5 σ + μv( 1113857elowastv + k6σev minus k6μvi

lowastv

minus k6σevelowastv ilowastvivelowastv

+ k6μvilowastv

(36)

We choose k3 k2(μh + εε) such that k3(ω + τ+ μh)ilowasth

k2(ε + μh)elowasth and choose k6 k5(μh + σσ) such thatk6μvilowastv k5(σ + μv)elowastv Now aphslowasth ilowastv (μh + ε)elowasth andapvslowastv ilowastv (μv + σ)elowastv because k1 k2 and k4 k5 re-

spectively We now obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

+ k1aphivslowasth minus k3 ω + τ + μh( 1113857ih + k3εminus k2 ε + μh( 11138571113858 1113859eh

minus k4sv minus slowastv( 1113857

2μvsv

+ k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsv

slowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891

+ k4apvihslowastv minus k6μviv + k6σ minus k5 σ + μv( 11138571113858 1113859ev

(37)

Suppose k6 (k1aphslowasthμv) and k3 (k4apvslowastv ω + τ+

μh) We then substitute and simplify to get

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

minus k4sv minus slowastv( 1113857

2μvsv

+ k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsvslowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891

(38)

From Lemma 1 the terms

k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsv

slowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891le 0

(39)

(erefore Vprime lt 0 for all (sh eh ih rh sv ev iv)ne (slowasth

elowasth ilowasth rlowasth slowastv elowastv ilowastv ) implying that the endemic equilibrium isglobally asymptotically stable if R0 gt 1

4 Analysis of Optimal Control Model

In this section we formulate the optimal control model bymodifying system (6) to an optimal control problem Wethus define some linear functions ci(t) 1 for i 1 2 3 It isimportant to note that controls are fully effective whenci(t) 1 and not effective when ci(t) 0 (e forces ofinfection ξh and xiv which correspond to the human andvector population respectively are reduced by the factor(1minus c1) where c1 measures the level of success obtained dueto the effort of educating people on the dangers of exposingtheir skin and encouraging them to wear long sleeves andlong pants during the day to minimize tsetse fly-humancontacts (e factor c2 represents the effort of treatment tocontrol the disease and the factor c3 also represents theeffort of using insecticides to ensure that the breeding sites ofthe tsetse fly are minimized Hence taking into account theassumptions and extensions made we try to find the mosteffective strategy that reduces the HAT infection in thepopulation at a veryminimum costWith the use of boundedLebesgue measurable control we define the objectivefunction to be minimized as

J c1 c2 c3( 1113857 1113946tF

0( M1eh + M2ih + M3ev + M4iv

+12k1c

21 +

12k2c

22 +

12k3c

231113857 dt

(40)

(us the dynamics of the controls that minimizes theobjective function is given by

dshdt

πh + ρrh minus 1minus c1( 1113857aphivsh minus μhsh

deh

dt 1minus c1( 1113857aphivsh minus εeh minus μheh

dih

dt εeh minus μhih minusωih minus c2τih

drh

dt c2τih minus ρrh minus μhrh

dsv

dt πv minus c3μvsv minus 1minus c1( 1113857apvihsv

dev

dt 1minus c1( 1113857apvihsv minus c3μvev minus σev

div

dt σev minus c3μviv

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

subject to the initial conditions sh ge 0 eh ge 0 ih ge 0 rh ge 0

sv ge 0 ev ge 0 and iv ge 0 (e associated effective reproduc-tion number for (41) denoted by RE is obtained as

RE

a2ε 1minus c1( 11138572phπhπvσ

c3μh ε + μh( 1113857μ2v σ + c3μv( 1113857 μh + c2τ + ω( 1113857

1113971

Rc

1113968

(42)

8 Computational and Mathematical Methods in Medicine

(e goal is to minimize the exposed and infectious humanpopulations (eh ih) the exposed and infectious vector pop-ulations (ev iv) and the cost of implementing the control bythe use of possible ci i 1 2 3 (e functional objectiveincludes the social cost which relates to the resources that areneeded for educating people on personal protection (12)k1c

21

the application of treatment (12)k2c22 and spraying of tsetse

fly operations (12)k3c23 (e quantities M1 and M2 re-

spectively represent the associated cost with minimizing theexposed and infected human population while M3 and M4also represent the cost associated withminimizing the exposedand infected vectors respectively (e quantity tF is the timeperiod of intervention As explained in [15] the costs cor-responding to M1eh M2ih M3ev and M4iv are linear whilethe cost control functions (12)k1c

21 (12)k2c

22 and (12)k3c

23

should be nonlinear and take a quadratic form (erefore weseek to minimize the objective function over the given timeinterval [0 tF] Pontryaginrsquos maximum principle is used tosolve this optimal control problem and the derivation of thenecessary conditions (e Lagrangian of the optimal controlproblem is given by

L M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22 +

12k3c

231113874 1113875

(43)

To determine the Lagrangian minimum value we definethe Hamiltonian H for the control problem as

H M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22

+12k3c

23 + λsh

dsh

dt+ λeh

deh

dt+ λih

dih

dt+ λrh

drh

dt

+ λsv

dsv

dt+ λev

dev

dt+ λiv

div

dt

(44)

where λsh λeh

λih λrh

λsv λev

and λivare adjoint variables or

costate variables (e differential equations of adjoint var-iables are obtained by taking the partial derivatives of theHamiltonian equation with respect to the state variableswhich gives

dλsh

dt λsh

μh + λshminus λeh

1113872 1113873 1minus c1( 1113857aphiv

dλeh

dt minusM1 + λeh

minus λih1113872 1113873ε + λeh

μh

dλih

dt minusM2 + λih

minus λrh1113872 1113873c2τ + λih

ω + μh( 1113857 + λsvminus λev

1113872 1113873 1minus c1( 1113857apvsv

dλrh

dt λrhminus λsh

1113872 1113873ρ + λrhμh

dλsv

dt λsvminus λev

1113872 1113873 1minus c1( 1113857apvih + λsvμvc3

dλev

dt minusM3 + λev

minus λiv1113872 1113873σ + λev

μvc3

dλiv

dt minusM4 + λsh

minus λeh1113872 1113873 1minus c1( 1113857aphsh + λiv

μvc3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Theorem 6 Given the optimal controls clowast1 clowast2 clowast3 and thesolutions sh eh ih rh sv ev iv of the corresponding stateequations (41) and (40) which minimize J(c1 c2 c3) over theregion Ω then there exist adjoint variablesλsh

λeh λih

λrh λsv

λev λiv

satisfying

minusdλi

dt

zH

zi i isin sh eh ih rh sv ev iv1113864 1113865 (46)

and the optimal solution clowast1 clowast2 clowast3 is given by

clowast1 min 1 max 0 1113954c1( 11138571113864 1113865

clowast2 min 1 max 0 1113954c2( 11138571113864 1113865

clowast3 min 1 max 0 1113954c3( 11138571113864 1113865

(47)

Proof (e Pontryaginrsquos maximum principle described in[16 17] is applied Corollary 41 in [17] shows the existenceof an optimal control due to the convexity of the integrand Jwith respect to c1 c2 c3 and Lipschitz property of the statesystem with respect to the state variables By using theoptimal conditions

zH

zc1 0

zH

zc2 0

zH

zc3 0

(48)

we obtain

zH

zc1 k1c1 + λsh

minus λeh1113872 1113873aphivsh + λsv

minus λev1113872 1113873apvihsv 0

zH

zc2 k2c2 + λrh

minus λih1113872 1113873τih 0

zH

zc3 k3c3 minus λsv

μvsv + λevμvev + λiv

μviv1113872 1113873 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)

Table 2 Parameters values used for simulations

Parameter Value Referenceπh 0000215day [14]πv 007day [14]ph 062 [6]pv 0065 [6 15]a Varying Assumedσ 0001 Assumedε 0083 [18]ω 0004 [3]ρ 002 [6]τ 0125 [3]μh 000044 Assumedμv 0034 [15]

Computational and Mathematical Methods in Medicine 9

Solving (49) we have

c1 λeh minus λsh( )aphivsh + λev minus λsv( )apvihsv

k1

c2 λih minus λrh( )τih

k2

c3 λsvμvsv + λevμvev + λivμviv

k3

(50)

As stated earlier the lower and upper boundaries for thecontrol parameters are 0 and 1 respectively If c1 c2 c3 lt 1then c1 c2 c3 0 and if c1 c2 c3 gt 1 then c1 c2 c3 1 otherwise c1 c1 c2 c2 c3 c3 erefore for thecontrol parameters clowast1 c

lowast2 clowast3 we obtain the optimum value of

the function J(c1 c2 c3)

41 Optimal Control Simulations e Octave programminglanguage is used to simulate the optimal control model usingthe set of parameters obtained from previously reportedstudies and datasets which have been cited Some of theseparameters are assumed for the sake of illustrations Table 2represents the values of the model parameters used forsimulationse following initial conditions were considered

sh(0) 30

eh(0) 7

ih(0) 2

rh(0) 0

sv(0) 40

ev(0) 10

iv(0) 3

(51)

and the weight constants were assumed to be

M1 1

M2 2

M3 2

M4 2

k1 2

k2 10

k3 5

(52)

Figures 2 and 3 represent the control proles at dierentvalues of c1 c2 and c3 while the rest of the plots are thegraphs of infectious human and vector population plottedagainst time in days and they represent the eect of optimalcontrols c1 c2 and c3 in reducing the number of individualsinfected From Figure 4 we observe that the use of treatmentand insecticides only has a signicant impact in reducing thenumber of infectious individuals and they show that thisstrategy is eective to control tsetse ies and infected humanpopulations In Figure 5 we observe that the use of edu-cation and insecticides reduce the number of infectiousindividuals but the results depicted in Figure 5(a) shows thatthis strategy is not eective and ecient to control theinfectious human population From Figure 6 we observethat the use of education and treatment reduces the numberof infectious individuals but the results from Figure 6(b)shows that this strategy is not eective and ecient tocontrol the population of infectious tsetse ies Lastly theresults depicted from Figure 7 shows that the strategy ofusing education treatment and insecticides is very ecientand eective to reduce the number of infected individualserefore the use of education treatment and insecticidessimultaneously is very ecient and eective to eliminateHAT in Africa

In epidemiology a reproduction number less than unityimplies that the disease can be eradicated in the long run

06

04

02

0

Con

trol p

rofil

e

c1 = 0c2 ne 0c3 ne 0

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 = 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 2 (a) Control prole when c1 0 c2 ne 0 and c3 ne 0 at Rc 00075 (b) Control prole when c1 ne 0 c2 0 and c3 ne 0 at Rc 000013

10 Computational and Mathematical Methods in Medicine

Hence choosing suitable parameters for the controls c1 c2and c3 it was observed that the eective reproductionnumbers obtained for Figures 2ndash7were 00075 000013 00440000075 00075 000013 0044 and 0000075 respectivelyis shows that incorporating all the control measures that iseducating individuals giving treatment and applying in-secticides is an eective method to help reduce the numbersecondary infections in the population which correspondswith eradicating the disease in the long run

5 Conclusion

In this paper we studied and analyzed the model fortransmission of HAT and determined the basic reproduction

number e local and global stabilities of disease-freeequilibrium and endemic equilibrium were also provedFor the optimal control model education treatment andinsecticides as control measures were used to optimize theobjective function dened by Equation (40) e numericalsimulations of the optimal control model show that the beststrategy to reduce the number of infected individuals isthrough the use of education treatment and insecticidesis is the eective and ecient method to eliminate thedisease Furthermore the national authorities non-governmental organizations (NGOs) and stakeholdersmust not lose their interest in controlling the disease be-cause neglecting this disease may cause rapid reoccurrenceand much eect to the people who are at risk

06

04

02

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(a)

4

35

3

25

15

1

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(b)

Figure 4 Simulations of the model showing the eorts of treatment and insecticides only on infectious individuals at Rc 00075

c1 ne 0c2 ne 0c3 = 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 ne 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 3 (a) Control prole when c1 ne 0 c2 ne 0 and c3 0 at Rc 0044 (b) Control prole when c1 ne 0 c2 ne 0 and c3 ne 0 at Rc 0000075

Computational and Mathematical Methods in Medicine 11

20

15

10

5Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(a)

12

10

8

6

4

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(b)

Figure 7 Simulations of the model showing the eorts of education treatment and insecticides on infectious individuals at Rc 0000075

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(a)

44

42

38

4

34

36

32

3

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(b)

Figure 6 Simulations of the model showing the eorts of education and treatment only on infectious individuals at Rc 0044

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

20

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

(a)

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

Infe

ctio

us v

ecto

r pop

ulat

ion

12

8

10

6

4

20 20 40 60 80 100 120

Time (days)

(b)

Figure 5 Simulations of the model showing the eorts of education and insecticides only on infectious individuals at Rc 000013

12 Computational and Mathematical Methods in Medicine

Data Availability

(e secondary data supporting this research are from pre-viously reported studies and datasets which have been cited(e processed data are available in cited references

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(e authors thank the support from Next Einstein Initiativeof the African Institute for Mathematical Sciences Ghana(Annual Grant for Graduate Studies) and MasterCardFoundation

References

[1] K S Rock S J Torr C Lumbala and M J KeelingldquoQuantitative evaluation of the strategy to eliminate humanafrican trypanosomiasis in the democratic republic of CongordquoParasites and Vectors vol 8 no 1 p 532 2015

[2] J R Franco P P Simarro A Diarra and J G JanninldquoEpidemiology of human african trypanosomiasisrdquo ClinicalEpidemiology vol 2014 no 6 pp 257ndash275 2014

[3] R Brun J Blum F Chappuis and C Burri ldquoHuman africantrypanosomiasisrdquo Lancet vol 375 no 9709 pp 148ndash1592010

[4] World Health Organization et al Control and Surveillance ofHuman African Trypanosomiasis Report of a WHO ExpertCommittee World Health Organization Geneva Switzerland2013

[5] M Martcheva An Introduction to Mathematical Epidemiol-ogy Springer Vol 61 Springer Berlin Germany 2015

[6] D J Rogers ldquoA general model for the african trypanoso-miasesrdquo Parasitology vol 97 no 1 pp 193ndash212 1988

[7] J W Hargrove R Ouifki D Kajunguri G A Vale andS J Torr ldquoModeling the control of trypanosomiasis usingtrypanocides or insecticide-treated livestockrdquo PLoS NeglectedTropical Diseases vol 6 no 5 p 1615 2012

[8] D Kajunguri ldquoModelling the control of tsetse and africantrypanosomiasis through application of insecticides on cattlein Southeastern Ugandardquo PhD thesis Stellenbosch Uni-versity Stellenbosch South Africa 2013

[9] S Olaniyi and O S Obabiyi ldquoMathematical model for malariatransmission dynamics in human and mosquito populationswith nonlinear forces of infectionrdquo International Journal ofPure and Applied Mathematics vol 88 no 1 pp 125ndash1562013

[10] R H Chisholm P T Campbell Y Wu S Y C TongJ McVernon and N Geard ldquoImplications of asymptomaticcarriers for infectious disease transmission and controlrdquoRoyal Society Open Science vol 5 no 2 p 172341 2018

[11] P Yan and S Liu ldquoSeir epidemic model with delayrdquo ANZIAMJournal vol 48 no 1 pp 119ndash134 2006

[12] G T Azu-Tungmah ldquoA mathematical model to control thespread of malaria in Ghanardquo PhD thesis Kwame NkrumahUniversity of Science and Technology Kumasi Ghana 2012

[13] S A Pedro S Abelman F T Ndjomatchoua R Sang andH E Z Tonnang ldquoStability bifurcation and chaos analysis ofvector-borne disease model with application to rift valleyfeverrdquo PloS One vol 9 no 10 Article ID 108172 2014

[14] S Olaniyi and O S Obabiyi ldquoQualitative analysis of malariadynamics with nonlinear incidence functionrdquo AppliedMathematical Sciences vol 8 no 78 pp 3889ndash3904 2014

[15] L S Pontryagin V G Boltyanskii and R V Gamkrelidze EfMishchenko the Mathematical 2eory of Optimal ProcessesInterscience New York NY USA 1962

[16] B Heimann ldquoFleming WHrishel RW deterministic andstochastic optimal control New York-heidelbergberlinSpringer-verlag 1975 xiii 222 s dm 60 60rdquo ZAMM-Journal of Applied Mathematics and Mechanics Zeitschriftfur Angewandte Mathematik und Mechanik vol 59 no 9p 494 1979

[17] S Davis S Aksoy and A Galvani ldquoA global sensitivityanalysis for african sleeping sicknessrdquo Parasitology vol 138no 4 pp 516ndash526 2011

[18] M Artzrouni and J P Gouteux ldquoA compartmental model ofsleeping sickness in central africardquo Journal of BiologicalSystems vol 4 no 4 pp 459ndash477 1996

Computational and Mathematical Methods in Medicine 13

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Submit your manuscripts atwwwhindawicom

(e spectral radius σ(FVminus1) gives

R0 σ FVminus1

1113872 1113873

a2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

1113971

(16)

One infected human in a population of susceptible vectorswill cause Rv infected vectors likewise one infected vector ina population will cause Rh infected humans [5](erefore thebasic reproduction number can be rewritten as R0

RhRv

1113968

where Rh (aεphπhμh(μh + ε)(μh + τ + ω)) and Rv

(σapvπvμ2v(σ + μv)) (us R0 can also be defined as thesquare root of the product of the number of infected humansin the susceptible population caused by one infected tsetsefly in its infectious lifetime and the number of infected tsetse

flies caused by one infected human during the infectiousperiod [12]

34 Local Stability of Disease-Free Equilibrium (DFE)

Theorem 2 If R0 le 1 the DFE given by E0 is locally as-ymptotically stable in the region defined by (7) and it isunstable when R0 gt 1

Proof (e DFE is locally stable if all eigenvalues of Jacobianmatrix JE0

are negative (e matrix has all eigenvalues neg-ative only if the trace of JE0

lt 0 and determinant of JE0gt 0 By

linearizing system (6) around E0 we obtain the followingJacobian matrix

JE0

minusμh 0 0 ρ 0 0 minusaphsh

0 minus ε + μh( 1113857 0 0 0 0 aphsh

0 ε minus ω + τ + μh( 1113857 0 0 0 00 0 τ minus ρ + μh( 1113857 0 0 00 0 minusapvsv 0 minusμv 0 00 0 apvsv 0 0 minus σ + μv( 1113857 00 0 0 0 0 σ minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(17)

(e trace of matrix JE0is such that

tr JE01113872 1113873 minus( μh + ε + μh + μh + ω + τ + ρ + μh

+ μv + σ + μv + μv1113857

minus 4μh + 3μv + ε + ω + τ + ρ + σ( 1113857lt 0

(18)

Using the basic properties of matrix algebra as in [13] itis clear that the eigenvalues λ1 minusμh and λ2 minusμv of thematrix JE0

have negative real parts (e reduced matrix is

JE1

minus ε + μh( 1113857 0 0 0 aphsh

ε minus ω + τ + μh( 1113857 0 0 0

0 τ minus ρ + μh( 1113857 0 0

0 apvsv 0 minus σ + μv( 1113857 0

0 0 0 σ minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(19)

From matrix JE1 the eigenvalue λ3 minus(ρ + μh) has

negative real part (e remaining matrix is further reducedby using the reduction techniques and we obtain

JE2

minus ε + μh( 1113857 0 0 aphsh

0 minus ω + τ + μh( 1113857 0aphshεε + μh

0 apvsv minus σ + μv( 1113857 0

0 0 σ minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(20)

Using the properties of matrix algebra the matrix JE2has

eigenvalue minus(ε + μh) which has negative real part Wefurther reduce to a 2 times 2 matrix by using the same reductiontechniques (e matrix is

JE3

minus ω + τ + μh( 1113857aphshεε + μh

apvsvσσ + μv

minusμv

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (21)

From the reduced 2 times 2 matrix the trace is negative andthe determinant is

Det JE31113872 1113873 ω + τ + μh( 1113857μv minus

aphshεε + μh

timesapvsvσσ + μv

ω + τ + μh( 1113857μv 1minusa2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 11138571113890 1113891

(22)

Since

R0

a2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

1113971

(23)

then by letting RT (a2εphpvπhπvσμ2vμh(ε + μh)(σ + μv)(μh + τ + ω)) we find our determinant as

Det JE31113872 1113873 ω + τ + μh( 1113857μv 1minusRT1113858 1113859 (24)

(e value of RT can be seen to be positive because all theparameters are positive As a result the determinant in (24)

Computational and Mathematical Methods in Medicine 5

is positive if and only if RT lt 1 (erefore the DFE is locallystable if RT le 1

35 Global Stability of Disease-Free Equilibrium (DFE)To show that the DFE is globally stable we apply Lyapunovrsquostheorem in [5]

Theorem 3 2eDFE defined by E0 is globally asymptoticallystable in the region defined by (7) if R0 le 1 Otherwise unstableif R0 gt 1

Proof We define Lyapunovrsquos function as

V k1 sh minus sh0 minus sh0 lnsh

sh01113888 1113889 + k2eh + k3ih

+ k4 sv minus sv0 minus sv0 lnsv

sv01113888 1113889 + k5ev + k6iv

(25)

satisfying system (6) where k1 k2 k3 k4 k5 k6 gt 0 are to bedetermined and sh0 (πhμh) and sv0 (πvμv) We firstshow that Vgt 0 for all Ene ((πhμh) 0 0 0 (πvμv) 0 0) It isenough to check that

k1sh0sh

sh0minus 1minus ln

sh

sh01113888 1113889gt 0

k4sv0svsv0minus 1minus ln

svsv0

1113888 1113889gt 0

(26)

(e function g(m) mminus 1minus lnm such thatm (shsh0) (svsv0) has minimum value equal to zerowhen m 1 hence g(m)gt 0 for all mgt 0 (us Lyapunovrsquosfunction Vgt 0 (e function V is radially unbounded be-cause as |m|⟶infin the function g(m)⟶infin We nowtake the derivative of V with respect to time and use system(6) to replace the derivatives in the right hand side such that

Vprime k1 1minussh0sh

1113888 1113889dsh

dt+ k2

deh

dt+ k3

dih

dt+ k4 1minus

sv0sv

1113888 1113889dsv

dt

+ k5dev

dt+ k6

div

dt

k1 1minussh0sh

1113888 1113889 πh + ρrh minus aphivsh minus μhsh1113858 1113859

+ k2 aphivsh minus εeh minus μheh1113858 1113859

+ k3 εeh minus μhih minusωih minus τih1113858 1113859

+ k4 1minussv0sv

1113888 1113889 πv minus μvsv minus apvihsv1113858 1113859

+ k5 apvihsv minus μvev minus σev1113858 1113859 + k6 σev minus μviv1113858 1113859

2k1πh minus aphshk1iv + ρrhk1 minus k1μhsh

minus k1π2h

μhsh+

aphπhiv

μhk1 minus

πhρrh

μhshk1 + k2aphshiv

minus k2 ε + μh( 1113857eh + k3εeh minus k3 τ + μh + ω( 1113857ih

+ 2k4πv minus k4apvsvih minus μvsvk4 minusπ2v

μvsvk4

+apvihπv

μvk4 minus k5 μv + σ( 1113857ev

+ k5apvsvih + k6σev minus k6μviv(27)

(e terms with rh are ignored because if sh eh ih areglobally stable then rh⟶ 0 at any time t and the DFE forsystem (6) is globally stable Taking k1 k2 (1μh + ε)k4 k5 (1μv + σ) k3 (1ε) and k6 (1σ) the de-rivative of V with respect to time becomes

Vprime minusπh

μh + επhμhsh

+μhsh

πhminus 21113888 1113889minus

τ + μh + ω( 1113857

εih

+aphπhμh + ε( 1113857μh

iv minusπv

μv + σπv

μvsv+μvsvπvminus 21113888 1113889

minusμvσ

iv +apvπv

μv μv + σ( 1113857ih

minusπh

μh + επh

μhsh+μhsh

πhminus 21113888 1113889 +

πv

μv + σπv

μvsv+μvsvπvminus 21113888 11138891113890 1113891

+τ + μh + ω( 1113857

εεapvπv

μv σ + μv( 1113857 μh + τ + ω( 1113857minus 11113888 1113889ih

+μvσ

aσphπhμvμh μh + ε( 1113857

minus 11113888 1113889iv

minusπh

μh + επh

μhsh+μhsh

πhminus 21113888 1113889 +

πv

μv + σπv

μvsv+μvsvπvminus 21113888 11138891113890 1113891

+τ + μh + ω( 1113857

εRv minus 1( 1113857ih +

μv

σRh minus 1( 1113857iv

(28)

(e terms ((πhμhsh) + (μhshπh)minus 2) and ((πvμvsv) +

(μvsvπv)minus 2) are positive because if we suppose m

(πhμhsh) (πvμvsv) we have m + (1m)minus 2 (m2 minus 2m +

1m) ((mminus 1)2m)gt 0 for all mgt 1 and since Rv le 1 andRh le 1 then Vprime is negative (us we have Vprime lt 0 for allE0 ne ((πhμh) 0 0 0 (πvμv) 0 0)

(us the largest compact invariant set in D is the sin-gleton set E0 Hence system (6) is globally asymptoticallystable

6 Computational and Mathematical Methods in Medicine

36 Local Stability of Endemic Equilibrium (EE)

Theorem 4 2e unique endemic equilibrium defined by Elowast

is locally asymptotically stable in the region defined by (7) ifR0 gt 1 but is unstable if R0 le 1

Proof We give the proof of this theorem based on the ap-proach used by Olaniyi and Obabiyi [9 14] From the EE

points defined in (12) since all values are positive we expressthe value of ilowasth in terms of R0 to obtain

ilowasth

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

B

middota2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857minus 11113890 1113891

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

BR20 minus 11113960 1113961

(29)

where

B ( apv( aσphπv( ερω + μh( ρ(τ + ω) + ε(ρ + τ + ω)

+ μh ε + ρ + τ + ω + μh( 111385711138571113857 + μh ε + μh( 1113857 ρ + μh( 1113857

middot τ + ω + μh( 1113857μv σ + μv( 111385711138571113857

(30)

Since the basic reproduction number R0 (a2εphpvπhπvσμ2vμh(ε + μh)(σ + μv)(μh + τ + ω))

1113968 if we

let

RT a2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857 (31)

then we can find ilowasth as

ilowasth

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

BRT minus 11113858 1113859

(32)

(e value of B is clearly positive because all parametersare positive Hence ilowasth gt 0 if and only if RT gt 1 implying thatthe EE is locally asymptotically stable if RT gt 1

37 Global Stability of Endemic Equilibrium (EE) To showthe global stability of the EE we use Lyapunovrsquos theoremtogether with the following lemma

Lemma 1 Suppose that y1 y2 middot middot middot yn are n positive num-bers 2en their arithmetic mean is greater than or equal tothe geometric mean that is (y1 + y2 + middot middot middot + ynn)ge(y1y2 middot middot middot yn)1n

Theorem 5 2e EE defined by Elowast is globally asymptoticallystable if R0 gt 1 otherwise unstable

Proof (e proof is based on the idea as explained byMartcheva [5] We define Lyapunovrsquos function as

V k1 sh minus slowasth minus slowasth ln

shslowasth

1113888 1113889 + k2 eh minus elowasth minus elowasth ln

ehelowasth

1113888 1113889

+ k3 ih minus ilowasth minus ilowasth ln

ih

ilowasth1113888 1113889 + k4 sv minus s

lowastv minus slowastv ln

sv

slowastv1113888 1113889

+ k5 ev minus elowastv minus elowastv ln

ev

elowastv1113888 1113889 + k6 iv minus i

lowastv minus ilowastv ln

iv

ilowastv1113888 1113889

(33)

satisfying system (6) with k1 k2 k3 k4 k5 k6 gt 0 to bedetermined (e function V is nonnegative for all(sh eh ih rh sv ev iv)ne (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) and radiallyunbounded

We need to prove that Vprime lt 0 for all(sh eh ih rh sv ev iv)ne (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) We find thederivative of V with respect to time and replace the de-rivatives sh

prime ehprime ihprime rhprime svprime evprime ivprime with system (6) We also ignore

the rh terms because if sh eh ih are globally stable thenrh⟶ 0 at any time t and EE is globally stable

Vprime k1 1minusslowasthsh

1113888 1113889 πh minus aphivsh minus μhsh1113858 1113859 + k2 1minuselowastheh

1113888 1113889

middot aphivsh minus μh + ε( 1113857eh1113858 1113859 + k3 1minusilowasthih

1113888 1113889

middot εeh minus ω + τ + uh( 1113857ih1113858 1113859 + k4 1minusslowastvsv

1113888 1113889

middot πv minus apvihsv minus μvsv1113858 1113859 + k5 1minuselowastvev

1113888 1113889

middot apvihsv minus μv + σ( 1113857ev1113858 1113859 + k6 1minusilowastviv

1113888 1113889 σev minus uviv1113858 1113859

(34)

We now substitute πh aphilowastv slowasth + μhslowasth and πv

apvilowasthslowastv + μvslowastv at the endemic equilibrium and then simplifyand put similar terms together to obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth minus k1aphivsh minus k1aph

slowast2h ilowastvsh

+ k1aphivslowasth + k2aphshiv minus k2 ε + μh( 1113857eh minus k2aphshiv

elowastheh

+ k2 ε + μh( 1113857elowasth + k3εeh minus k3 ω + τ + μh( 1113857i

lowasth minus k3εeh

ilowasthih

+ k3 ω + τ + μh( 1113857ilowasth minus k4

sv minus slowastv( 11138572μv

sv+ k4apvi

lowasthslowastv

minus k4apvihsv minus k4apvslowast2v ilowasth

sv+ k4apvihs

lowastv + k5apvsvih

minus k5 σ + μv( 1113857ev minus k5apvsvihelowastvev

+ k5 σ + μv( 1113857elowastv + k6σev

minus k6μvilowastv minus k6σev

ilowastviv

+ k6μvilowastv

(35)

Computational and Mathematical Methods in Medicine 7

We suppose k1 k2 and k4 k5 and multiply and dividethe same equilibrium value to some of the fractions to obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth minus k1aph

slowast2h ilowastvsh

+ k1aphivslowasth

minus k2 ε + μh( 1113857eh minus k2aphshivelowasthslowasth ilowastvslowasth ilowastv eh

+ k2 ε + μh( 1113857elowasth

+ k3εeh minus k3 ω + τ + μh( 1113857ih minus k3εehilowasthelowasthihelowasth

+ k3 ω + τ + μh( 1113857ilowasth minus k4

sv minus slowastv( 11138572μv

sv+ k4apvi

lowasthslowastv

minus k4apvslowast2v ilowasth

sv+ k4apvihs

lowastv minus k5 σ + μv( 1113857ev

minus k5apvsvihelowastv slowastv ilowasthslowastv ilowasthev

+ k5 σ + μv( 1113857elowastv + k6σev minus k6μvi

lowastv

minus k6σevelowastv ilowastvivelowastv

+ k6μvilowastv

(36)

We choose k3 k2(μh + εε) such that k3(ω + τ+ μh)ilowasth

k2(ε + μh)elowasth and choose k6 k5(μh + σσ) such thatk6μvilowastv k5(σ + μv)elowastv Now aphslowasth ilowastv (μh + ε)elowasth andapvslowastv ilowastv (μv + σ)elowastv because k1 k2 and k4 k5 re-

spectively We now obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

+ k1aphivslowasth minus k3 ω + τ + μh( 1113857ih + k3εminus k2 ε + μh( 11138571113858 1113859eh

minus k4sv minus slowastv( 1113857

2μvsv

+ k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsv

slowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891

+ k4apvihslowastv minus k6μviv + k6σ minus k5 σ + μv( 11138571113858 1113859ev

(37)

Suppose k6 (k1aphslowasthμv) and k3 (k4apvslowastv ω + τ+

μh) We then substitute and simplify to get

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

minus k4sv minus slowastv( 1113857

2μvsv

+ k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsvslowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891

(38)

From Lemma 1 the terms

k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsv

slowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891le 0

(39)

(erefore Vprime lt 0 for all (sh eh ih rh sv ev iv)ne (slowasth

elowasth ilowasth rlowasth slowastv elowastv ilowastv ) implying that the endemic equilibrium isglobally asymptotically stable if R0 gt 1

4 Analysis of Optimal Control Model

In this section we formulate the optimal control model bymodifying system (6) to an optimal control problem Wethus define some linear functions ci(t) 1 for i 1 2 3 It isimportant to note that controls are fully effective whenci(t) 1 and not effective when ci(t) 0 (e forces ofinfection ξh and xiv which correspond to the human andvector population respectively are reduced by the factor(1minus c1) where c1 measures the level of success obtained dueto the effort of educating people on the dangers of exposingtheir skin and encouraging them to wear long sleeves andlong pants during the day to minimize tsetse fly-humancontacts (e factor c2 represents the effort of treatment tocontrol the disease and the factor c3 also represents theeffort of using insecticides to ensure that the breeding sites ofthe tsetse fly are minimized Hence taking into account theassumptions and extensions made we try to find the mosteffective strategy that reduces the HAT infection in thepopulation at a veryminimum costWith the use of boundedLebesgue measurable control we define the objectivefunction to be minimized as

J c1 c2 c3( 1113857 1113946tF

0( M1eh + M2ih + M3ev + M4iv

+12k1c

21 +

12k2c

22 +

12k3c

231113857 dt

(40)

(us the dynamics of the controls that minimizes theobjective function is given by

dshdt

πh + ρrh minus 1minus c1( 1113857aphivsh minus μhsh

deh

dt 1minus c1( 1113857aphivsh minus εeh minus μheh

dih

dt εeh minus μhih minusωih minus c2τih

drh

dt c2τih minus ρrh minus μhrh

dsv

dt πv minus c3μvsv minus 1minus c1( 1113857apvihsv

dev

dt 1minus c1( 1113857apvihsv minus c3μvev minus σev

div

dt σev minus c3μviv

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

subject to the initial conditions sh ge 0 eh ge 0 ih ge 0 rh ge 0

sv ge 0 ev ge 0 and iv ge 0 (e associated effective reproduc-tion number for (41) denoted by RE is obtained as

RE

a2ε 1minus c1( 11138572phπhπvσ

c3μh ε + μh( 1113857μ2v σ + c3μv( 1113857 μh + c2τ + ω( 1113857

1113971

Rc

1113968

(42)

8 Computational and Mathematical Methods in Medicine

(e goal is to minimize the exposed and infectious humanpopulations (eh ih) the exposed and infectious vector pop-ulations (ev iv) and the cost of implementing the control bythe use of possible ci i 1 2 3 (e functional objectiveincludes the social cost which relates to the resources that areneeded for educating people on personal protection (12)k1c

21

the application of treatment (12)k2c22 and spraying of tsetse

fly operations (12)k3c23 (e quantities M1 and M2 re-

spectively represent the associated cost with minimizing theexposed and infected human population while M3 and M4also represent the cost associated withminimizing the exposedand infected vectors respectively (e quantity tF is the timeperiod of intervention As explained in [15] the costs cor-responding to M1eh M2ih M3ev and M4iv are linear whilethe cost control functions (12)k1c

21 (12)k2c

22 and (12)k3c

23

should be nonlinear and take a quadratic form (erefore weseek to minimize the objective function over the given timeinterval [0 tF] Pontryaginrsquos maximum principle is used tosolve this optimal control problem and the derivation of thenecessary conditions (e Lagrangian of the optimal controlproblem is given by

L M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22 +

12k3c

231113874 1113875

(43)

To determine the Lagrangian minimum value we definethe Hamiltonian H for the control problem as

H M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22

+12k3c

23 + λsh

dsh

dt+ λeh

deh

dt+ λih

dih

dt+ λrh

drh

dt

+ λsv

dsv

dt+ λev

dev

dt+ λiv

div

dt

(44)

where λsh λeh

λih λrh

λsv λev

and λivare adjoint variables or

costate variables (e differential equations of adjoint var-iables are obtained by taking the partial derivatives of theHamiltonian equation with respect to the state variableswhich gives

dλsh

dt λsh

μh + λshminus λeh

1113872 1113873 1minus c1( 1113857aphiv

dλeh

dt minusM1 + λeh

minus λih1113872 1113873ε + λeh

μh

dλih

dt minusM2 + λih

minus λrh1113872 1113873c2τ + λih

ω + μh( 1113857 + λsvminus λev

1113872 1113873 1minus c1( 1113857apvsv

dλrh

dt λrhminus λsh

1113872 1113873ρ + λrhμh

dλsv

dt λsvminus λev

1113872 1113873 1minus c1( 1113857apvih + λsvμvc3

dλev

dt minusM3 + λev

minus λiv1113872 1113873σ + λev

μvc3

dλiv

dt minusM4 + λsh

minus λeh1113872 1113873 1minus c1( 1113857aphsh + λiv

μvc3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Theorem 6 Given the optimal controls clowast1 clowast2 clowast3 and thesolutions sh eh ih rh sv ev iv of the corresponding stateequations (41) and (40) which minimize J(c1 c2 c3) over theregion Ω then there exist adjoint variablesλsh

λeh λih

λrh λsv

λev λiv

satisfying

minusdλi

dt

zH

zi i isin sh eh ih rh sv ev iv1113864 1113865 (46)

and the optimal solution clowast1 clowast2 clowast3 is given by

clowast1 min 1 max 0 1113954c1( 11138571113864 1113865

clowast2 min 1 max 0 1113954c2( 11138571113864 1113865

clowast3 min 1 max 0 1113954c3( 11138571113864 1113865

(47)

Proof (e Pontryaginrsquos maximum principle described in[16 17] is applied Corollary 41 in [17] shows the existenceof an optimal control due to the convexity of the integrand Jwith respect to c1 c2 c3 and Lipschitz property of the statesystem with respect to the state variables By using theoptimal conditions

zH

zc1 0

zH

zc2 0

zH

zc3 0

(48)

we obtain

zH

zc1 k1c1 + λsh

minus λeh1113872 1113873aphivsh + λsv

minus λev1113872 1113873apvihsv 0

zH

zc2 k2c2 + λrh

minus λih1113872 1113873τih 0

zH

zc3 k3c3 minus λsv

μvsv + λevμvev + λiv

μviv1113872 1113873 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)

Table 2 Parameters values used for simulations

Parameter Value Referenceπh 0000215day [14]πv 007day [14]ph 062 [6]pv 0065 [6 15]a Varying Assumedσ 0001 Assumedε 0083 [18]ω 0004 [3]ρ 002 [6]τ 0125 [3]μh 000044 Assumedμv 0034 [15]

Computational and Mathematical Methods in Medicine 9

Solving (49) we have

c1 λeh minus λsh( )aphivsh + λev minus λsv( )apvihsv

k1

c2 λih minus λrh( )τih

k2

c3 λsvμvsv + λevμvev + λivμviv

k3

(50)

As stated earlier the lower and upper boundaries for thecontrol parameters are 0 and 1 respectively If c1 c2 c3 lt 1then c1 c2 c3 0 and if c1 c2 c3 gt 1 then c1 c2 c3 1 otherwise c1 c1 c2 c2 c3 c3 erefore for thecontrol parameters clowast1 c

lowast2 clowast3 we obtain the optimum value of

the function J(c1 c2 c3)

41 Optimal Control Simulations e Octave programminglanguage is used to simulate the optimal control model usingthe set of parameters obtained from previously reportedstudies and datasets which have been cited Some of theseparameters are assumed for the sake of illustrations Table 2represents the values of the model parameters used forsimulationse following initial conditions were considered

sh(0) 30

eh(0) 7

ih(0) 2

rh(0) 0

sv(0) 40

ev(0) 10

iv(0) 3

(51)

and the weight constants were assumed to be

M1 1

M2 2

M3 2

M4 2

k1 2

k2 10

k3 5

(52)

Figures 2 and 3 represent the control proles at dierentvalues of c1 c2 and c3 while the rest of the plots are thegraphs of infectious human and vector population plottedagainst time in days and they represent the eect of optimalcontrols c1 c2 and c3 in reducing the number of individualsinfected From Figure 4 we observe that the use of treatmentand insecticides only has a signicant impact in reducing thenumber of infectious individuals and they show that thisstrategy is eective to control tsetse ies and infected humanpopulations In Figure 5 we observe that the use of edu-cation and insecticides reduce the number of infectiousindividuals but the results depicted in Figure 5(a) shows thatthis strategy is not eective and ecient to control theinfectious human population From Figure 6 we observethat the use of education and treatment reduces the numberof infectious individuals but the results from Figure 6(b)shows that this strategy is not eective and ecient tocontrol the population of infectious tsetse ies Lastly theresults depicted from Figure 7 shows that the strategy ofusing education treatment and insecticides is very ecientand eective to reduce the number of infected individualserefore the use of education treatment and insecticidessimultaneously is very ecient and eective to eliminateHAT in Africa

In epidemiology a reproduction number less than unityimplies that the disease can be eradicated in the long run

06

04

02

0

Con

trol p

rofil

e

c1 = 0c2 ne 0c3 ne 0

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 = 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 2 (a) Control prole when c1 0 c2 ne 0 and c3 ne 0 at Rc 00075 (b) Control prole when c1 ne 0 c2 0 and c3 ne 0 at Rc 000013

10 Computational and Mathematical Methods in Medicine

Hence choosing suitable parameters for the controls c1 c2and c3 it was observed that the eective reproductionnumbers obtained for Figures 2ndash7were 00075 000013 00440000075 00075 000013 0044 and 0000075 respectivelyis shows that incorporating all the control measures that iseducating individuals giving treatment and applying in-secticides is an eective method to help reduce the numbersecondary infections in the population which correspondswith eradicating the disease in the long run

5 Conclusion

In this paper we studied and analyzed the model fortransmission of HAT and determined the basic reproduction

number e local and global stabilities of disease-freeequilibrium and endemic equilibrium were also provedFor the optimal control model education treatment andinsecticides as control measures were used to optimize theobjective function dened by Equation (40) e numericalsimulations of the optimal control model show that the beststrategy to reduce the number of infected individuals isthrough the use of education treatment and insecticidesis is the eective and ecient method to eliminate thedisease Furthermore the national authorities non-governmental organizations (NGOs) and stakeholdersmust not lose their interest in controlling the disease be-cause neglecting this disease may cause rapid reoccurrenceand much eect to the people who are at risk

06

04

02

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(a)

4

35

3

25

15

1

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(b)

Figure 4 Simulations of the model showing the eorts of treatment and insecticides only on infectious individuals at Rc 00075

c1 ne 0c2 ne 0c3 = 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 ne 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 3 (a) Control prole when c1 ne 0 c2 ne 0 and c3 0 at Rc 0044 (b) Control prole when c1 ne 0 c2 ne 0 and c3 ne 0 at Rc 0000075

Computational and Mathematical Methods in Medicine 11

20

15

10

5Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(a)

12

10

8

6

4

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(b)

Figure 7 Simulations of the model showing the eorts of education treatment and insecticides on infectious individuals at Rc 0000075

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(a)

44

42

38

4

34

36

32

3

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(b)

Figure 6 Simulations of the model showing the eorts of education and treatment only on infectious individuals at Rc 0044

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

20

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

(a)

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

Infe

ctio

us v

ecto

r pop

ulat

ion

12

8

10

6

4

20 20 40 60 80 100 120

Time (days)

(b)

Figure 5 Simulations of the model showing the eorts of education and insecticides only on infectious individuals at Rc 000013

12 Computational and Mathematical Methods in Medicine

Data Availability

(e secondary data supporting this research are from pre-viously reported studies and datasets which have been cited(e processed data are available in cited references

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(e authors thank the support from Next Einstein Initiativeof the African Institute for Mathematical Sciences Ghana(Annual Grant for Graduate Studies) and MasterCardFoundation

References

[1] K S Rock S J Torr C Lumbala and M J KeelingldquoQuantitative evaluation of the strategy to eliminate humanafrican trypanosomiasis in the democratic republic of CongordquoParasites and Vectors vol 8 no 1 p 532 2015

[2] J R Franco P P Simarro A Diarra and J G JanninldquoEpidemiology of human african trypanosomiasisrdquo ClinicalEpidemiology vol 2014 no 6 pp 257ndash275 2014

[3] R Brun J Blum F Chappuis and C Burri ldquoHuman africantrypanosomiasisrdquo Lancet vol 375 no 9709 pp 148ndash1592010

[4] World Health Organization et al Control and Surveillance ofHuman African Trypanosomiasis Report of a WHO ExpertCommittee World Health Organization Geneva Switzerland2013

[5] M Martcheva An Introduction to Mathematical Epidemiol-ogy Springer Vol 61 Springer Berlin Germany 2015

[6] D J Rogers ldquoA general model for the african trypanoso-miasesrdquo Parasitology vol 97 no 1 pp 193ndash212 1988

[7] J W Hargrove R Ouifki D Kajunguri G A Vale andS J Torr ldquoModeling the control of trypanosomiasis usingtrypanocides or insecticide-treated livestockrdquo PLoS NeglectedTropical Diseases vol 6 no 5 p 1615 2012

[8] D Kajunguri ldquoModelling the control of tsetse and africantrypanosomiasis through application of insecticides on cattlein Southeastern Ugandardquo PhD thesis Stellenbosch Uni-versity Stellenbosch South Africa 2013

[9] S Olaniyi and O S Obabiyi ldquoMathematical model for malariatransmission dynamics in human and mosquito populationswith nonlinear forces of infectionrdquo International Journal ofPure and Applied Mathematics vol 88 no 1 pp 125ndash1562013

[10] R H Chisholm P T Campbell Y Wu S Y C TongJ McVernon and N Geard ldquoImplications of asymptomaticcarriers for infectious disease transmission and controlrdquoRoyal Society Open Science vol 5 no 2 p 172341 2018

[11] P Yan and S Liu ldquoSeir epidemic model with delayrdquo ANZIAMJournal vol 48 no 1 pp 119ndash134 2006

[12] G T Azu-Tungmah ldquoA mathematical model to control thespread of malaria in Ghanardquo PhD thesis Kwame NkrumahUniversity of Science and Technology Kumasi Ghana 2012

[13] S A Pedro S Abelman F T Ndjomatchoua R Sang andH E Z Tonnang ldquoStability bifurcation and chaos analysis ofvector-borne disease model with application to rift valleyfeverrdquo PloS One vol 9 no 10 Article ID 108172 2014

[14] S Olaniyi and O S Obabiyi ldquoQualitative analysis of malariadynamics with nonlinear incidence functionrdquo AppliedMathematical Sciences vol 8 no 78 pp 3889ndash3904 2014

[15] L S Pontryagin V G Boltyanskii and R V Gamkrelidze EfMishchenko the Mathematical 2eory of Optimal ProcessesInterscience New York NY USA 1962

[16] B Heimann ldquoFleming WHrishel RW deterministic andstochastic optimal control New York-heidelbergberlinSpringer-verlag 1975 xiii 222 s dm 60 60rdquo ZAMM-Journal of Applied Mathematics and Mechanics Zeitschriftfur Angewandte Mathematik und Mechanik vol 59 no 9p 494 1979

[17] S Davis S Aksoy and A Galvani ldquoA global sensitivityanalysis for african sleeping sicknessrdquo Parasitology vol 138no 4 pp 516ndash526 2011

[18] M Artzrouni and J P Gouteux ldquoA compartmental model ofsleeping sickness in central africardquo Journal of BiologicalSystems vol 4 no 4 pp 459ndash477 1996

Computational and Mathematical Methods in Medicine 13

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of

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Disease Markers

Hindawiwwwhindawicom Volume 2018

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Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom

is positive if and only if RT lt 1 (erefore the DFE is locallystable if RT le 1

35 Global Stability of Disease-Free Equilibrium (DFE)To show that the DFE is globally stable we apply Lyapunovrsquostheorem in [5]

Theorem 3 2eDFE defined by E0 is globally asymptoticallystable in the region defined by (7) if R0 le 1 Otherwise unstableif R0 gt 1

Proof We define Lyapunovrsquos function as

V k1 sh minus sh0 minus sh0 lnsh

sh01113888 1113889 + k2eh + k3ih

+ k4 sv minus sv0 minus sv0 lnsv

sv01113888 1113889 + k5ev + k6iv

(25)

satisfying system (6) where k1 k2 k3 k4 k5 k6 gt 0 are to bedetermined and sh0 (πhμh) and sv0 (πvμv) We firstshow that Vgt 0 for all Ene ((πhμh) 0 0 0 (πvμv) 0 0) It isenough to check that

k1sh0sh

sh0minus 1minus ln

sh

sh01113888 1113889gt 0

k4sv0svsv0minus 1minus ln

svsv0

1113888 1113889gt 0

(26)

(e function g(m) mminus 1minus lnm such thatm (shsh0) (svsv0) has minimum value equal to zerowhen m 1 hence g(m)gt 0 for all mgt 0 (us Lyapunovrsquosfunction Vgt 0 (e function V is radially unbounded be-cause as |m|⟶infin the function g(m)⟶infin We nowtake the derivative of V with respect to time and use system(6) to replace the derivatives in the right hand side such that

Vprime k1 1minussh0sh

1113888 1113889dsh

dt+ k2

deh

dt+ k3

dih

dt+ k4 1minus

sv0sv

1113888 1113889dsv

dt

+ k5dev

dt+ k6

div

dt

k1 1minussh0sh

1113888 1113889 πh + ρrh minus aphivsh minus μhsh1113858 1113859

+ k2 aphivsh minus εeh minus μheh1113858 1113859

+ k3 εeh minus μhih minusωih minus τih1113858 1113859

+ k4 1minussv0sv

1113888 1113889 πv minus μvsv minus apvihsv1113858 1113859

+ k5 apvihsv minus μvev minus σev1113858 1113859 + k6 σev minus μviv1113858 1113859

2k1πh minus aphshk1iv + ρrhk1 minus k1μhsh

minus k1π2h

μhsh+

aphπhiv

μhk1 minus

πhρrh

μhshk1 + k2aphshiv

minus k2 ε + μh( 1113857eh + k3εeh minus k3 τ + μh + ω( 1113857ih

+ 2k4πv minus k4apvsvih minus μvsvk4 minusπ2v

μvsvk4

+apvihπv

μvk4 minus k5 μv + σ( 1113857ev

+ k5apvsvih + k6σev minus k6μviv(27)

(e terms with rh are ignored because if sh eh ih areglobally stable then rh⟶ 0 at any time t and the DFE forsystem (6) is globally stable Taking k1 k2 (1μh + ε)k4 k5 (1μv + σ) k3 (1ε) and k6 (1σ) the de-rivative of V with respect to time becomes

Vprime minusπh

μh + επhμhsh

+μhsh

πhminus 21113888 1113889minus

τ + μh + ω( 1113857

εih

+aphπhμh + ε( 1113857μh

iv minusπv

μv + σπv

μvsv+μvsvπvminus 21113888 1113889

minusμvσ

iv +apvπv

μv μv + σ( 1113857ih

minusπh

μh + επh

μhsh+μhsh

πhminus 21113888 1113889 +

πv

μv + σπv

μvsv+μvsvπvminus 21113888 11138891113890 1113891

+τ + μh + ω( 1113857

εεapvπv

μv σ + μv( 1113857 μh + τ + ω( 1113857minus 11113888 1113889ih

+μvσ

aσphπhμvμh μh + ε( 1113857

minus 11113888 1113889iv

minusπh

μh + επh

μhsh+μhsh

πhminus 21113888 1113889 +

πv

μv + σπv

μvsv+μvsvπvminus 21113888 11138891113890 1113891

+τ + μh + ω( 1113857

εRv minus 1( 1113857ih +

μv

σRh minus 1( 1113857iv

(28)

(e terms ((πhμhsh) + (μhshπh)minus 2) and ((πvμvsv) +

(μvsvπv)minus 2) are positive because if we suppose m

(πhμhsh) (πvμvsv) we have m + (1m)minus 2 (m2 minus 2m +

1m) ((mminus 1)2m)gt 0 for all mgt 1 and since Rv le 1 andRh le 1 then Vprime is negative (us we have Vprime lt 0 for allE0 ne ((πhμh) 0 0 0 (πvμv) 0 0)

(us the largest compact invariant set in D is the sin-gleton set E0 Hence system (6) is globally asymptoticallystable

6 Computational and Mathematical Methods in Medicine

36 Local Stability of Endemic Equilibrium (EE)

Theorem 4 2e unique endemic equilibrium defined by Elowast

is locally asymptotically stable in the region defined by (7) ifR0 gt 1 but is unstable if R0 le 1

Proof We give the proof of this theorem based on the ap-proach used by Olaniyi and Obabiyi [9 14] From the EE

points defined in (12) since all values are positive we expressthe value of ilowasth in terms of R0 to obtain

ilowasth

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

B

middota2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857minus 11113890 1113891

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

BR20 minus 11113960 1113961

(29)

where

B ( apv( aσphπv( ερω + μh( ρ(τ + ω) + ε(ρ + τ + ω)

+ μh ε + ρ + τ + ω + μh( 111385711138571113857 + μh ε + μh( 1113857 ρ + μh( 1113857

middot τ + ω + μh( 1113857μv σ + μv( 111385711138571113857

(30)

Since the basic reproduction number R0 (a2εphpvπhπvσμ2vμh(ε + μh)(σ + μv)(μh + τ + ω))

1113968 if we

let

RT a2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857 (31)

then we can find ilowasth as

ilowasth

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

BRT minus 11113858 1113859

(32)

(e value of B is clearly positive because all parametersare positive Hence ilowasth gt 0 if and only if RT gt 1 implying thatthe EE is locally asymptotically stable if RT gt 1

37 Global Stability of Endemic Equilibrium (EE) To showthe global stability of the EE we use Lyapunovrsquos theoremtogether with the following lemma

Lemma 1 Suppose that y1 y2 middot middot middot yn are n positive num-bers 2en their arithmetic mean is greater than or equal tothe geometric mean that is (y1 + y2 + middot middot middot + ynn)ge(y1y2 middot middot middot yn)1n

Theorem 5 2e EE defined by Elowast is globally asymptoticallystable if R0 gt 1 otherwise unstable

Proof (e proof is based on the idea as explained byMartcheva [5] We define Lyapunovrsquos function as

V k1 sh minus slowasth minus slowasth ln

shslowasth

1113888 1113889 + k2 eh minus elowasth minus elowasth ln

ehelowasth

1113888 1113889

+ k3 ih minus ilowasth minus ilowasth ln

ih

ilowasth1113888 1113889 + k4 sv minus s

lowastv minus slowastv ln

sv

slowastv1113888 1113889

+ k5 ev minus elowastv minus elowastv ln

ev

elowastv1113888 1113889 + k6 iv minus i

lowastv minus ilowastv ln

iv

ilowastv1113888 1113889

(33)

satisfying system (6) with k1 k2 k3 k4 k5 k6 gt 0 to bedetermined (e function V is nonnegative for all(sh eh ih rh sv ev iv)ne (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) and radiallyunbounded

We need to prove that Vprime lt 0 for all(sh eh ih rh sv ev iv)ne (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) We find thederivative of V with respect to time and replace the de-rivatives sh

prime ehprime ihprime rhprime svprime evprime ivprime with system (6) We also ignore

the rh terms because if sh eh ih are globally stable thenrh⟶ 0 at any time t and EE is globally stable

Vprime k1 1minusslowasthsh

1113888 1113889 πh minus aphivsh minus μhsh1113858 1113859 + k2 1minuselowastheh

1113888 1113889

middot aphivsh minus μh + ε( 1113857eh1113858 1113859 + k3 1minusilowasthih

1113888 1113889

middot εeh minus ω + τ + uh( 1113857ih1113858 1113859 + k4 1minusslowastvsv

1113888 1113889

middot πv minus apvihsv minus μvsv1113858 1113859 + k5 1minuselowastvev

1113888 1113889

middot apvihsv minus μv + σ( 1113857ev1113858 1113859 + k6 1minusilowastviv

1113888 1113889 σev minus uviv1113858 1113859

(34)

We now substitute πh aphilowastv slowasth + μhslowasth and πv

apvilowasthslowastv + μvslowastv at the endemic equilibrium and then simplifyand put similar terms together to obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth minus k1aphivsh minus k1aph

slowast2h ilowastvsh

+ k1aphivslowasth + k2aphshiv minus k2 ε + μh( 1113857eh minus k2aphshiv

elowastheh

+ k2 ε + μh( 1113857elowasth + k3εeh minus k3 ω + τ + μh( 1113857i

lowasth minus k3εeh

ilowasthih

+ k3 ω + τ + μh( 1113857ilowasth minus k4

sv minus slowastv( 11138572μv

sv+ k4apvi

lowasthslowastv

minus k4apvihsv minus k4apvslowast2v ilowasth

sv+ k4apvihs

lowastv + k5apvsvih

minus k5 σ + μv( 1113857ev minus k5apvsvihelowastvev

+ k5 σ + μv( 1113857elowastv + k6σev

minus k6μvilowastv minus k6σev

ilowastviv

+ k6μvilowastv

(35)

Computational and Mathematical Methods in Medicine 7

We suppose k1 k2 and k4 k5 and multiply and dividethe same equilibrium value to some of the fractions to obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth minus k1aph

slowast2h ilowastvsh

+ k1aphivslowasth

minus k2 ε + μh( 1113857eh minus k2aphshivelowasthslowasth ilowastvslowasth ilowastv eh

+ k2 ε + μh( 1113857elowasth

+ k3εeh minus k3 ω + τ + μh( 1113857ih minus k3εehilowasthelowasthihelowasth

+ k3 ω + τ + μh( 1113857ilowasth minus k4

sv minus slowastv( 11138572μv

sv+ k4apvi

lowasthslowastv

minus k4apvslowast2v ilowasth

sv+ k4apvihs

lowastv minus k5 σ + μv( 1113857ev

minus k5apvsvihelowastv slowastv ilowasthslowastv ilowasthev

+ k5 σ + μv( 1113857elowastv + k6σev minus k6μvi

lowastv

minus k6σevelowastv ilowastvivelowastv

+ k6μvilowastv

(36)

We choose k3 k2(μh + εε) such that k3(ω + τ+ μh)ilowasth

k2(ε + μh)elowasth and choose k6 k5(μh + σσ) such thatk6μvilowastv k5(σ + μv)elowastv Now aphslowasth ilowastv (μh + ε)elowasth andapvslowastv ilowastv (μv + σ)elowastv because k1 k2 and k4 k5 re-

spectively We now obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

+ k1aphivslowasth minus k3 ω + τ + μh( 1113857ih + k3εminus k2 ε + μh( 11138571113858 1113859eh

minus k4sv minus slowastv( 1113857

2μvsv

+ k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsv

slowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891

+ k4apvihslowastv minus k6μviv + k6σ minus k5 σ + μv( 11138571113858 1113859ev

(37)

Suppose k6 (k1aphslowasthμv) and k3 (k4apvslowastv ω + τ+

μh) We then substitute and simplify to get

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

minus k4sv minus slowastv( 1113857

2μvsv

+ k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsvslowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891

(38)

From Lemma 1 the terms

k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsv

slowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891le 0

(39)

(erefore Vprime lt 0 for all (sh eh ih rh sv ev iv)ne (slowasth

elowasth ilowasth rlowasth slowastv elowastv ilowastv ) implying that the endemic equilibrium isglobally asymptotically stable if R0 gt 1

4 Analysis of Optimal Control Model

In this section we formulate the optimal control model bymodifying system (6) to an optimal control problem Wethus define some linear functions ci(t) 1 for i 1 2 3 It isimportant to note that controls are fully effective whenci(t) 1 and not effective when ci(t) 0 (e forces ofinfection ξh and xiv which correspond to the human andvector population respectively are reduced by the factor(1minus c1) where c1 measures the level of success obtained dueto the effort of educating people on the dangers of exposingtheir skin and encouraging them to wear long sleeves andlong pants during the day to minimize tsetse fly-humancontacts (e factor c2 represents the effort of treatment tocontrol the disease and the factor c3 also represents theeffort of using insecticides to ensure that the breeding sites ofthe tsetse fly are minimized Hence taking into account theassumptions and extensions made we try to find the mosteffective strategy that reduces the HAT infection in thepopulation at a veryminimum costWith the use of boundedLebesgue measurable control we define the objectivefunction to be minimized as

J c1 c2 c3( 1113857 1113946tF

0( M1eh + M2ih + M3ev + M4iv

+12k1c

21 +

12k2c

22 +

12k3c

231113857 dt

(40)

(us the dynamics of the controls that minimizes theobjective function is given by

dshdt

πh + ρrh minus 1minus c1( 1113857aphivsh minus μhsh

deh

dt 1minus c1( 1113857aphivsh minus εeh minus μheh

dih

dt εeh minus μhih minusωih minus c2τih

drh

dt c2τih minus ρrh minus μhrh

dsv

dt πv minus c3μvsv minus 1minus c1( 1113857apvihsv

dev

dt 1minus c1( 1113857apvihsv minus c3μvev minus σev

div

dt σev minus c3μviv

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

subject to the initial conditions sh ge 0 eh ge 0 ih ge 0 rh ge 0

sv ge 0 ev ge 0 and iv ge 0 (e associated effective reproduc-tion number for (41) denoted by RE is obtained as

RE

a2ε 1minus c1( 11138572phπhπvσ

c3μh ε + μh( 1113857μ2v σ + c3μv( 1113857 μh + c2τ + ω( 1113857

1113971

Rc

1113968

(42)

8 Computational and Mathematical Methods in Medicine

(e goal is to minimize the exposed and infectious humanpopulations (eh ih) the exposed and infectious vector pop-ulations (ev iv) and the cost of implementing the control bythe use of possible ci i 1 2 3 (e functional objectiveincludes the social cost which relates to the resources that areneeded for educating people on personal protection (12)k1c

21

the application of treatment (12)k2c22 and spraying of tsetse

fly operations (12)k3c23 (e quantities M1 and M2 re-

spectively represent the associated cost with minimizing theexposed and infected human population while M3 and M4also represent the cost associated withminimizing the exposedand infected vectors respectively (e quantity tF is the timeperiod of intervention As explained in [15] the costs cor-responding to M1eh M2ih M3ev and M4iv are linear whilethe cost control functions (12)k1c

21 (12)k2c

22 and (12)k3c

23

should be nonlinear and take a quadratic form (erefore weseek to minimize the objective function over the given timeinterval [0 tF] Pontryaginrsquos maximum principle is used tosolve this optimal control problem and the derivation of thenecessary conditions (e Lagrangian of the optimal controlproblem is given by

L M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22 +

12k3c

231113874 1113875

(43)

To determine the Lagrangian minimum value we definethe Hamiltonian H for the control problem as

H M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22

+12k3c

23 + λsh

dsh

dt+ λeh

deh

dt+ λih

dih

dt+ λrh

drh

dt

+ λsv

dsv

dt+ λev

dev

dt+ λiv

div

dt

(44)

where λsh λeh

λih λrh

λsv λev

and λivare adjoint variables or

costate variables (e differential equations of adjoint var-iables are obtained by taking the partial derivatives of theHamiltonian equation with respect to the state variableswhich gives

dλsh

dt λsh

μh + λshminus λeh

1113872 1113873 1minus c1( 1113857aphiv

dλeh

dt minusM1 + λeh

minus λih1113872 1113873ε + λeh

μh

dλih

dt minusM2 + λih

minus λrh1113872 1113873c2τ + λih

ω + μh( 1113857 + λsvminus λev

1113872 1113873 1minus c1( 1113857apvsv

dλrh

dt λrhminus λsh

1113872 1113873ρ + λrhμh

dλsv

dt λsvminus λev

1113872 1113873 1minus c1( 1113857apvih + λsvμvc3

dλev

dt minusM3 + λev

minus λiv1113872 1113873σ + λev

μvc3

dλiv

dt minusM4 + λsh

minus λeh1113872 1113873 1minus c1( 1113857aphsh + λiv

μvc3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Theorem 6 Given the optimal controls clowast1 clowast2 clowast3 and thesolutions sh eh ih rh sv ev iv of the corresponding stateequations (41) and (40) which minimize J(c1 c2 c3) over theregion Ω then there exist adjoint variablesλsh

λeh λih

λrh λsv

λev λiv

satisfying

minusdλi

dt

zH

zi i isin sh eh ih rh sv ev iv1113864 1113865 (46)

and the optimal solution clowast1 clowast2 clowast3 is given by

clowast1 min 1 max 0 1113954c1( 11138571113864 1113865

clowast2 min 1 max 0 1113954c2( 11138571113864 1113865

clowast3 min 1 max 0 1113954c3( 11138571113864 1113865

(47)

Proof (e Pontryaginrsquos maximum principle described in[16 17] is applied Corollary 41 in [17] shows the existenceof an optimal control due to the convexity of the integrand Jwith respect to c1 c2 c3 and Lipschitz property of the statesystem with respect to the state variables By using theoptimal conditions

zH

zc1 0

zH

zc2 0

zH

zc3 0

(48)

we obtain

zH

zc1 k1c1 + λsh

minus λeh1113872 1113873aphivsh + λsv

minus λev1113872 1113873apvihsv 0

zH

zc2 k2c2 + λrh

minus λih1113872 1113873τih 0

zH

zc3 k3c3 minus λsv

μvsv + λevμvev + λiv

μviv1113872 1113873 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)

Table 2 Parameters values used for simulations

Parameter Value Referenceπh 0000215day [14]πv 007day [14]ph 062 [6]pv 0065 [6 15]a Varying Assumedσ 0001 Assumedε 0083 [18]ω 0004 [3]ρ 002 [6]τ 0125 [3]μh 000044 Assumedμv 0034 [15]

Computational and Mathematical Methods in Medicine 9

Solving (49) we have

c1 λeh minus λsh( )aphivsh + λev minus λsv( )apvihsv

k1

c2 λih minus λrh( )τih

k2

c3 λsvμvsv + λevμvev + λivμviv

k3

(50)

As stated earlier the lower and upper boundaries for thecontrol parameters are 0 and 1 respectively If c1 c2 c3 lt 1then c1 c2 c3 0 and if c1 c2 c3 gt 1 then c1 c2 c3 1 otherwise c1 c1 c2 c2 c3 c3 erefore for thecontrol parameters clowast1 c

lowast2 clowast3 we obtain the optimum value of

the function J(c1 c2 c3)

41 Optimal Control Simulations e Octave programminglanguage is used to simulate the optimal control model usingthe set of parameters obtained from previously reportedstudies and datasets which have been cited Some of theseparameters are assumed for the sake of illustrations Table 2represents the values of the model parameters used forsimulationse following initial conditions were considered

sh(0) 30

eh(0) 7

ih(0) 2

rh(0) 0

sv(0) 40

ev(0) 10

iv(0) 3

(51)

and the weight constants were assumed to be

M1 1

M2 2

M3 2

M4 2

k1 2

k2 10

k3 5

(52)

Figures 2 and 3 represent the control proles at dierentvalues of c1 c2 and c3 while the rest of the plots are thegraphs of infectious human and vector population plottedagainst time in days and they represent the eect of optimalcontrols c1 c2 and c3 in reducing the number of individualsinfected From Figure 4 we observe that the use of treatmentand insecticides only has a signicant impact in reducing thenumber of infectious individuals and they show that thisstrategy is eective to control tsetse ies and infected humanpopulations In Figure 5 we observe that the use of edu-cation and insecticides reduce the number of infectiousindividuals but the results depicted in Figure 5(a) shows thatthis strategy is not eective and ecient to control theinfectious human population From Figure 6 we observethat the use of education and treatment reduces the numberof infectious individuals but the results from Figure 6(b)shows that this strategy is not eective and ecient tocontrol the population of infectious tsetse ies Lastly theresults depicted from Figure 7 shows that the strategy ofusing education treatment and insecticides is very ecientand eective to reduce the number of infected individualserefore the use of education treatment and insecticidessimultaneously is very ecient and eective to eliminateHAT in Africa

In epidemiology a reproduction number less than unityimplies that the disease can be eradicated in the long run

06

04

02

0

Con

trol p

rofil

e

c1 = 0c2 ne 0c3 ne 0

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 = 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 2 (a) Control prole when c1 0 c2 ne 0 and c3 ne 0 at Rc 00075 (b) Control prole when c1 ne 0 c2 0 and c3 ne 0 at Rc 000013

10 Computational and Mathematical Methods in Medicine

Hence choosing suitable parameters for the controls c1 c2and c3 it was observed that the eective reproductionnumbers obtained for Figures 2ndash7were 00075 000013 00440000075 00075 000013 0044 and 0000075 respectivelyis shows that incorporating all the control measures that iseducating individuals giving treatment and applying in-secticides is an eective method to help reduce the numbersecondary infections in the population which correspondswith eradicating the disease in the long run

5 Conclusion

In this paper we studied and analyzed the model fortransmission of HAT and determined the basic reproduction

number e local and global stabilities of disease-freeequilibrium and endemic equilibrium were also provedFor the optimal control model education treatment andinsecticides as control measures were used to optimize theobjective function dened by Equation (40) e numericalsimulations of the optimal control model show that the beststrategy to reduce the number of infected individuals isthrough the use of education treatment and insecticidesis is the eective and ecient method to eliminate thedisease Furthermore the national authorities non-governmental organizations (NGOs) and stakeholdersmust not lose their interest in controlling the disease be-cause neglecting this disease may cause rapid reoccurrenceand much eect to the people who are at risk

06

04

02

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(a)

4

35

3

25

15

1

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(b)

Figure 4 Simulations of the model showing the eorts of treatment and insecticides only on infectious individuals at Rc 00075

c1 ne 0c2 ne 0c3 = 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 ne 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 3 (a) Control prole when c1 ne 0 c2 ne 0 and c3 0 at Rc 0044 (b) Control prole when c1 ne 0 c2 ne 0 and c3 ne 0 at Rc 0000075

Computational and Mathematical Methods in Medicine 11

20

15

10

5Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(a)

12

10

8

6

4

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(b)

Figure 7 Simulations of the model showing the eorts of education treatment and insecticides on infectious individuals at Rc 0000075

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(a)

44

42

38

4

34

36

32

3

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(b)

Figure 6 Simulations of the model showing the eorts of education and treatment only on infectious individuals at Rc 0044

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

20

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

(a)

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

Infe

ctio

us v

ecto

r pop

ulat

ion

12

8

10

6

4

20 20 40 60 80 100 120

Time (days)

(b)

Figure 5 Simulations of the model showing the eorts of education and insecticides only on infectious individuals at Rc 000013

12 Computational and Mathematical Methods in Medicine

Data Availability

(e secondary data supporting this research are from pre-viously reported studies and datasets which have been cited(e processed data are available in cited references

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(e authors thank the support from Next Einstein Initiativeof the African Institute for Mathematical Sciences Ghana(Annual Grant for Graduate Studies) and MasterCardFoundation

References

[1] K S Rock S J Torr C Lumbala and M J KeelingldquoQuantitative evaluation of the strategy to eliminate humanafrican trypanosomiasis in the democratic republic of CongordquoParasites and Vectors vol 8 no 1 p 532 2015

[2] J R Franco P P Simarro A Diarra and J G JanninldquoEpidemiology of human african trypanosomiasisrdquo ClinicalEpidemiology vol 2014 no 6 pp 257ndash275 2014

[3] R Brun J Blum F Chappuis and C Burri ldquoHuman africantrypanosomiasisrdquo Lancet vol 375 no 9709 pp 148ndash1592010

[4] World Health Organization et al Control and Surveillance ofHuman African Trypanosomiasis Report of a WHO ExpertCommittee World Health Organization Geneva Switzerland2013

[5] M Martcheva An Introduction to Mathematical Epidemiol-ogy Springer Vol 61 Springer Berlin Germany 2015

[6] D J Rogers ldquoA general model for the african trypanoso-miasesrdquo Parasitology vol 97 no 1 pp 193ndash212 1988

[7] J W Hargrove R Ouifki D Kajunguri G A Vale andS J Torr ldquoModeling the control of trypanosomiasis usingtrypanocides or insecticide-treated livestockrdquo PLoS NeglectedTropical Diseases vol 6 no 5 p 1615 2012

[8] D Kajunguri ldquoModelling the control of tsetse and africantrypanosomiasis through application of insecticides on cattlein Southeastern Ugandardquo PhD thesis Stellenbosch Uni-versity Stellenbosch South Africa 2013

[9] S Olaniyi and O S Obabiyi ldquoMathematical model for malariatransmission dynamics in human and mosquito populationswith nonlinear forces of infectionrdquo International Journal ofPure and Applied Mathematics vol 88 no 1 pp 125ndash1562013

[10] R H Chisholm P T Campbell Y Wu S Y C TongJ McVernon and N Geard ldquoImplications of asymptomaticcarriers for infectious disease transmission and controlrdquoRoyal Society Open Science vol 5 no 2 p 172341 2018

[11] P Yan and S Liu ldquoSeir epidemic model with delayrdquo ANZIAMJournal vol 48 no 1 pp 119ndash134 2006

[12] G T Azu-Tungmah ldquoA mathematical model to control thespread of malaria in Ghanardquo PhD thesis Kwame NkrumahUniversity of Science and Technology Kumasi Ghana 2012

[13] S A Pedro S Abelman F T Ndjomatchoua R Sang andH E Z Tonnang ldquoStability bifurcation and chaos analysis ofvector-borne disease model with application to rift valleyfeverrdquo PloS One vol 9 no 10 Article ID 108172 2014

[14] S Olaniyi and O S Obabiyi ldquoQualitative analysis of malariadynamics with nonlinear incidence functionrdquo AppliedMathematical Sciences vol 8 no 78 pp 3889ndash3904 2014

[15] L S Pontryagin V G Boltyanskii and R V Gamkrelidze EfMishchenko the Mathematical 2eory of Optimal ProcessesInterscience New York NY USA 1962

[16] B Heimann ldquoFleming WHrishel RW deterministic andstochastic optimal control New York-heidelbergberlinSpringer-verlag 1975 xiii 222 s dm 60 60rdquo ZAMM-Journal of Applied Mathematics and Mechanics Zeitschriftfur Angewandte Mathematik und Mechanik vol 59 no 9p 494 1979

[17] S Davis S Aksoy and A Galvani ldquoA global sensitivityanalysis for african sleeping sicknessrdquo Parasitology vol 138no 4 pp 516ndash526 2011

[18] M Artzrouni and J P Gouteux ldquoA compartmental model ofsleeping sickness in central africardquo Journal of BiologicalSystems vol 4 no 4 pp 459ndash477 1996

Computational and Mathematical Methods in Medicine 13

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Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

MEDIATORSINFLAMMATION

of

EndocrinologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

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Disease Markers

Hindawiwwwhindawicom Volume 2018

BioMed Research International

OncologyJournal of

Hindawiwwwhindawicom Volume 2013

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Hindawiwwwhindawicom Volume 2018

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ObesityJournal of

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Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom

36 Local Stability of Endemic Equilibrium (EE)

Theorem 4 2e unique endemic equilibrium defined by Elowast

is locally asymptotically stable in the region defined by (7) ifR0 gt 1 but is unstable if R0 le 1

Proof We give the proof of this theorem based on the ap-proach used by Olaniyi and Obabiyi [9 14] From the EE

points defined in (12) since all values are positive we expressthe value of ilowasth in terms of R0 to obtain

ilowasth

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

B

middota2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857minus 11113890 1113891

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

BR20 minus 11113960 1113961

(29)

where

B ( apv( aσphπv( ερω + μh( ρ(τ + ω) + ε(ρ + τ + ω)

+ μh ε + ρ + τ + ω + μh( 111385711138571113857 + μh ε + μh( 1113857 ρ + μh( 1113857

middot τ + ω + μh( 1113857μv σ + μv( 111385711138571113857

(30)

Since the basic reproduction number R0 (a2εphpvπhπvσμ2vμh(ε + μh)(σ + μv)(μh + τ + ω))

1113968 if we

let

RT a2εphpvπhπvσ

μ2vμh ε + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857 (31)

then we can find ilowasth as

ilowasth

μ2vμh ε + μh( 1113857 ρ + μh( 1113857 σ + μv( 1113857 μh + τ + ω( 1113857

BRT minus 11113858 1113859

(32)

(e value of B is clearly positive because all parametersare positive Hence ilowasth gt 0 if and only if RT gt 1 implying thatthe EE is locally asymptotically stable if RT gt 1

37 Global Stability of Endemic Equilibrium (EE) To showthe global stability of the EE we use Lyapunovrsquos theoremtogether with the following lemma

Lemma 1 Suppose that y1 y2 middot middot middot yn are n positive num-bers 2en their arithmetic mean is greater than or equal tothe geometric mean that is (y1 + y2 + middot middot middot + ynn)ge(y1y2 middot middot middot yn)1n

Theorem 5 2e EE defined by Elowast is globally asymptoticallystable if R0 gt 1 otherwise unstable

Proof (e proof is based on the idea as explained byMartcheva [5] We define Lyapunovrsquos function as

V k1 sh minus slowasth minus slowasth ln

shslowasth

1113888 1113889 + k2 eh minus elowasth minus elowasth ln

ehelowasth

1113888 1113889

+ k3 ih minus ilowasth minus ilowasth ln

ih

ilowasth1113888 1113889 + k4 sv minus s

lowastv minus slowastv ln

sv

slowastv1113888 1113889

+ k5 ev minus elowastv minus elowastv ln

ev

elowastv1113888 1113889 + k6 iv minus i

lowastv minus ilowastv ln

iv

ilowastv1113888 1113889

(33)

satisfying system (6) with k1 k2 k3 k4 k5 k6 gt 0 to bedetermined (e function V is nonnegative for all(sh eh ih rh sv ev iv)ne (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) and radiallyunbounded

We need to prove that Vprime lt 0 for all(sh eh ih rh sv ev iv)ne (slowasth elowasth ilowasth rlowasth slowastv elowastv ilowastv ) We find thederivative of V with respect to time and replace the de-rivatives sh

prime ehprime ihprime rhprime svprime evprime ivprime with system (6) We also ignore

the rh terms because if sh eh ih are globally stable thenrh⟶ 0 at any time t and EE is globally stable

Vprime k1 1minusslowasthsh

1113888 1113889 πh minus aphivsh minus μhsh1113858 1113859 + k2 1minuselowastheh

1113888 1113889

middot aphivsh minus μh + ε( 1113857eh1113858 1113859 + k3 1minusilowasthih

1113888 1113889

middot εeh minus ω + τ + uh( 1113857ih1113858 1113859 + k4 1minusslowastvsv

1113888 1113889

middot πv minus apvihsv minus μvsv1113858 1113859 + k5 1minuselowastvev

1113888 1113889

middot apvihsv minus μv + σ( 1113857ev1113858 1113859 + k6 1minusilowastviv

1113888 1113889 σev minus uviv1113858 1113859

(34)

We now substitute πh aphilowastv slowasth + μhslowasth and πv

apvilowasthslowastv + μvslowastv at the endemic equilibrium and then simplifyand put similar terms together to obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth minus k1aphivsh minus k1aph

slowast2h ilowastvsh

+ k1aphivslowasth + k2aphshiv minus k2 ε + μh( 1113857eh minus k2aphshiv

elowastheh

+ k2 ε + μh( 1113857elowasth + k3εeh minus k3 ω + τ + μh( 1113857i

lowasth minus k3εeh

ilowasthih

+ k3 ω + τ + μh( 1113857ilowasth minus k4

sv minus slowastv( 11138572μv

sv+ k4apvi

lowasthslowastv

minus k4apvihsv minus k4apvslowast2v ilowasth

sv+ k4apvihs

lowastv + k5apvsvih

minus k5 σ + μv( 1113857ev minus k5apvsvihelowastvev

+ k5 σ + μv( 1113857elowastv + k6σev

minus k6μvilowastv minus k6σev

ilowastviv

+ k6μvilowastv

(35)

Computational and Mathematical Methods in Medicine 7

We suppose k1 k2 and k4 k5 and multiply and dividethe same equilibrium value to some of the fractions to obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth minus k1aph

slowast2h ilowastvsh

+ k1aphivslowasth

minus k2 ε + μh( 1113857eh minus k2aphshivelowasthslowasth ilowastvslowasth ilowastv eh

+ k2 ε + μh( 1113857elowasth

+ k3εeh minus k3 ω + τ + μh( 1113857ih minus k3εehilowasthelowasthihelowasth

+ k3 ω + τ + μh( 1113857ilowasth minus k4

sv minus slowastv( 11138572μv

sv+ k4apvi

lowasthslowastv

minus k4apvslowast2v ilowasth

sv+ k4apvihs

lowastv minus k5 σ + μv( 1113857ev

minus k5apvsvihelowastv slowastv ilowasthslowastv ilowasthev

+ k5 σ + μv( 1113857elowastv + k6σev minus k6μvi

lowastv

minus k6σevelowastv ilowastvivelowastv

+ k6μvilowastv

(36)

We choose k3 k2(μh + εε) such that k3(ω + τ+ μh)ilowasth

k2(ε + μh)elowasth and choose k6 k5(μh + σσ) such thatk6μvilowastv k5(σ + μv)elowastv Now aphslowasth ilowastv (μh + ε)elowasth andapvslowastv ilowastv (μv + σ)elowastv because k1 k2 and k4 k5 re-

spectively We now obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

+ k1aphivslowasth minus k3 ω + τ + μh( 1113857ih + k3εminus k2 ε + μh( 11138571113858 1113859eh

minus k4sv minus slowastv( 1113857

2μvsv

+ k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsv

slowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891

+ k4apvihslowastv minus k6μviv + k6σ minus k5 σ + μv( 11138571113858 1113859ev

(37)

Suppose k6 (k1aphslowasthμv) and k3 (k4apvslowastv ω + τ+

μh) We then substitute and simplify to get

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

minus k4sv minus slowastv( 1113857

2μvsv

+ k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsvslowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891

(38)

From Lemma 1 the terms

k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsv

slowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891le 0

(39)

(erefore Vprime lt 0 for all (sh eh ih rh sv ev iv)ne (slowasth

elowasth ilowasth rlowasth slowastv elowastv ilowastv ) implying that the endemic equilibrium isglobally asymptotically stable if R0 gt 1

4 Analysis of Optimal Control Model

In this section we formulate the optimal control model bymodifying system (6) to an optimal control problem Wethus define some linear functions ci(t) 1 for i 1 2 3 It isimportant to note that controls are fully effective whenci(t) 1 and not effective when ci(t) 0 (e forces ofinfection ξh and xiv which correspond to the human andvector population respectively are reduced by the factor(1minus c1) where c1 measures the level of success obtained dueto the effort of educating people on the dangers of exposingtheir skin and encouraging them to wear long sleeves andlong pants during the day to minimize tsetse fly-humancontacts (e factor c2 represents the effort of treatment tocontrol the disease and the factor c3 also represents theeffort of using insecticides to ensure that the breeding sites ofthe tsetse fly are minimized Hence taking into account theassumptions and extensions made we try to find the mosteffective strategy that reduces the HAT infection in thepopulation at a veryminimum costWith the use of boundedLebesgue measurable control we define the objectivefunction to be minimized as

J c1 c2 c3( 1113857 1113946tF

0( M1eh + M2ih + M3ev + M4iv

+12k1c

21 +

12k2c

22 +

12k3c

231113857 dt

(40)

(us the dynamics of the controls that minimizes theobjective function is given by

dshdt

πh + ρrh minus 1minus c1( 1113857aphivsh minus μhsh

deh

dt 1minus c1( 1113857aphivsh minus εeh minus μheh

dih

dt εeh minus μhih minusωih minus c2τih

drh

dt c2τih minus ρrh minus μhrh

dsv

dt πv minus c3μvsv minus 1minus c1( 1113857apvihsv

dev

dt 1minus c1( 1113857apvihsv minus c3μvev minus σev

div

dt σev minus c3μviv

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

subject to the initial conditions sh ge 0 eh ge 0 ih ge 0 rh ge 0

sv ge 0 ev ge 0 and iv ge 0 (e associated effective reproduc-tion number for (41) denoted by RE is obtained as

RE

a2ε 1minus c1( 11138572phπhπvσ

c3μh ε + μh( 1113857μ2v σ + c3μv( 1113857 μh + c2τ + ω( 1113857

1113971

Rc

1113968

(42)

8 Computational and Mathematical Methods in Medicine

(e goal is to minimize the exposed and infectious humanpopulations (eh ih) the exposed and infectious vector pop-ulations (ev iv) and the cost of implementing the control bythe use of possible ci i 1 2 3 (e functional objectiveincludes the social cost which relates to the resources that areneeded for educating people on personal protection (12)k1c

21

the application of treatment (12)k2c22 and spraying of tsetse

fly operations (12)k3c23 (e quantities M1 and M2 re-

spectively represent the associated cost with minimizing theexposed and infected human population while M3 and M4also represent the cost associated withminimizing the exposedand infected vectors respectively (e quantity tF is the timeperiod of intervention As explained in [15] the costs cor-responding to M1eh M2ih M3ev and M4iv are linear whilethe cost control functions (12)k1c

21 (12)k2c

22 and (12)k3c

23

should be nonlinear and take a quadratic form (erefore weseek to minimize the objective function over the given timeinterval [0 tF] Pontryaginrsquos maximum principle is used tosolve this optimal control problem and the derivation of thenecessary conditions (e Lagrangian of the optimal controlproblem is given by

L M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22 +

12k3c

231113874 1113875

(43)

To determine the Lagrangian minimum value we definethe Hamiltonian H for the control problem as

H M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22

+12k3c

23 + λsh

dsh

dt+ λeh

deh

dt+ λih

dih

dt+ λrh

drh

dt

+ λsv

dsv

dt+ λev

dev

dt+ λiv

div

dt

(44)

where λsh λeh

λih λrh

λsv λev

and λivare adjoint variables or

costate variables (e differential equations of adjoint var-iables are obtained by taking the partial derivatives of theHamiltonian equation with respect to the state variableswhich gives

dλsh

dt λsh

μh + λshminus λeh

1113872 1113873 1minus c1( 1113857aphiv

dλeh

dt minusM1 + λeh

minus λih1113872 1113873ε + λeh

μh

dλih

dt minusM2 + λih

minus λrh1113872 1113873c2τ + λih

ω + μh( 1113857 + λsvminus λev

1113872 1113873 1minus c1( 1113857apvsv

dλrh

dt λrhminus λsh

1113872 1113873ρ + λrhμh

dλsv

dt λsvminus λev

1113872 1113873 1minus c1( 1113857apvih + λsvμvc3

dλev

dt minusM3 + λev

minus λiv1113872 1113873σ + λev

μvc3

dλiv

dt minusM4 + λsh

minus λeh1113872 1113873 1minus c1( 1113857aphsh + λiv

μvc3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Theorem 6 Given the optimal controls clowast1 clowast2 clowast3 and thesolutions sh eh ih rh sv ev iv of the corresponding stateequations (41) and (40) which minimize J(c1 c2 c3) over theregion Ω then there exist adjoint variablesλsh

λeh λih

λrh λsv

λev λiv

satisfying

minusdλi

dt

zH

zi i isin sh eh ih rh sv ev iv1113864 1113865 (46)

and the optimal solution clowast1 clowast2 clowast3 is given by

clowast1 min 1 max 0 1113954c1( 11138571113864 1113865

clowast2 min 1 max 0 1113954c2( 11138571113864 1113865

clowast3 min 1 max 0 1113954c3( 11138571113864 1113865

(47)

Proof (e Pontryaginrsquos maximum principle described in[16 17] is applied Corollary 41 in [17] shows the existenceof an optimal control due to the convexity of the integrand Jwith respect to c1 c2 c3 and Lipschitz property of the statesystem with respect to the state variables By using theoptimal conditions

zH

zc1 0

zH

zc2 0

zH

zc3 0

(48)

we obtain

zH

zc1 k1c1 + λsh

minus λeh1113872 1113873aphivsh + λsv

minus λev1113872 1113873apvihsv 0

zH

zc2 k2c2 + λrh

minus λih1113872 1113873τih 0

zH

zc3 k3c3 minus λsv

μvsv + λevμvev + λiv

μviv1113872 1113873 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)

Table 2 Parameters values used for simulations

Parameter Value Referenceπh 0000215day [14]πv 007day [14]ph 062 [6]pv 0065 [6 15]a Varying Assumedσ 0001 Assumedε 0083 [18]ω 0004 [3]ρ 002 [6]τ 0125 [3]μh 000044 Assumedμv 0034 [15]

Computational and Mathematical Methods in Medicine 9

Solving (49) we have

c1 λeh minus λsh( )aphivsh + λev minus λsv( )apvihsv

k1

c2 λih minus λrh( )τih

k2

c3 λsvμvsv + λevμvev + λivμviv

k3

(50)

As stated earlier the lower and upper boundaries for thecontrol parameters are 0 and 1 respectively If c1 c2 c3 lt 1then c1 c2 c3 0 and if c1 c2 c3 gt 1 then c1 c2 c3 1 otherwise c1 c1 c2 c2 c3 c3 erefore for thecontrol parameters clowast1 c

lowast2 clowast3 we obtain the optimum value of

the function J(c1 c2 c3)

41 Optimal Control Simulations e Octave programminglanguage is used to simulate the optimal control model usingthe set of parameters obtained from previously reportedstudies and datasets which have been cited Some of theseparameters are assumed for the sake of illustrations Table 2represents the values of the model parameters used forsimulationse following initial conditions were considered

sh(0) 30

eh(0) 7

ih(0) 2

rh(0) 0

sv(0) 40

ev(0) 10

iv(0) 3

(51)

and the weight constants were assumed to be

M1 1

M2 2

M3 2

M4 2

k1 2

k2 10

k3 5

(52)

Figures 2 and 3 represent the control proles at dierentvalues of c1 c2 and c3 while the rest of the plots are thegraphs of infectious human and vector population plottedagainst time in days and they represent the eect of optimalcontrols c1 c2 and c3 in reducing the number of individualsinfected From Figure 4 we observe that the use of treatmentand insecticides only has a signicant impact in reducing thenumber of infectious individuals and they show that thisstrategy is eective to control tsetse ies and infected humanpopulations In Figure 5 we observe that the use of edu-cation and insecticides reduce the number of infectiousindividuals but the results depicted in Figure 5(a) shows thatthis strategy is not eective and ecient to control theinfectious human population From Figure 6 we observethat the use of education and treatment reduces the numberof infectious individuals but the results from Figure 6(b)shows that this strategy is not eective and ecient tocontrol the population of infectious tsetse ies Lastly theresults depicted from Figure 7 shows that the strategy ofusing education treatment and insecticides is very ecientand eective to reduce the number of infected individualserefore the use of education treatment and insecticidessimultaneously is very ecient and eective to eliminateHAT in Africa

In epidemiology a reproduction number less than unityimplies that the disease can be eradicated in the long run

06

04

02

0

Con

trol p

rofil

e

c1 = 0c2 ne 0c3 ne 0

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 = 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 2 (a) Control prole when c1 0 c2 ne 0 and c3 ne 0 at Rc 00075 (b) Control prole when c1 ne 0 c2 0 and c3 ne 0 at Rc 000013

10 Computational and Mathematical Methods in Medicine

Hence choosing suitable parameters for the controls c1 c2and c3 it was observed that the eective reproductionnumbers obtained for Figures 2ndash7were 00075 000013 00440000075 00075 000013 0044 and 0000075 respectivelyis shows that incorporating all the control measures that iseducating individuals giving treatment and applying in-secticides is an eective method to help reduce the numbersecondary infections in the population which correspondswith eradicating the disease in the long run

5 Conclusion

In this paper we studied and analyzed the model fortransmission of HAT and determined the basic reproduction

number e local and global stabilities of disease-freeequilibrium and endemic equilibrium were also provedFor the optimal control model education treatment andinsecticides as control measures were used to optimize theobjective function dened by Equation (40) e numericalsimulations of the optimal control model show that the beststrategy to reduce the number of infected individuals isthrough the use of education treatment and insecticidesis is the eective and ecient method to eliminate thedisease Furthermore the national authorities non-governmental organizations (NGOs) and stakeholdersmust not lose their interest in controlling the disease be-cause neglecting this disease may cause rapid reoccurrenceand much eect to the people who are at risk

06

04

02

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(a)

4

35

3

25

15

1

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(b)

Figure 4 Simulations of the model showing the eorts of treatment and insecticides only on infectious individuals at Rc 00075

c1 ne 0c2 ne 0c3 = 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 ne 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 3 (a) Control prole when c1 ne 0 c2 ne 0 and c3 0 at Rc 0044 (b) Control prole when c1 ne 0 c2 ne 0 and c3 ne 0 at Rc 0000075

Computational and Mathematical Methods in Medicine 11

20

15

10

5Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(a)

12

10

8

6

4

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(b)

Figure 7 Simulations of the model showing the eorts of education treatment and insecticides on infectious individuals at Rc 0000075

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(a)

44

42

38

4

34

36

32

3

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(b)

Figure 6 Simulations of the model showing the eorts of education and treatment only on infectious individuals at Rc 0044

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

20

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

(a)

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

Infe

ctio

us v

ecto

r pop

ulat

ion

12

8

10

6

4

20 20 40 60 80 100 120

Time (days)

(b)

Figure 5 Simulations of the model showing the eorts of education and insecticides only on infectious individuals at Rc 000013

12 Computational and Mathematical Methods in Medicine

Data Availability

(e secondary data supporting this research are from pre-viously reported studies and datasets which have been cited(e processed data are available in cited references

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(e authors thank the support from Next Einstein Initiativeof the African Institute for Mathematical Sciences Ghana(Annual Grant for Graduate Studies) and MasterCardFoundation

References

[1] K S Rock S J Torr C Lumbala and M J KeelingldquoQuantitative evaluation of the strategy to eliminate humanafrican trypanosomiasis in the democratic republic of CongordquoParasites and Vectors vol 8 no 1 p 532 2015

[2] J R Franco P P Simarro A Diarra and J G JanninldquoEpidemiology of human african trypanosomiasisrdquo ClinicalEpidemiology vol 2014 no 6 pp 257ndash275 2014

[3] R Brun J Blum F Chappuis and C Burri ldquoHuman africantrypanosomiasisrdquo Lancet vol 375 no 9709 pp 148ndash1592010

[4] World Health Organization et al Control and Surveillance ofHuman African Trypanosomiasis Report of a WHO ExpertCommittee World Health Organization Geneva Switzerland2013

[5] M Martcheva An Introduction to Mathematical Epidemiol-ogy Springer Vol 61 Springer Berlin Germany 2015

[6] D J Rogers ldquoA general model for the african trypanoso-miasesrdquo Parasitology vol 97 no 1 pp 193ndash212 1988

[7] J W Hargrove R Ouifki D Kajunguri G A Vale andS J Torr ldquoModeling the control of trypanosomiasis usingtrypanocides or insecticide-treated livestockrdquo PLoS NeglectedTropical Diseases vol 6 no 5 p 1615 2012

[8] D Kajunguri ldquoModelling the control of tsetse and africantrypanosomiasis through application of insecticides on cattlein Southeastern Ugandardquo PhD thesis Stellenbosch Uni-versity Stellenbosch South Africa 2013

[9] S Olaniyi and O S Obabiyi ldquoMathematical model for malariatransmission dynamics in human and mosquito populationswith nonlinear forces of infectionrdquo International Journal ofPure and Applied Mathematics vol 88 no 1 pp 125ndash1562013

[10] R H Chisholm P T Campbell Y Wu S Y C TongJ McVernon and N Geard ldquoImplications of asymptomaticcarriers for infectious disease transmission and controlrdquoRoyal Society Open Science vol 5 no 2 p 172341 2018

[11] P Yan and S Liu ldquoSeir epidemic model with delayrdquo ANZIAMJournal vol 48 no 1 pp 119ndash134 2006

[12] G T Azu-Tungmah ldquoA mathematical model to control thespread of malaria in Ghanardquo PhD thesis Kwame NkrumahUniversity of Science and Technology Kumasi Ghana 2012

[13] S A Pedro S Abelman F T Ndjomatchoua R Sang andH E Z Tonnang ldquoStability bifurcation and chaos analysis ofvector-borne disease model with application to rift valleyfeverrdquo PloS One vol 9 no 10 Article ID 108172 2014

[14] S Olaniyi and O S Obabiyi ldquoQualitative analysis of malariadynamics with nonlinear incidence functionrdquo AppliedMathematical Sciences vol 8 no 78 pp 3889ndash3904 2014

[15] L S Pontryagin V G Boltyanskii and R V Gamkrelidze EfMishchenko the Mathematical 2eory of Optimal ProcessesInterscience New York NY USA 1962

[16] B Heimann ldquoFleming WHrishel RW deterministic andstochastic optimal control New York-heidelbergberlinSpringer-verlag 1975 xiii 222 s dm 60 60rdquo ZAMM-Journal of Applied Mathematics and Mechanics Zeitschriftfur Angewandte Mathematik und Mechanik vol 59 no 9p 494 1979

[17] S Davis S Aksoy and A Galvani ldquoA global sensitivityanalysis for african sleeping sicknessrdquo Parasitology vol 138no 4 pp 516ndash526 2011

[18] M Artzrouni and J P Gouteux ldquoA compartmental model ofsleeping sickness in central africardquo Journal of BiologicalSystems vol 4 no 4 pp 459ndash477 1996

Computational and Mathematical Methods in Medicine 13

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of

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Disease Markers

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OncologyJournal of

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Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom

We suppose k1 k2 and k4 k5 and multiply and dividethe same equilibrium value to some of the fractions to obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth minus k1aph

slowast2h ilowastvsh

+ k1aphivslowasth

minus k2 ε + μh( 1113857eh minus k2aphshivelowasthslowasth ilowastvslowasth ilowastv eh

+ k2 ε + μh( 1113857elowasth

+ k3εeh minus k3 ω + τ + μh( 1113857ih minus k3εehilowasthelowasthihelowasth

+ k3 ω + τ + μh( 1113857ilowasth minus k4

sv minus slowastv( 11138572μv

sv+ k4apvi

lowasthslowastv

minus k4apvslowast2v ilowasth

sv+ k4apvihs

lowastv minus k5 σ + μv( 1113857ev

minus k5apvsvihelowastv slowastv ilowasthslowastv ilowasthev

+ k5 σ + μv( 1113857elowastv + k6σev minus k6μvi

lowastv

minus k6σevelowastv ilowastvivelowastv

+ k6μvilowastv

(36)

We choose k3 k2(μh + εε) such that k3(ω + τ+ μh)ilowasth

k2(ε + μh)elowasth and choose k6 k5(μh + σσ) such thatk6μvilowastv k5(σ + μv)elowastv Now aphslowasth ilowastv (μh + ε)elowasth andapvslowastv ilowastv (μv + σ)elowastv because k1 k2 and k4 k5 re-

spectively We now obtain

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

+ k1aphivslowasth minus k3 ω + τ + μh( 1113857ih + k3εminus k2 ε + μh( 11138571113858 1113859eh

minus k4sv minus slowastv( 1113857

2μvsv

+ k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsv

slowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891

+ k4apvihslowastv minus k6μviv + k6σ minus k5 σ + μv( 11138571113858 1113859ev

(37)

Suppose k6 (k1aphslowasthμv) and k3 (k4apvslowastv ω + τ+

μh) We then substitute and simplify to get

Vprime minusk1sh minus slowasth( 1113857

2μhsh

+ k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

minus k4sv minus slowastv( 1113857

2μvsv

+ k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsvslowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891

(38)

From Lemma 1 the terms

k1aphilowastv slowasth 3minus

slowasthshminus

elowasth ivsh

slowasthehilowastvminus

ehilowasthihelowasth

1113890 1113891

k4apvilowasthslowastv 3minus

slowastvsvminus

elowastv ihsv

slowastv evilowasthminus

evilowastv

ivelowastv

1113890 1113891le 0

(39)

(erefore Vprime lt 0 for all (sh eh ih rh sv ev iv)ne (slowasth

elowasth ilowasth rlowasth slowastv elowastv ilowastv ) implying that the endemic equilibrium isglobally asymptotically stable if R0 gt 1

4 Analysis of Optimal Control Model

In this section we formulate the optimal control model bymodifying system (6) to an optimal control problem Wethus define some linear functions ci(t) 1 for i 1 2 3 It isimportant to note that controls are fully effective whenci(t) 1 and not effective when ci(t) 0 (e forces ofinfection ξh and xiv which correspond to the human andvector population respectively are reduced by the factor(1minus c1) where c1 measures the level of success obtained dueto the effort of educating people on the dangers of exposingtheir skin and encouraging them to wear long sleeves andlong pants during the day to minimize tsetse fly-humancontacts (e factor c2 represents the effort of treatment tocontrol the disease and the factor c3 also represents theeffort of using insecticides to ensure that the breeding sites ofthe tsetse fly are minimized Hence taking into account theassumptions and extensions made we try to find the mosteffective strategy that reduces the HAT infection in thepopulation at a veryminimum costWith the use of boundedLebesgue measurable control we define the objectivefunction to be minimized as

J c1 c2 c3( 1113857 1113946tF

0( M1eh + M2ih + M3ev + M4iv

+12k1c

21 +

12k2c

22 +

12k3c

231113857 dt

(40)

(us the dynamics of the controls that minimizes theobjective function is given by

dshdt

πh + ρrh minus 1minus c1( 1113857aphivsh minus μhsh

deh

dt 1minus c1( 1113857aphivsh minus εeh minus μheh

dih

dt εeh minus μhih minusωih minus c2τih

drh

dt c2τih minus ρrh minus μhrh

dsv

dt πv minus c3μvsv minus 1minus c1( 1113857apvihsv

dev

dt 1minus c1( 1113857apvihsv minus c3μvev minus σev

div

dt σev minus c3μviv

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

subject to the initial conditions sh ge 0 eh ge 0 ih ge 0 rh ge 0

sv ge 0 ev ge 0 and iv ge 0 (e associated effective reproduc-tion number for (41) denoted by RE is obtained as

RE

a2ε 1minus c1( 11138572phπhπvσ

c3μh ε + μh( 1113857μ2v σ + c3μv( 1113857 μh + c2τ + ω( 1113857

1113971

Rc

1113968

(42)

8 Computational and Mathematical Methods in Medicine

(e goal is to minimize the exposed and infectious humanpopulations (eh ih) the exposed and infectious vector pop-ulations (ev iv) and the cost of implementing the control bythe use of possible ci i 1 2 3 (e functional objectiveincludes the social cost which relates to the resources that areneeded for educating people on personal protection (12)k1c

21

the application of treatment (12)k2c22 and spraying of tsetse

fly operations (12)k3c23 (e quantities M1 and M2 re-

spectively represent the associated cost with minimizing theexposed and infected human population while M3 and M4also represent the cost associated withminimizing the exposedand infected vectors respectively (e quantity tF is the timeperiod of intervention As explained in [15] the costs cor-responding to M1eh M2ih M3ev and M4iv are linear whilethe cost control functions (12)k1c

21 (12)k2c

22 and (12)k3c

23

should be nonlinear and take a quadratic form (erefore weseek to minimize the objective function over the given timeinterval [0 tF] Pontryaginrsquos maximum principle is used tosolve this optimal control problem and the derivation of thenecessary conditions (e Lagrangian of the optimal controlproblem is given by

L M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22 +

12k3c

231113874 1113875

(43)

To determine the Lagrangian minimum value we definethe Hamiltonian H for the control problem as

H M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22

+12k3c

23 + λsh

dsh

dt+ λeh

deh

dt+ λih

dih

dt+ λrh

drh

dt

+ λsv

dsv

dt+ λev

dev

dt+ λiv

div

dt

(44)

where λsh λeh

λih λrh

λsv λev

and λivare adjoint variables or

costate variables (e differential equations of adjoint var-iables are obtained by taking the partial derivatives of theHamiltonian equation with respect to the state variableswhich gives

dλsh

dt λsh

μh + λshminus λeh

1113872 1113873 1minus c1( 1113857aphiv

dλeh

dt minusM1 + λeh

minus λih1113872 1113873ε + λeh

μh

dλih

dt minusM2 + λih

minus λrh1113872 1113873c2τ + λih

ω + μh( 1113857 + λsvminus λev

1113872 1113873 1minus c1( 1113857apvsv

dλrh

dt λrhminus λsh

1113872 1113873ρ + λrhμh

dλsv

dt λsvminus λev

1113872 1113873 1minus c1( 1113857apvih + λsvμvc3

dλev

dt minusM3 + λev

minus λiv1113872 1113873σ + λev

μvc3

dλiv

dt minusM4 + λsh

minus λeh1113872 1113873 1minus c1( 1113857aphsh + λiv

μvc3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Theorem 6 Given the optimal controls clowast1 clowast2 clowast3 and thesolutions sh eh ih rh sv ev iv of the corresponding stateequations (41) and (40) which minimize J(c1 c2 c3) over theregion Ω then there exist adjoint variablesλsh

λeh λih

λrh λsv

λev λiv

satisfying

minusdλi

dt

zH

zi i isin sh eh ih rh sv ev iv1113864 1113865 (46)

and the optimal solution clowast1 clowast2 clowast3 is given by

clowast1 min 1 max 0 1113954c1( 11138571113864 1113865

clowast2 min 1 max 0 1113954c2( 11138571113864 1113865

clowast3 min 1 max 0 1113954c3( 11138571113864 1113865

(47)

Proof (e Pontryaginrsquos maximum principle described in[16 17] is applied Corollary 41 in [17] shows the existenceof an optimal control due to the convexity of the integrand Jwith respect to c1 c2 c3 and Lipschitz property of the statesystem with respect to the state variables By using theoptimal conditions

zH

zc1 0

zH

zc2 0

zH

zc3 0

(48)

we obtain

zH

zc1 k1c1 + λsh

minus λeh1113872 1113873aphivsh + λsv

minus λev1113872 1113873apvihsv 0

zH

zc2 k2c2 + λrh

minus λih1113872 1113873τih 0

zH

zc3 k3c3 minus λsv

μvsv + λevμvev + λiv

μviv1113872 1113873 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)

Table 2 Parameters values used for simulations

Parameter Value Referenceπh 0000215day [14]πv 007day [14]ph 062 [6]pv 0065 [6 15]a Varying Assumedσ 0001 Assumedε 0083 [18]ω 0004 [3]ρ 002 [6]τ 0125 [3]μh 000044 Assumedμv 0034 [15]

Computational and Mathematical Methods in Medicine 9

Solving (49) we have

c1 λeh minus λsh( )aphivsh + λev minus λsv( )apvihsv

k1

c2 λih minus λrh( )τih

k2

c3 λsvμvsv + λevμvev + λivμviv

k3

(50)

As stated earlier the lower and upper boundaries for thecontrol parameters are 0 and 1 respectively If c1 c2 c3 lt 1then c1 c2 c3 0 and if c1 c2 c3 gt 1 then c1 c2 c3 1 otherwise c1 c1 c2 c2 c3 c3 erefore for thecontrol parameters clowast1 c

lowast2 clowast3 we obtain the optimum value of

the function J(c1 c2 c3)

41 Optimal Control Simulations e Octave programminglanguage is used to simulate the optimal control model usingthe set of parameters obtained from previously reportedstudies and datasets which have been cited Some of theseparameters are assumed for the sake of illustrations Table 2represents the values of the model parameters used forsimulationse following initial conditions were considered

sh(0) 30

eh(0) 7

ih(0) 2

rh(0) 0

sv(0) 40

ev(0) 10

iv(0) 3

(51)

and the weight constants were assumed to be

M1 1

M2 2

M3 2

M4 2

k1 2

k2 10

k3 5

(52)

Figures 2 and 3 represent the control proles at dierentvalues of c1 c2 and c3 while the rest of the plots are thegraphs of infectious human and vector population plottedagainst time in days and they represent the eect of optimalcontrols c1 c2 and c3 in reducing the number of individualsinfected From Figure 4 we observe that the use of treatmentand insecticides only has a signicant impact in reducing thenumber of infectious individuals and they show that thisstrategy is eective to control tsetse ies and infected humanpopulations In Figure 5 we observe that the use of edu-cation and insecticides reduce the number of infectiousindividuals but the results depicted in Figure 5(a) shows thatthis strategy is not eective and ecient to control theinfectious human population From Figure 6 we observethat the use of education and treatment reduces the numberof infectious individuals but the results from Figure 6(b)shows that this strategy is not eective and ecient tocontrol the population of infectious tsetse ies Lastly theresults depicted from Figure 7 shows that the strategy ofusing education treatment and insecticides is very ecientand eective to reduce the number of infected individualserefore the use of education treatment and insecticidessimultaneously is very ecient and eective to eliminateHAT in Africa

In epidemiology a reproduction number less than unityimplies that the disease can be eradicated in the long run

06

04

02

0

Con

trol p

rofil

e

c1 = 0c2 ne 0c3 ne 0

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 = 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 2 (a) Control prole when c1 0 c2 ne 0 and c3 ne 0 at Rc 00075 (b) Control prole when c1 ne 0 c2 0 and c3 ne 0 at Rc 000013

10 Computational and Mathematical Methods in Medicine

Hence choosing suitable parameters for the controls c1 c2and c3 it was observed that the eective reproductionnumbers obtained for Figures 2ndash7were 00075 000013 00440000075 00075 000013 0044 and 0000075 respectivelyis shows that incorporating all the control measures that iseducating individuals giving treatment and applying in-secticides is an eective method to help reduce the numbersecondary infections in the population which correspondswith eradicating the disease in the long run

5 Conclusion

In this paper we studied and analyzed the model fortransmission of HAT and determined the basic reproduction

number e local and global stabilities of disease-freeequilibrium and endemic equilibrium were also provedFor the optimal control model education treatment andinsecticides as control measures were used to optimize theobjective function dened by Equation (40) e numericalsimulations of the optimal control model show that the beststrategy to reduce the number of infected individuals isthrough the use of education treatment and insecticidesis is the eective and ecient method to eliminate thedisease Furthermore the national authorities non-governmental organizations (NGOs) and stakeholdersmust not lose their interest in controlling the disease be-cause neglecting this disease may cause rapid reoccurrenceand much eect to the people who are at risk

06

04

02

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(a)

4

35

3

25

15

1

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(b)

Figure 4 Simulations of the model showing the eorts of treatment and insecticides only on infectious individuals at Rc 00075

c1 ne 0c2 ne 0c3 = 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 ne 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 3 (a) Control prole when c1 ne 0 c2 ne 0 and c3 0 at Rc 0044 (b) Control prole when c1 ne 0 c2 ne 0 and c3 ne 0 at Rc 0000075

Computational and Mathematical Methods in Medicine 11

20

15

10

5Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(a)

12

10

8

6

4

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(b)

Figure 7 Simulations of the model showing the eorts of education treatment and insecticides on infectious individuals at Rc 0000075

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(a)

44

42

38

4

34

36

32

3

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(b)

Figure 6 Simulations of the model showing the eorts of education and treatment only on infectious individuals at Rc 0044

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

20

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

(a)

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

Infe

ctio

us v

ecto

r pop

ulat

ion

12

8

10

6

4

20 20 40 60 80 100 120

Time (days)

(b)

Figure 5 Simulations of the model showing the eorts of education and insecticides only on infectious individuals at Rc 000013

12 Computational and Mathematical Methods in Medicine

Data Availability

(e secondary data supporting this research are from pre-viously reported studies and datasets which have been cited(e processed data are available in cited references

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(e authors thank the support from Next Einstein Initiativeof the African Institute for Mathematical Sciences Ghana(Annual Grant for Graduate Studies) and MasterCardFoundation

References

[1] K S Rock S J Torr C Lumbala and M J KeelingldquoQuantitative evaluation of the strategy to eliminate humanafrican trypanosomiasis in the democratic republic of CongordquoParasites and Vectors vol 8 no 1 p 532 2015

[2] J R Franco P P Simarro A Diarra and J G JanninldquoEpidemiology of human african trypanosomiasisrdquo ClinicalEpidemiology vol 2014 no 6 pp 257ndash275 2014

[3] R Brun J Blum F Chappuis and C Burri ldquoHuman africantrypanosomiasisrdquo Lancet vol 375 no 9709 pp 148ndash1592010

[4] World Health Organization et al Control and Surveillance ofHuman African Trypanosomiasis Report of a WHO ExpertCommittee World Health Organization Geneva Switzerland2013

[5] M Martcheva An Introduction to Mathematical Epidemiol-ogy Springer Vol 61 Springer Berlin Germany 2015

[6] D J Rogers ldquoA general model for the african trypanoso-miasesrdquo Parasitology vol 97 no 1 pp 193ndash212 1988

[7] J W Hargrove R Ouifki D Kajunguri G A Vale andS J Torr ldquoModeling the control of trypanosomiasis usingtrypanocides or insecticide-treated livestockrdquo PLoS NeglectedTropical Diseases vol 6 no 5 p 1615 2012

[8] D Kajunguri ldquoModelling the control of tsetse and africantrypanosomiasis through application of insecticides on cattlein Southeastern Ugandardquo PhD thesis Stellenbosch Uni-versity Stellenbosch South Africa 2013

[9] S Olaniyi and O S Obabiyi ldquoMathematical model for malariatransmission dynamics in human and mosquito populationswith nonlinear forces of infectionrdquo International Journal ofPure and Applied Mathematics vol 88 no 1 pp 125ndash1562013

[10] R H Chisholm P T Campbell Y Wu S Y C TongJ McVernon and N Geard ldquoImplications of asymptomaticcarriers for infectious disease transmission and controlrdquoRoyal Society Open Science vol 5 no 2 p 172341 2018

[11] P Yan and S Liu ldquoSeir epidemic model with delayrdquo ANZIAMJournal vol 48 no 1 pp 119ndash134 2006

[12] G T Azu-Tungmah ldquoA mathematical model to control thespread of malaria in Ghanardquo PhD thesis Kwame NkrumahUniversity of Science and Technology Kumasi Ghana 2012

[13] S A Pedro S Abelman F T Ndjomatchoua R Sang andH E Z Tonnang ldquoStability bifurcation and chaos analysis ofvector-borne disease model with application to rift valleyfeverrdquo PloS One vol 9 no 10 Article ID 108172 2014

[14] S Olaniyi and O S Obabiyi ldquoQualitative analysis of malariadynamics with nonlinear incidence functionrdquo AppliedMathematical Sciences vol 8 no 78 pp 3889ndash3904 2014

[15] L S Pontryagin V G Boltyanskii and R V Gamkrelidze EfMishchenko the Mathematical 2eory of Optimal ProcessesInterscience New York NY USA 1962

[16] B Heimann ldquoFleming WHrishel RW deterministic andstochastic optimal control New York-heidelbergberlinSpringer-verlag 1975 xiii 222 s dm 60 60rdquo ZAMM-Journal of Applied Mathematics and Mechanics Zeitschriftfur Angewandte Mathematik und Mechanik vol 59 no 9p 494 1979

[17] S Davis S Aksoy and A Galvani ldquoA global sensitivityanalysis for african sleeping sicknessrdquo Parasitology vol 138no 4 pp 516ndash526 2011

[18] M Artzrouni and J P Gouteux ldquoA compartmental model ofsleeping sickness in central africardquo Journal of BiologicalSystems vol 4 no 4 pp 459ndash477 1996

Computational and Mathematical Methods in Medicine 13

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Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

MEDIATORSINFLAMMATION

of

EndocrinologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

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Disease Markers

Hindawiwwwhindawicom Volume 2018

BioMed Research International

OncologyJournal of

Hindawiwwwhindawicom Volume 2013

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Oxidative Medicine and Cellular Longevity

Hindawiwwwhindawicom Volume 2018

PPAR Research

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Volume 2018

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ObesityJournal of

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Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom

(e goal is to minimize the exposed and infectious humanpopulations (eh ih) the exposed and infectious vector pop-ulations (ev iv) and the cost of implementing the control bythe use of possible ci i 1 2 3 (e functional objectiveincludes the social cost which relates to the resources that areneeded for educating people on personal protection (12)k1c

21

the application of treatment (12)k2c22 and spraying of tsetse

fly operations (12)k3c23 (e quantities M1 and M2 re-

spectively represent the associated cost with minimizing theexposed and infected human population while M3 and M4also represent the cost associated withminimizing the exposedand infected vectors respectively (e quantity tF is the timeperiod of intervention As explained in [15] the costs cor-responding to M1eh M2ih M3ev and M4iv are linear whilethe cost control functions (12)k1c

21 (12)k2c

22 and (12)k3c

23

should be nonlinear and take a quadratic form (erefore weseek to minimize the objective function over the given timeinterval [0 tF] Pontryaginrsquos maximum principle is used tosolve this optimal control problem and the derivation of thenecessary conditions (e Lagrangian of the optimal controlproblem is given by

L M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22 +

12k3c

231113874 1113875

(43)

To determine the Lagrangian minimum value we definethe Hamiltonian H for the control problem as

H M1eh + M2ih + M3ev + M4iv +12k1c

21 +

12k2c

22

+12k3c

23 + λsh

dsh

dt+ λeh

deh

dt+ λih

dih

dt+ λrh

drh

dt

+ λsv

dsv

dt+ λev

dev

dt+ λiv

div

dt

(44)

where λsh λeh

λih λrh

λsv λev

and λivare adjoint variables or

costate variables (e differential equations of adjoint var-iables are obtained by taking the partial derivatives of theHamiltonian equation with respect to the state variableswhich gives

dλsh

dt λsh

μh + λshminus λeh

1113872 1113873 1minus c1( 1113857aphiv

dλeh

dt minusM1 + λeh

minus λih1113872 1113873ε + λeh

μh

dλih

dt minusM2 + λih

minus λrh1113872 1113873c2τ + λih

ω + μh( 1113857 + λsvminus λev

1113872 1113873 1minus c1( 1113857apvsv

dλrh

dt λrhminus λsh

1113872 1113873ρ + λrhμh

dλsv

dt λsvminus λev

1113872 1113873 1minus c1( 1113857apvih + λsvμvc3

dλev

dt minusM3 + λev

minus λiv1113872 1113873σ + λev

μvc3

dλiv

dt minusM4 + λsh

minus λeh1113872 1113873 1minus c1( 1113857aphsh + λiv

μvc3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

Theorem 6 Given the optimal controls clowast1 clowast2 clowast3 and thesolutions sh eh ih rh sv ev iv of the corresponding stateequations (41) and (40) which minimize J(c1 c2 c3) over theregion Ω then there exist adjoint variablesλsh

λeh λih

λrh λsv

λev λiv

satisfying

minusdλi

dt

zH

zi i isin sh eh ih rh sv ev iv1113864 1113865 (46)

and the optimal solution clowast1 clowast2 clowast3 is given by

clowast1 min 1 max 0 1113954c1( 11138571113864 1113865

clowast2 min 1 max 0 1113954c2( 11138571113864 1113865

clowast3 min 1 max 0 1113954c3( 11138571113864 1113865

(47)

Proof (e Pontryaginrsquos maximum principle described in[16 17] is applied Corollary 41 in [17] shows the existenceof an optimal control due to the convexity of the integrand Jwith respect to c1 c2 c3 and Lipschitz property of the statesystem with respect to the state variables By using theoptimal conditions

zH

zc1 0

zH

zc2 0

zH

zc3 0

(48)

we obtain

zH

zc1 k1c1 + λsh

minus λeh1113872 1113873aphivsh + λsv

minus λev1113872 1113873apvihsv 0

zH

zc2 k2c2 + λrh

minus λih1113872 1113873τih 0

zH

zc3 k3c3 minus λsv

μvsv + λevμvev + λiv

μviv1113872 1113873 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(49)

Table 2 Parameters values used for simulations

Parameter Value Referenceπh 0000215day [14]πv 007day [14]ph 062 [6]pv 0065 [6 15]a Varying Assumedσ 0001 Assumedε 0083 [18]ω 0004 [3]ρ 002 [6]τ 0125 [3]μh 000044 Assumedμv 0034 [15]

Computational and Mathematical Methods in Medicine 9

Solving (49) we have

c1 λeh minus λsh( )aphivsh + λev minus λsv( )apvihsv

k1

c2 λih minus λrh( )τih

k2

c3 λsvμvsv + λevμvev + λivμviv

k3

(50)

As stated earlier the lower and upper boundaries for thecontrol parameters are 0 and 1 respectively If c1 c2 c3 lt 1then c1 c2 c3 0 and if c1 c2 c3 gt 1 then c1 c2 c3 1 otherwise c1 c1 c2 c2 c3 c3 erefore for thecontrol parameters clowast1 c

lowast2 clowast3 we obtain the optimum value of

the function J(c1 c2 c3)

41 Optimal Control Simulations e Octave programminglanguage is used to simulate the optimal control model usingthe set of parameters obtained from previously reportedstudies and datasets which have been cited Some of theseparameters are assumed for the sake of illustrations Table 2represents the values of the model parameters used forsimulationse following initial conditions were considered

sh(0) 30

eh(0) 7

ih(0) 2

rh(0) 0

sv(0) 40

ev(0) 10

iv(0) 3

(51)

and the weight constants were assumed to be

M1 1

M2 2

M3 2

M4 2

k1 2

k2 10

k3 5

(52)

Figures 2 and 3 represent the control proles at dierentvalues of c1 c2 and c3 while the rest of the plots are thegraphs of infectious human and vector population plottedagainst time in days and they represent the eect of optimalcontrols c1 c2 and c3 in reducing the number of individualsinfected From Figure 4 we observe that the use of treatmentand insecticides only has a signicant impact in reducing thenumber of infectious individuals and they show that thisstrategy is eective to control tsetse ies and infected humanpopulations In Figure 5 we observe that the use of edu-cation and insecticides reduce the number of infectiousindividuals but the results depicted in Figure 5(a) shows thatthis strategy is not eective and ecient to control theinfectious human population From Figure 6 we observethat the use of education and treatment reduces the numberof infectious individuals but the results from Figure 6(b)shows that this strategy is not eective and ecient tocontrol the population of infectious tsetse ies Lastly theresults depicted from Figure 7 shows that the strategy ofusing education treatment and insecticides is very ecientand eective to reduce the number of infected individualserefore the use of education treatment and insecticidessimultaneously is very ecient and eective to eliminateHAT in Africa

In epidemiology a reproduction number less than unityimplies that the disease can be eradicated in the long run

06

04

02

0

Con

trol p

rofil

e

c1 = 0c2 ne 0c3 ne 0

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 = 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 2 (a) Control prole when c1 0 c2 ne 0 and c3 ne 0 at Rc 00075 (b) Control prole when c1 ne 0 c2 0 and c3 ne 0 at Rc 000013

10 Computational and Mathematical Methods in Medicine

Hence choosing suitable parameters for the controls c1 c2and c3 it was observed that the eective reproductionnumbers obtained for Figures 2ndash7were 00075 000013 00440000075 00075 000013 0044 and 0000075 respectivelyis shows that incorporating all the control measures that iseducating individuals giving treatment and applying in-secticides is an eective method to help reduce the numbersecondary infections in the population which correspondswith eradicating the disease in the long run

5 Conclusion

In this paper we studied and analyzed the model fortransmission of HAT and determined the basic reproduction

number e local and global stabilities of disease-freeequilibrium and endemic equilibrium were also provedFor the optimal control model education treatment andinsecticides as control measures were used to optimize theobjective function dened by Equation (40) e numericalsimulations of the optimal control model show that the beststrategy to reduce the number of infected individuals isthrough the use of education treatment and insecticidesis is the eective and ecient method to eliminate thedisease Furthermore the national authorities non-governmental organizations (NGOs) and stakeholdersmust not lose their interest in controlling the disease be-cause neglecting this disease may cause rapid reoccurrenceand much eect to the people who are at risk

06

04

02

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(a)

4

35

3

25

15

1

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(b)

Figure 4 Simulations of the model showing the eorts of treatment and insecticides only on infectious individuals at Rc 00075

c1 ne 0c2 ne 0c3 = 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 ne 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 3 (a) Control prole when c1 ne 0 c2 ne 0 and c3 0 at Rc 0044 (b) Control prole when c1 ne 0 c2 ne 0 and c3 ne 0 at Rc 0000075

Computational and Mathematical Methods in Medicine 11

20

15

10

5Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(a)

12

10

8

6

4

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(b)

Figure 7 Simulations of the model showing the eorts of education treatment and insecticides on infectious individuals at Rc 0000075

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(a)

44

42

38

4

34

36

32

3

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(b)

Figure 6 Simulations of the model showing the eorts of education and treatment only on infectious individuals at Rc 0044

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

20

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

(a)

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

Infe

ctio

us v

ecto

r pop

ulat

ion

12

8

10

6

4

20 20 40 60 80 100 120

Time (days)

(b)

Figure 5 Simulations of the model showing the eorts of education and insecticides only on infectious individuals at Rc 000013

12 Computational and Mathematical Methods in Medicine

Data Availability

(e secondary data supporting this research are from pre-viously reported studies and datasets which have been cited(e processed data are available in cited references

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(e authors thank the support from Next Einstein Initiativeof the African Institute for Mathematical Sciences Ghana(Annual Grant for Graduate Studies) and MasterCardFoundation

References

[1] K S Rock S J Torr C Lumbala and M J KeelingldquoQuantitative evaluation of the strategy to eliminate humanafrican trypanosomiasis in the democratic republic of CongordquoParasites and Vectors vol 8 no 1 p 532 2015

[2] J R Franco P P Simarro A Diarra and J G JanninldquoEpidemiology of human african trypanosomiasisrdquo ClinicalEpidemiology vol 2014 no 6 pp 257ndash275 2014

[3] R Brun J Blum F Chappuis and C Burri ldquoHuman africantrypanosomiasisrdquo Lancet vol 375 no 9709 pp 148ndash1592010

[4] World Health Organization et al Control and Surveillance ofHuman African Trypanosomiasis Report of a WHO ExpertCommittee World Health Organization Geneva Switzerland2013

[5] M Martcheva An Introduction to Mathematical Epidemiol-ogy Springer Vol 61 Springer Berlin Germany 2015

[6] D J Rogers ldquoA general model for the african trypanoso-miasesrdquo Parasitology vol 97 no 1 pp 193ndash212 1988

[7] J W Hargrove R Ouifki D Kajunguri G A Vale andS J Torr ldquoModeling the control of trypanosomiasis usingtrypanocides or insecticide-treated livestockrdquo PLoS NeglectedTropical Diseases vol 6 no 5 p 1615 2012

[8] D Kajunguri ldquoModelling the control of tsetse and africantrypanosomiasis through application of insecticides on cattlein Southeastern Ugandardquo PhD thesis Stellenbosch Uni-versity Stellenbosch South Africa 2013

[9] S Olaniyi and O S Obabiyi ldquoMathematical model for malariatransmission dynamics in human and mosquito populationswith nonlinear forces of infectionrdquo International Journal ofPure and Applied Mathematics vol 88 no 1 pp 125ndash1562013

[10] R H Chisholm P T Campbell Y Wu S Y C TongJ McVernon and N Geard ldquoImplications of asymptomaticcarriers for infectious disease transmission and controlrdquoRoyal Society Open Science vol 5 no 2 p 172341 2018

[11] P Yan and S Liu ldquoSeir epidemic model with delayrdquo ANZIAMJournal vol 48 no 1 pp 119ndash134 2006

[12] G T Azu-Tungmah ldquoA mathematical model to control thespread of malaria in Ghanardquo PhD thesis Kwame NkrumahUniversity of Science and Technology Kumasi Ghana 2012

[13] S A Pedro S Abelman F T Ndjomatchoua R Sang andH E Z Tonnang ldquoStability bifurcation and chaos analysis ofvector-borne disease model with application to rift valleyfeverrdquo PloS One vol 9 no 10 Article ID 108172 2014

[14] S Olaniyi and O S Obabiyi ldquoQualitative analysis of malariadynamics with nonlinear incidence functionrdquo AppliedMathematical Sciences vol 8 no 78 pp 3889ndash3904 2014

[15] L S Pontryagin V G Boltyanskii and R V Gamkrelidze EfMishchenko the Mathematical 2eory of Optimal ProcessesInterscience New York NY USA 1962

[16] B Heimann ldquoFleming WHrishel RW deterministic andstochastic optimal control New York-heidelbergberlinSpringer-verlag 1975 xiii 222 s dm 60 60rdquo ZAMM-Journal of Applied Mathematics and Mechanics Zeitschriftfur Angewandte Mathematik und Mechanik vol 59 no 9p 494 1979

[17] S Davis S Aksoy and A Galvani ldquoA global sensitivityanalysis for african sleeping sicknessrdquo Parasitology vol 138no 4 pp 516ndash526 2011

[18] M Artzrouni and J P Gouteux ldquoA compartmental model ofsleeping sickness in central africardquo Journal of BiologicalSystems vol 4 no 4 pp 459ndash477 1996

Computational and Mathematical Methods in Medicine 13

Stem Cells International

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

MEDIATORSINFLAMMATION

of

EndocrinologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Disease Markers

Hindawiwwwhindawicom Volume 2018

BioMed Research International

OncologyJournal of

Hindawiwwwhindawicom Volume 2013

Hindawiwwwhindawicom Volume 2018

Oxidative Medicine and Cellular Longevity

Hindawiwwwhindawicom Volume 2018

PPAR Research

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Immunology ResearchHindawiwwwhindawicom Volume 2018

Journal of

ObesityJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational and Mathematical Methods in Medicine

Hindawiwwwhindawicom Volume 2018

Behavioural Neurology

OphthalmologyJournal of

Hindawiwwwhindawicom Volume 2018

Diabetes ResearchJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Research and TreatmentAIDS

Hindawiwwwhindawicom Volume 2018

Gastroenterology Research and Practice

Hindawiwwwhindawicom Volume 2018

Parkinsonrsquos Disease

Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom

Solving (49) we have

c1 λeh minus λsh( )aphivsh + λev minus λsv( )apvihsv

k1

c2 λih minus λrh( )τih

k2

c3 λsvμvsv + λevμvev + λivμviv

k3

(50)

As stated earlier the lower and upper boundaries for thecontrol parameters are 0 and 1 respectively If c1 c2 c3 lt 1then c1 c2 c3 0 and if c1 c2 c3 gt 1 then c1 c2 c3 1 otherwise c1 c1 c2 c2 c3 c3 erefore for thecontrol parameters clowast1 c

lowast2 clowast3 we obtain the optimum value of

the function J(c1 c2 c3)

41 Optimal Control Simulations e Octave programminglanguage is used to simulate the optimal control model usingthe set of parameters obtained from previously reportedstudies and datasets which have been cited Some of theseparameters are assumed for the sake of illustrations Table 2represents the values of the model parameters used forsimulationse following initial conditions were considered

sh(0) 30

eh(0) 7

ih(0) 2

rh(0) 0

sv(0) 40

ev(0) 10

iv(0) 3

(51)

and the weight constants were assumed to be

M1 1

M2 2

M3 2

M4 2

k1 2

k2 10

k3 5

(52)

Figures 2 and 3 represent the control proles at dierentvalues of c1 c2 and c3 while the rest of the plots are thegraphs of infectious human and vector population plottedagainst time in days and they represent the eect of optimalcontrols c1 c2 and c3 in reducing the number of individualsinfected From Figure 4 we observe that the use of treatmentand insecticides only has a signicant impact in reducing thenumber of infectious individuals and they show that thisstrategy is eective to control tsetse ies and infected humanpopulations In Figure 5 we observe that the use of edu-cation and insecticides reduce the number of infectiousindividuals but the results depicted in Figure 5(a) shows thatthis strategy is not eective and ecient to control theinfectious human population From Figure 6 we observethat the use of education and treatment reduces the numberof infectious individuals but the results from Figure 6(b)shows that this strategy is not eective and ecient tocontrol the population of infectious tsetse ies Lastly theresults depicted from Figure 7 shows that the strategy ofusing education treatment and insecticides is very ecientand eective to reduce the number of infected individualserefore the use of education treatment and insecticidessimultaneously is very ecient and eective to eliminateHAT in Africa

In epidemiology a reproduction number less than unityimplies that the disease can be eradicated in the long run

06

04

02

0

Con

trol p

rofil

e

c1 = 0c2 ne 0c3 ne 0

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 = 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 2 (a) Control prole when c1 0 c2 ne 0 and c3 ne 0 at Rc 00075 (b) Control prole when c1 ne 0 c2 0 and c3 ne 0 at Rc 000013

10 Computational and Mathematical Methods in Medicine

Hence choosing suitable parameters for the controls c1 c2and c3 it was observed that the eective reproductionnumbers obtained for Figures 2ndash7were 00075 000013 00440000075 00075 000013 0044 and 0000075 respectivelyis shows that incorporating all the control measures that iseducating individuals giving treatment and applying in-secticides is an eective method to help reduce the numbersecondary infections in the population which correspondswith eradicating the disease in the long run

5 Conclusion

In this paper we studied and analyzed the model fortransmission of HAT and determined the basic reproduction

number e local and global stabilities of disease-freeequilibrium and endemic equilibrium were also provedFor the optimal control model education treatment andinsecticides as control measures were used to optimize theobjective function dened by Equation (40) e numericalsimulations of the optimal control model show that the beststrategy to reduce the number of infected individuals isthrough the use of education treatment and insecticidesis is the eective and ecient method to eliminate thedisease Furthermore the national authorities non-governmental organizations (NGOs) and stakeholdersmust not lose their interest in controlling the disease be-cause neglecting this disease may cause rapid reoccurrenceand much eect to the people who are at risk

06

04

02

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(a)

4

35

3

25

15

1

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(b)

Figure 4 Simulations of the model showing the eorts of treatment and insecticides only on infectious individuals at Rc 00075

c1 ne 0c2 ne 0c3 = 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 ne 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 3 (a) Control prole when c1 ne 0 c2 ne 0 and c3 0 at Rc 0044 (b) Control prole when c1 ne 0 c2 ne 0 and c3 ne 0 at Rc 0000075

Computational and Mathematical Methods in Medicine 11

20

15

10

5Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(a)

12

10

8

6

4

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(b)

Figure 7 Simulations of the model showing the eorts of education treatment and insecticides on infectious individuals at Rc 0000075

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(a)

44

42

38

4

34

36

32

3

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(b)

Figure 6 Simulations of the model showing the eorts of education and treatment only on infectious individuals at Rc 0044

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

20

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

(a)

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

Infe

ctio

us v

ecto

r pop

ulat

ion

12

8

10

6

4

20 20 40 60 80 100 120

Time (days)

(b)

Figure 5 Simulations of the model showing the eorts of education and insecticides only on infectious individuals at Rc 000013

12 Computational and Mathematical Methods in Medicine

Data Availability

(e secondary data supporting this research are from pre-viously reported studies and datasets which have been cited(e processed data are available in cited references

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(e authors thank the support from Next Einstein Initiativeof the African Institute for Mathematical Sciences Ghana(Annual Grant for Graduate Studies) and MasterCardFoundation

References

[1] K S Rock S J Torr C Lumbala and M J KeelingldquoQuantitative evaluation of the strategy to eliminate humanafrican trypanosomiasis in the democratic republic of CongordquoParasites and Vectors vol 8 no 1 p 532 2015

[2] J R Franco P P Simarro A Diarra and J G JanninldquoEpidemiology of human african trypanosomiasisrdquo ClinicalEpidemiology vol 2014 no 6 pp 257ndash275 2014

[3] R Brun J Blum F Chappuis and C Burri ldquoHuman africantrypanosomiasisrdquo Lancet vol 375 no 9709 pp 148ndash1592010

[4] World Health Organization et al Control and Surveillance ofHuman African Trypanosomiasis Report of a WHO ExpertCommittee World Health Organization Geneva Switzerland2013

[5] M Martcheva An Introduction to Mathematical Epidemiol-ogy Springer Vol 61 Springer Berlin Germany 2015

[6] D J Rogers ldquoA general model for the african trypanoso-miasesrdquo Parasitology vol 97 no 1 pp 193ndash212 1988

[7] J W Hargrove R Ouifki D Kajunguri G A Vale andS J Torr ldquoModeling the control of trypanosomiasis usingtrypanocides or insecticide-treated livestockrdquo PLoS NeglectedTropical Diseases vol 6 no 5 p 1615 2012

[8] D Kajunguri ldquoModelling the control of tsetse and africantrypanosomiasis through application of insecticides on cattlein Southeastern Ugandardquo PhD thesis Stellenbosch Uni-versity Stellenbosch South Africa 2013

[9] S Olaniyi and O S Obabiyi ldquoMathematical model for malariatransmission dynamics in human and mosquito populationswith nonlinear forces of infectionrdquo International Journal ofPure and Applied Mathematics vol 88 no 1 pp 125ndash1562013

[10] R H Chisholm P T Campbell Y Wu S Y C TongJ McVernon and N Geard ldquoImplications of asymptomaticcarriers for infectious disease transmission and controlrdquoRoyal Society Open Science vol 5 no 2 p 172341 2018

[11] P Yan and S Liu ldquoSeir epidemic model with delayrdquo ANZIAMJournal vol 48 no 1 pp 119ndash134 2006

[12] G T Azu-Tungmah ldquoA mathematical model to control thespread of malaria in Ghanardquo PhD thesis Kwame NkrumahUniversity of Science and Technology Kumasi Ghana 2012

[13] S A Pedro S Abelman F T Ndjomatchoua R Sang andH E Z Tonnang ldquoStability bifurcation and chaos analysis ofvector-borne disease model with application to rift valleyfeverrdquo PloS One vol 9 no 10 Article ID 108172 2014

[14] S Olaniyi and O S Obabiyi ldquoQualitative analysis of malariadynamics with nonlinear incidence functionrdquo AppliedMathematical Sciences vol 8 no 78 pp 3889ndash3904 2014

[15] L S Pontryagin V G Boltyanskii and R V Gamkrelidze EfMishchenko the Mathematical 2eory of Optimal ProcessesInterscience New York NY USA 1962

[16] B Heimann ldquoFleming WHrishel RW deterministic andstochastic optimal control New York-heidelbergberlinSpringer-verlag 1975 xiii 222 s dm 60 60rdquo ZAMM-Journal of Applied Mathematics and Mechanics Zeitschriftfur Angewandte Mathematik und Mechanik vol 59 no 9p 494 1979

[17] S Davis S Aksoy and A Galvani ldquoA global sensitivityanalysis for african sleeping sicknessrdquo Parasitology vol 138no 4 pp 516ndash526 2011

[18] M Artzrouni and J P Gouteux ldquoA compartmental model ofsleeping sickness in central africardquo Journal of BiologicalSystems vol 4 no 4 pp 459ndash477 1996

Computational and Mathematical Methods in Medicine 13

Stem Cells International

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

MEDIATORSINFLAMMATION

of

EndocrinologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Disease Markers

Hindawiwwwhindawicom Volume 2018

BioMed Research International

OncologyJournal of

Hindawiwwwhindawicom Volume 2013

Hindawiwwwhindawicom Volume 2018

Oxidative Medicine and Cellular Longevity

Hindawiwwwhindawicom Volume 2018

PPAR Research

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Immunology ResearchHindawiwwwhindawicom Volume 2018

Journal of

ObesityJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational and Mathematical Methods in Medicine

Hindawiwwwhindawicom Volume 2018

Behavioural Neurology

OphthalmologyJournal of

Hindawiwwwhindawicom Volume 2018

Diabetes ResearchJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Research and TreatmentAIDS

Hindawiwwwhindawicom Volume 2018

Gastroenterology Research and Practice

Hindawiwwwhindawicom Volume 2018

Parkinsonrsquos Disease

Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom

Hence choosing suitable parameters for the controls c1 c2and c3 it was observed that the eective reproductionnumbers obtained for Figures 2ndash7were 00075 000013 00440000075 00075 000013 0044 and 0000075 respectivelyis shows that incorporating all the control measures that iseducating individuals giving treatment and applying in-secticides is an eective method to help reduce the numbersecondary infections in the population which correspondswith eradicating the disease in the long run

5 Conclusion

In this paper we studied and analyzed the model fortransmission of HAT and determined the basic reproduction

number e local and global stabilities of disease-freeequilibrium and endemic equilibrium were also provedFor the optimal control model education treatment andinsecticides as control measures were used to optimize theobjective function dened by Equation (40) e numericalsimulations of the optimal control model show that the beststrategy to reduce the number of infected individuals isthrough the use of education treatment and insecticidesis is the eective and ecient method to eliminate thedisease Furthermore the national authorities non-governmental organizations (NGOs) and stakeholdersmust not lose their interest in controlling the disease be-cause neglecting this disease may cause rapid reoccurrenceand much eect to the people who are at risk

06

04

02

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(a)

4

35

3

25

15

1

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 = 0 c2 ne 0 c3 ne 0

(b)

Figure 4 Simulations of the model showing the eorts of treatment and insecticides only on infectious individuals at Rc 00075

c1 ne 0c2 ne 0c3 = 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(a)

c1 ne 0c2 ne 0c3 ne 0

06

04

02

0

Con

trol p

rofil

e

0 20 40 60 80 100 120Time (days)

(b)

Figure 3 (a) Control prole when c1 ne 0 c2 ne 0 and c3 0 at Rc 0044 (b) Control prole when c1 ne 0 c2 ne 0 and c3 ne 0 at Rc 0000075

Computational and Mathematical Methods in Medicine 11

20

15

10

5Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(a)

12

10

8

6

4

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(b)

Figure 7 Simulations of the model showing the eorts of education treatment and insecticides on infectious individuals at Rc 0000075

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(a)

44

42

38

4

34

36

32

3

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(b)

Figure 6 Simulations of the model showing the eorts of education and treatment only on infectious individuals at Rc 0044

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

20

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

(a)

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

Infe

ctio

us v

ecto

r pop

ulat

ion

12

8

10

6

4

20 20 40 60 80 100 120

Time (days)

(b)

Figure 5 Simulations of the model showing the eorts of education and insecticides only on infectious individuals at Rc 000013

12 Computational and Mathematical Methods in Medicine

Data Availability

(e secondary data supporting this research are from pre-viously reported studies and datasets which have been cited(e processed data are available in cited references

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(e authors thank the support from Next Einstein Initiativeof the African Institute for Mathematical Sciences Ghana(Annual Grant for Graduate Studies) and MasterCardFoundation

References

[1] K S Rock S J Torr C Lumbala and M J KeelingldquoQuantitative evaluation of the strategy to eliminate humanafrican trypanosomiasis in the democratic republic of CongordquoParasites and Vectors vol 8 no 1 p 532 2015

[2] J R Franco P P Simarro A Diarra and J G JanninldquoEpidemiology of human african trypanosomiasisrdquo ClinicalEpidemiology vol 2014 no 6 pp 257ndash275 2014

[3] R Brun J Blum F Chappuis and C Burri ldquoHuman africantrypanosomiasisrdquo Lancet vol 375 no 9709 pp 148ndash1592010

[4] World Health Organization et al Control and Surveillance ofHuman African Trypanosomiasis Report of a WHO ExpertCommittee World Health Organization Geneva Switzerland2013

[5] M Martcheva An Introduction to Mathematical Epidemiol-ogy Springer Vol 61 Springer Berlin Germany 2015

[6] D J Rogers ldquoA general model for the african trypanoso-miasesrdquo Parasitology vol 97 no 1 pp 193ndash212 1988

[7] J W Hargrove R Ouifki D Kajunguri G A Vale andS J Torr ldquoModeling the control of trypanosomiasis usingtrypanocides or insecticide-treated livestockrdquo PLoS NeglectedTropical Diseases vol 6 no 5 p 1615 2012

[8] D Kajunguri ldquoModelling the control of tsetse and africantrypanosomiasis through application of insecticides on cattlein Southeastern Ugandardquo PhD thesis Stellenbosch Uni-versity Stellenbosch South Africa 2013

[9] S Olaniyi and O S Obabiyi ldquoMathematical model for malariatransmission dynamics in human and mosquito populationswith nonlinear forces of infectionrdquo International Journal ofPure and Applied Mathematics vol 88 no 1 pp 125ndash1562013

[10] R H Chisholm P T Campbell Y Wu S Y C TongJ McVernon and N Geard ldquoImplications of asymptomaticcarriers for infectious disease transmission and controlrdquoRoyal Society Open Science vol 5 no 2 p 172341 2018

[11] P Yan and S Liu ldquoSeir epidemic model with delayrdquo ANZIAMJournal vol 48 no 1 pp 119ndash134 2006

[12] G T Azu-Tungmah ldquoA mathematical model to control thespread of malaria in Ghanardquo PhD thesis Kwame NkrumahUniversity of Science and Technology Kumasi Ghana 2012

[13] S A Pedro S Abelman F T Ndjomatchoua R Sang andH E Z Tonnang ldquoStability bifurcation and chaos analysis ofvector-borne disease model with application to rift valleyfeverrdquo PloS One vol 9 no 10 Article ID 108172 2014

[14] S Olaniyi and O S Obabiyi ldquoQualitative analysis of malariadynamics with nonlinear incidence functionrdquo AppliedMathematical Sciences vol 8 no 78 pp 3889ndash3904 2014

[15] L S Pontryagin V G Boltyanskii and R V Gamkrelidze EfMishchenko the Mathematical 2eory of Optimal ProcessesInterscience New York NY USA 1962

[16] B Heimann ldquoFleming WHrishel RW deterministic andstochastic optimal control New York-heidelbergberlinSpringer-verlag 1975 xiii 222 s dm 60 60rdquo ZAMM-Journal of Applied Mathematics and Mechanics Zeitschriftfur Angewandte Mathematik und Mechanik vol 59 no 9p 494 1979

[17] S Davis S Aksoy and A Galvani ldquoA global sensitivityanalysis for african sleeping sicknessrdquo Parasitology vol 138no 4 pp 516ndash526 2011

[18] M Artzrouni and J P Gouteux ldquoA compartmental model ofsleeping sickness in central africardquo Journal of BiologicalSystems vol 4 no 4 pp 459ndash477 1996

Computational and Mathematical Methods in Medicine 13

Stem Cells International

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

MEDIATORSINFLAMMATION

of

EndocrinologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Disease Markers

Hindawiwwwhindawicom Volume 2018

BioMed Research International

OncologyJournal of

Hindawiwwwhindawicom Volume 2013

Hindawiwwwhindawicom Volume 2018

Oxidative Medicine and Cellular Longevity

Hindawiwwwhindawicom Volume 2018

PPAR Research

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Immunology ResearchHindawiwwwhindawicom Volume 2018

Journal of

ObesityJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational and Mathematical Methods in Medicine

Hindawiwwwhindawicom Volume 2018

Behavioural Neurology

OphthalmologyJournal of

Hindawiwwwhindawicom Volume 2018

Diabetes ResearchJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Research and TreatmentAIDS

Hindawiwwwhindawicom Volume 2018

Gastroenterology Research and Practice

Hindawiwwwhindawicom Volume 2018

Parkinsonrsquos Disease

Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom

20

15

10

5Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(a)

12

10

8

6

4

2

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 ne 0

(b)

Figure 7 Simulations of the model showing the eorts of education treatment and insecticides on infectious individuals at Rc 0000075

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(a)

44

42

38

4

34

36

32

3

Infe

ctio

us v

ecto

r pop

ulat

ion

0 20 40 60 80 100 120Time (days)

c1 = c2 = c3 = 0c1 ne 0 c2 ne 0 c3 = 0

(b)

Figure 6 Simulations of the model showing the eorts of education and treatment only on infectious individuals at Rc 0044

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

20

15

10

5

Infe

ctio

us h

uman

pop

ulat

ion

0 20 40 60 80 100 120Time (days)

(a)

c1 = c2 = c3 = 0c1 ne 0 c2 = 0 c3 ne 0

Infe

ctio

us v

ecto

r pop

ulat

ion

12

8

10

6

4

20 20 40 60 80 100 120

Time (days)

(b)

Figure 5 Simulations of the model showing the eorts of education and insecticides only on infectious individuals at Rc 000013

12 Computational and Mathematical Methods in Medicine

Data Availability

(e secondary data supporting this research are from pre-viously reported studies and datasets which have been cited(e processed data are available in cited references

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(e authors thank the support from Next Einstein Initiativeof the African Institute for Mathematical Sciences Ghana(Annual Grant for Graduate Studies) and MasterCardFoundation

References

[1] K S Rock S J Torr C Lumbala and M J KeelingldquoQuantitative evaluation of the strategy to eliminate humanafrican trypanosomiasis in the democratic republic of CongordquoParasites and Vectors vol 8 no 1 p 532 2015

[2] J R Franco P P Simarro A Diarra and J G JanninldquoEpidemiology of human african trypanosomiasisrdquo ClinicalEpidemiology vol 2014 no 6 pp 257ndash275 2014

[3] R Brun J Blum F Chappuis and C Burri ldquoHuman africantrypanosomiasisrdquo Lancet vol 375 no 9709 pp 148ndash1592010

[4] World Health Organization et al Control and Surveillance ofHuman African Trypanosomiasis Report of a WHO ExpertCommittee World Health Organization Geneva Switzerland2013

[5] M Martcheva An Introduction to Mathematical Epidemiol-ogy Springer Vol 61 Springer Berlin Germany 2015

[6] D J Rogers ldquoA general model for the african trypanoso-miasesrdquo Parasitology vol 97 no 1 pp 193ndash212 1988

[7] J W Hargrove R Ouifki D Kajunguri G A Vale andS J Torr ldquoModeling the control of trypanosomiasis usingtrypanocides or insecticide-treated livestockrdquo PLoS NeglectedTropical Diseases vol 6 no 5 p 1615 2012

[8] D Kajunguri ldquoModelling the control of tsetse and africantrypanosomiasis through application of insecticides on cattlein Southeastern Ugandardquo PhD thesis Stellenbosch Uni-versity Stellenbosch South Africa 2013

[9] S Olaniyi and O S Obabiyi ldquoMathematical model for malariatransmission dynamics in human and mosquito populationswith nonlinear forces of infectionrdquo International Journal ofPure and Applied Mathematics vol 88 no 1 pp 125ndash1562013

[10] R H Chisholm P T Campbell Y Wu S Y C TongJ McVernon and N Geard ldquoImplications of asymptomaticcarriers for infectious disease transmission and controlrdquoRoyal Society Open Science vol 5 no 2 p 172341 2018

[11] P Yan and S Liu ldquoSeir epidemic model with delayrdquo ANZIAMJournal vol 48 no 1 pp 119ndash134 2006

[12] G T Azu-Tungmah ldquoA mathematical model to control thespread of malaria in Ghanardquo PhD thesis Kwame NkrumahUniversity of Science and Technology Kumasi Ghana 2012

[13] S A Pedro S Abelman F T Ndjomatchoua R Sang andH E Z Tonnang ldquoStability bifurcation and chaos analysis ofvector-borne disease model with application to rift valleyfeverrdquo PloS One vol 9 no 10 Article ID 108172 2014

[14] S Olaniyi and O S Obabiyi ldquoQualitative analysis of malariadynamics with nonlinear incidence functionrdquo AppliedMathematical Sciences vol 8 no 78 pp 3889ndash3904 2014

[15] L S Pontryagin V G Boltyanskii and R V Gamkrelidze EfMishchenko the Mathematical 2eory of Optimal ProcessesInterscience New York NY USA 1962

[16] B Heimann ldquoFleming WHrishel RW deterministic andstochastic optimal control New York-heidelbergberlinSpringer-verlag 1975 xiii 222 s dm 60 60rdquo ZAMM-Journal of Applied Mathematics and Mechanics Zeitschriftfur Angewandte Mathematik und Mechanik vol 59 no 9p 494 1979

[17] S Davis S Aksoy and A Galvani ldquoA global sensitivityanalysis for african sleeping sicknessrdquo Parasitology vol 138no 4 pp 516ndash526 2011

[18] M Artzrouni and J P Gouteux ldquoA compartmental model ofsleeping sickness in central africardquo Journal of BiologicalSystems vol 4 no 4 pp 459ndash477 1996

Computational and Mathematical Methods in Medicine 13

Stem Cells International

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

MEDIATORSINFLAMMATION

of

EndocrinologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Disease Markers

Hindawiwwwhindawicom Volume 2018

BioMed Research International

OncologyJournal of

Hindawiwwwhindawicom Volume 2013

Hindawiwwwhindawicom Volume 2018

Oxidative Medicine and Cellular Longevity

Hindawiwwwhindawicom Volume 2018

PPAR Research

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Immunology ResearchHindawiwwwhindawicom Volume 2018

Journal of

ObesityJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational and Mathematical Methods in Medicine

Hindawiwwwhindawicom Volume 2018

Behavioural Neurology

OphthalmologyJournal of

Hindawiwwwhindawicom Volume 2018

Diabetes ResearchJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Research and TreatmentAIDS

Hindawiwwwhindawicom Volume 2018

Gastroenterology Research and Practice

Hindawiwwwhindawicom Volume 2018

Parkinsonrsquos Disease

Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom

Data Availability

(e secondary data supporting this research are from pre-viously reported studies and datasets which have been cited(e processed data are available in cited references

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(e authors thank the support from Next Einstein Initiativeof the African Institute for Mathematical Sciences Ghana(Annual Grant for Graduate Studies) and MasterCardFoundation

References

[1] K S Rock S J Torr C Lumbala and M J KeelingldquoQuantitative evaluation of the strategy to eliminate humanafrican trypanosomiasis in the democratic republic of CongordquoParasites and Vectors vol 8 no 1 p 532 2015

[2] J R Franco P P Simarro A Diarra and J G JanninldquoEpidemiology of human african trypanosomiasisrdquo ClinicalEpidemiology vol 2014 no 6 pp 257ndash275 2014

[3] R Brun J Blum F Chappuis and C Burri ldquoHuman africantrypanosomiasisrdquo Lancet vol 375 no 9709 pp 148ndash1592010

[4] World Health Organization et al Control and Surveillance ofHuman African Trypanosomiasis Report of a WHO ExpertCommittee World Health Organization Geneva Switzerland2013

[5] M Martcheva An Introduction to Mathematical Epidemiol-ogy Springer Vol 61 Springer Berlin Germany 2015

[6] D J Rogers ldquoA general model for the african trypanoso-miasesrdquo Parasitology vol 97 no 1 pp 193ndash212 1988

[7] J W Hargrove R Ouifki D Kajunguri G A Vale andS J Torr ldquoModeling the control of trypanosomiasis usingtrypanocides or insecticide-treated livestockrdquo PLoS NeglectedTropical Diseases vol 6 no 5 p 1615 2012

[8] D Kajunguri ldquoModelling the control of tsetse and africantrypanosomiasis through application of insecticides on cattlein Southeastern Ugandardquo PhD thesis Stellenbosch Uni-versity Stellenbosch South Africa 2013

[9] S Olaniyi and O S Obabiyi ldquoMathematical model for malariatransmission dynamics in human and mosquito populationswith nonlinear forces of infectionrdquo International Journal ofPure and Applied Mathematics vol 88 no 1 pp 125ndash1562013

[10] R H Chisholm P T Campbell Y Wu S Y C TongJ McVernon and N Geard ldquoImplications of asymptomaticcarriers for infectious disease transmission and controlrdquoRoyal Society Open Science vol 5 no 2 p 172341 2018

[11] P Yan and S Liu ldquoSeir epidemic model with delayrdquo ANZIAMJournal vol 48 no 1 pp 119ndash134 2006

[12] G T Azu-Tungmah ldquoA mathematical model to control thespread of malaria in Ghanardquo PhD thesis Kwame NkrumahUniversity of Science and Technology Kumasi Ghana 2012

[13] S A Pedro S Abelman F T Ndjomatchoua R Sang andH E Z Tonnang ldquoStability bifurcation and chaos analysis ofvector-borne disease model with application to rift valleyfeverrdquo PloS One vol 9 no 10 Article ID 108172 2014

[14] S Olaniyi and O S Obabiyi ldquoQualitative analysis of malariadynamics with nonlinear incidence functionrdquo AppliedMathematical Sciences vol 8 no 78 pp 3889ndash3904 2014

[15] L S Pontryagin V G Boltyanskii and R V Gamkrelidze EfMishchenko the Mathematical 2eory of Optimal ProcessesInterscience New York NY USA 1962

[16] B Heimann ldquoFleming WHrishel RW deterministic andstochastic optimal control New York-heidelbergberlinSpringer-verlag 1975 xiii 222 s dm 60 60rdquo ZAMM-Journal of Applied Mathematics and Mechanics Zeitschriftfur Angewandte Mathematik und Mechanik vol 59 no 9p 494 1979

[17] S Davis S Aksoy and A Galvani ldquoA global sensitivityanalysis for african sleeping sicknessrdquo Parasitology vol 138no 4 pp 516ndash526 2011

[18] M Artzrouni and J P Gouteux ldquoA compartmental model ofsleeping sickness in central africardquo Journal of BiologicalSystems vol 4 no 4 pp 459ndash477 1996

Computational and Mathematical Methods in Medicine 13

Stem Cells International

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

MEDIATORSINFLAMMATION

of

EndocrinologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Disease Markers

Hindawiwwwhindawicom Volume 2018

BioMed Research International

OncologyJournal of

Hindawiwwwhindawicom Volume 2013

Hindawiwwwhindawicom Volume 2018

Oxidative Medicine and Cellular Longevity

Hindawiwwwhindawicom Volume 2018

PPAR Research

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Immunology ResearchHindawiwwwhindawicom Volume 2018

Journal of

ObesityJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational and Mathematical Methods in Medicine

Hindawiwwwhindawicom Volume 2018

Behavioural Neurology

OphthalmologyJournal of

Hindawiwwwhindawicom Volume 2018

Diabetes ResearchJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Research and TreatmentAIDS

Hindawiwwwhindawicom Volume 2018

Gastroenterology Research and Practice

Hindawiwwwhindawicom Volume 2018

Parkinsonrsquos Disease

Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom

Stem Cells International

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

MEDIATORSINFLAMMATION

of

EndocrinologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Disease Markers

Hindawiwwwhindawicom Volume 2018

BioMed Research International

OncologyJournal of

Hindawiwwwhindawicom Volume 2013

Hindawiwwwhindawicom Volume 2018

Oxidative Medicine and Cellular Longevity

Hindawiwwwhindawicom Volume 2018

PPAR Research

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Immunology ResearchHindawiwwwhindawicom Volume 2018

Journal of

ObesityJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational and Mathematical Methods in Medicine

Hindawiwwwhindawicom Volume 2018

Behavioural Neurology

OphthalmologyJournal of

Hindawiwwwhindawicom Volume 2018

Diabetes ResearchJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Research and TreatmentAIDS

Hindawiwwwhindawicom Volume 2018

Gastroenterology Research and Practice

Hindawiwwwhindawicom Volume 2018

Parkinsonrsquos Disease

Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom