Research Article Optimal Control of an SIR Model with ...downloads.hindawi.com/archive/2013/403549.pdf · Hindawi Publishing Corporation ISRN Biomathematics Volume , Article ID, pages

Hindawi Publishing CorporationISRN BiomathematicsVolume 2013 Article ID 403549 7 pageshttpdxdoiorg1011552013403549

Research ArticleOptimal Control of an SIR Model with Delay in State andControl Variables

Mohamed Elhia Mostafa Rachik and Elhabib Benlahmar

Laboratory of Analysis Modeling and Simulation Department of Mathematics and Computer Science Faculty of Sciences Ben MrsquoSikHassan II University Mohammedia BP 7955 Sidi Othman Casablanca Morocco

Correspondence should be addressed to Mohamed Elhia elhia mohamedyahoofr

Received 30 May 2013 Accepted 11 July 2013

Academic Editors H Ishikawa and M A Panteleev

Copyright copy 2013 Mohamed Elhia et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We will investigate the optimal control strategy of an SIR epidemic model with time delay in state and control variables We usea vaccination program to minimize the number of susceptible and infected individuals and to maximize the number of recoveredindividuals Existence for the optimal control is established Pontryaginrsquos maximum principle is used to characterize thisoptimal control and the optimality system is solved by a discretization method based on the forward and backward differenceapproximations The numerical simulation is carried out using data regarding the course of influenza A (H1N1) in Morocco Theobtained results confirm the performance of the optimization strategy

1 Introduction

For a long time infectious diseases have caused several epi-demics leaving behind them not only millions of dead andinfected individuals but also severe socioeconomic conse-quences Nowadays mathematical modeling of infectiousdiseases is one of the most important research areas Indeedmathematical epidemiology has contributed to a betterunderstanding of the dynamical behavior of infectious dis-eases its impacts and possible future predictions about itsspreading Mathematical models are used in comparingplanning implementing evaluating and optimizing variousdetection prevention therapy and control programs Manyinfluential results related to the development and analysisof epidemiological models have been established and can befound in many articles and books (see eg [1ndash3])

Epidemiologicalmodels often take the formof a systemofnonlinear ordinary and differential equations without timedelay However for various biological reasons the realdynamic behavior of an epidemic depends not only on its cur-rent state but also on its past history Thus to reflect the realbehavior of some diseases many researchers have proposedand analyzedmore realisticmodels including delays tomodel

differentmechanisms in the dynamics of epidemics like latentperiod temporary immunity and length of infection (see eg[4ndash8] and the references therein)

To the best of our knowledge including time delay inboth state and control variables in the context of an epidemiccontrolledmodel has not been studiedThere have been someworks (like [9 10]) which study an optimal control problemwith time delay but only in the state variable In this paper wewill investigate the effect of a vaccination program in the caseof an SIR (susceptible-infected-recovered) epidemic modelwith time delay in the control and the state variables To dothis we will consider an optimally controlled SIR epidemicmodel with time delay where the control means the percent-age of susceptible individuals being vaccinated per time unitand the time delay represents the required time so that a vac-cinated susceptible person moves from the susceptibles classto the recovered class We use optimal control approach tominimize the number of susceptible and infected individualsand to maximize the number of recovered individuals duringthe course of an epidemic

This paper is organized as follows In Section 2 we willpresent a mathematical model with time delay and a controlterm The analysis of optimization problem is presented in

2 ISRN Biomathematics

Section 3 In Section 4 we will give a numerical appropriatemethod and the corresponding simulation results Finally theconclusions are summarized in Section 5

2 Mathematical Model

We consider the SIR epidemic model with constant totalpopulation sizeThe population is divided into three disease-state compartments susceptible individuals (119878) people whocan catch the disease infectious (infective) individuals (119868)people who have the disease and can transmit the diseaserecovered individuals (119877) people who have recovered fromthe disease We assume that an individual can be infectedonly through contacts with infectious individuals and thatimmunity is permanent The transitions between differentstates are described by the following parameters

(i) Λ is the recruitment rate of susceptibles

(ii) 120573 is the effective contact rate

(iii) 119889 is the natural mortality rate

(iv) 120574 is the recovery rate

(v) 120598 is the disease induced death rate

The population dynamics is given by the following system ofordinary differential equations subject to nonnegative initialconditions

119889119878 (119905)

119889119905= Λ minus 120573119878 (119905)

119868 (119905)

119873 (119905)minus 119889119878 (119905)

119889119868 (119905)

119889119905= 120573119878 (119905)

119868 (119905)

119873 (119905)minus (120574 + 119889 + 120598) 119868 (119905)

119889119877 (119905)

119889119905= 120574119868 (119905) minus 119889119877 (119905)

(1)

where 119878(0) = 1198780 119868(0) = 119868

0119877(0) = 119877

0 and119873(119905) = 119878(119905)+119868(119905)+

119877(119905) is the total population number at time 119905The strategy of the control we adopt consists of a vaccina-

tion program our goal is to minimize the level of susceptibleand infected individuals and to maximize the recoveredindividuals Into the model (1) we include a control 119906 thatrepresents the percentage of susceptible individuals beingvaccinated per time unit In order to have a realisticmodel weneed to take into account that themovement of the vaccinatedsusceptible individuals from the class of susceptibles into therecovered class is subject to delay Thus the time delay isintroduced in the system as follows at time 119905 only a per-centage of susceptible individuals that have been vaccinated120591 time unit ago that is at time 119905 minus 120591 are removed from thesusceptible class and added to the recovered class So themathematical system with time delay in state and control

variables is given by the nonlinear retarded differentialequations

119889119878 (119905)

119889119905= Λ minus 120573119878 (119905)

119868 (119905)

119873 (119905)minus 119889119878 (119905) minus 119906 (119905 minus 120591) 119878 (119905 minus 120591)

119889119868 (119905)

119889119905= 120573119878 (119905)

119868 (119905)

119873 (119905)minus (120574 + 119889 + 120598) 119868 (119905)

119889119877 (119905)

119889119905= 120574119868 (119905) minus 119889119877 (119905) + 119906 (119905 minus 120591) 119878 (119905 minus 120591)

(2)

In addition for biological reasons we assume for 120579 isin [minus120591 0]that 119878(120579) 119868(120579) and 119877(120579) are nonnegative continuous func-tions and 119906(120579) = 0 Note that the control 119906 is assumed to beintegrable in the sense of Lebesgue bounded with 0 le 119906 le119887 lt 1 and 119887 is a given constant

To show the existence of solutions for the control system(2) we first prove that the system (2) is dissipative that is allsolutions are uniformly bounded in a proper subset Ω isin R3

+

Let (119878 119868 119877) isin R3+be any solution with nonnegative initial

conditions Adding equations of (2) we get

119889119873

119889119905= Λ minus 119889119873 minus 120598119868 lt Λ minus 119889119873 (3)

After integration using the constant variation formula wehave

119873(119905) leΛ

119889+ 119873 (0) 119890

minus119889119905 (4)

It then follows that

0 le 119873 (119905) leΛ

119889as 119905 997888rarr infin (5)

Therefore all feasible solutions of the system (2) enter into theregion

Ω = (119878 119868 119877) isin R3

+ 119873 le

Λ

119889 (6)

Then we can rewrite (2) in the following form

119889119883

119889119905= 119860119883 + 119865 (119883119883

120591) = 119866 (119883119883

120591) (7)

where

119883 (119905) = [

[

119878 (119905)

119868 (119905)

119877 (119905)

]

]

119860 = [

[

minus119889 0 0

0 minus (119889 + 120574 + 120576) 0

0 120574 minus119889

]

]

119865 (119883 (119905) 119883120591(119905)) =

[[[[[[

[

Λ minus120573119878 (119905) 119868 (119905)

119873 (119905)minus 119906120591(119905) 119878120591(119905)

120573119878 (119905) 119868 (119905)

119873 (119905)

119906120591 (119905) 119878120591 (119905)

]]]]]]

]

119906120591(119905) = 119906 (119905 minus 120591) 119883

120591(119905) = 119883 (119905 minus 120591)

(8)

ISRN Biomathematics 3

The second term on the right-hand side of (7) satisfies1003816100381610038161003816119865 (1198831 (119905) 1198831120591 (119905)) minus 119865 (1198832 (119905) 1198832120591 (119905))

1003816100381610038161003816

le 1198721

10038161003816100381610038161198831 (119905) minus 1198832 (119905)1003816100381610038161003816

+ 1198722

10038161003816100381610038161198831120591 (119905) minus 1198832120591 (119905)1003816100381610038161003816

(9)

where1198721and119872

2are some positive constants independent

of the state variables 119878(119905) 119868(119905) and 119877(119905) and10038161003816100381610038161198831 (119905) minus 1198832 (119905)

1003816100381610038161003816 =10038161003816100381610038161198781 (119905) minus 1198782 (119905)

1003816100381610038161003816

+10038161003816100381610038161198681 (119905) minus 1198682 (119905)

1003816100381610038161003816 +10038161003816100381610038161198771 (119905) minus 1198772 (119905)

1003816100381610038161003816

10038161003816100381610038161198831120591 (119905) minus 1198832120591 (119905)1003816100381610038161003816 =

10038161003816100381610038161198781120591 (119905) minus 1198782120591 (119905)1003816100381610038161003816

+10038161003816100381610038161198681120591 (119905) minus 1198682120591 (119905)

1003816100381610038161003816 +10038161003816100381610038161198771120591 (119905) minus 1198772120591 (119905)

1003816100381610038161003816

(10)

Here

119878119894120591 (119905) = 119878119894 (119905 minus 120591) 119868

119894120591 (119905) = 119868119894 (119905 minus 120591)

119877119894120591(119905) = 119877

119894(119905 minus 120591) for 119894 = 1 2

(11)

Moreover we get1003816100381610038161003816119866 (1198831 1198831120591) minus 119866 (1198832 1198832120591)

1003816100381610038161003816 le 119872 (10038161003816100381610038161198831 minus 1198832

1003816100381610038161003816 +10038161003816100381610038161198831120591 minus 1198832120591

1003816100381610038161003816)

(12)

where

119872 = max (1198721+ 119860 1198722) lt infin (13)

Thus it follows that the function119866 is uniformly Lipschitzcontinuous From the definition of the control 119906(119905) and therestriction on 119878(119905) 119868(119905) and 119877(119905) ge 0 we see that a solutionof the system (7) exists (see [11])

3 The Optimal Control Problem

Our goal is to minimize the number of susceptible andinfected individuals and to maximize the number of recov-ered individuals during the course of an epidemic Mathe-matically for a fixed terminal time 119905

119891 the problem is to min-

imize the objective functional

119869 (119906) = int

119905119891

0

1198601119878 (119905) + 119860

2119868 (119905) minus 119860

3119877 (119905) +

1198604

21199062(119905) 119889119905

(14)

where 119860119894ge 0 (for 119894 = 1 4) denote weights that balance

the size of the terms In other words we seek the optimalcontrol 119906lowast such that

119869 (119906lowast) = min 119869 (119906) 119906 isin U (15)

whereU is the set of admissible controls defined by

U = 119906 0 le 119906 le 119887 lt 1 119905 isin [0 119905119891]

119906 is Lebesgue measurable (16)

31 Existence of an Optimal Control The existence of theoptimal control can be obtained using a result by Fleming andRishel in [12]

Theorem 1 Consider the control problem with system (2)There exists an optimal control 119906lowast isin U such that

119869 (119906lowast) = min119906isinU

119869 (119906) (17)

Proof To use an existence result in [12] we must check thefollowing properties

(1) The set of controls and corresponding state variablesis nonempty

(2) The control setU is convex and closed

(3) The right-hand side of the state system is bounded bya linear function in the state and control variables

(4) The integrand of the objective functional is convex onU

(5) There exist constants 1198881 1198882gt 0 and 120588 gt 1 such that

the integrand 119871(119878 119868 119877 119906) of the objective functionalsatisfies

119871 (119878 119868 119877 119906) ge 1198882+ 1198881(|119906|2)1205882

(18)

An existence result by Lukes [13] was used to give the exis-tence of solution of system (2) with bounded coefficientswhich gives condition 1 The control set is convex and closedby definition Since the state system is linear in 119906 the rightside of (2) satisfies condition 3 using the boundedness ofthe solution The integrand in the objective functional (14) isconvex onU In addition we can easily see that there exist aconstant 120588 gt 1 and positive numbers 119888

1and 1198882satisfying

119871 (119878 119868 119877 119906) ge 1198882+ 1198881(|119906|2)1205882

(19)

32 Characterization of the Optimal Control In order toderive the necessary condition for the optimal controlPontryaginrsquos maximum principal with delay given in [14] wasusedThis principal converts (2) (14) and (15) into a problemof minimizing a Hamiltonian119867 defined by

119867 = 1198601119878 (119905) + 119860

2119868 (119905) minus 119860

3119877 (119905) +

1198604

21199062(119905) +

3

sum

119894=1

120582119894119891119894

(20)

where 119891119894is the right side of the differential equation of the 119894th

state variable


Theorem 2 Given an optimal control 119906lowast isin U and solutions119878lowast 119868lowast and 119877lowast of the corresponding state system (2) there existadjoint functions 120582

1 1205822 and 120582

3satisfying

1= minus 119860

1+ 1205821119889 + (120582

1minus 1205822) 120573

119868lowast

119873lowast

+ 120594[0119905119891minus120591]

(119905) (120582+

1minus 120582+

3) 119906lowast

2= minus119860

2+ (1205821minus 1205822) 120573

119878lowast

119873lowast+ (119889 + 120574 + 120598) 120582

2minus 1205741205823

3= 1198603+ 1205823119889

(21)

with the transversality conditions

1205821(119905119891) = 1205822(119905119891) = 1205823(119905119891) = 0 (22)

Furthermore the optimal control 119906lowast is given by

119906lowast(119905) = min(119887max(0

(120582+

1minus 120582+

3)

1198604

120594[0119905119891minus120591]

(119905) 119878lowast)) (23)

where 120582+119894(119905) = 120582

119894(119905 + 120591) for 119894 = 1 3

Proof The adjoint equations and transversality conditionscan be obtained by using Pontryaginrsquos maximum principlewith delay in the state and control variables [14] such that

1= minus

120597119867 (119905)

120597119878 (119905)minus 120594[0119905119891minus120591]

(119905)120597119867 (119905 + 120591)

120597119878 (119905 minus 120591) 120582

1(119905119891) = 0

2= minus

120597119867

120597119868 120582

2(119905119891) = 0

3= minus

120597119867

120597119877 120582

3(119905119891) = 0

(24)

The optimal control 119906lowast can be solved from the optimalitycondition

120597119867

120597119906+ 120594[0119905119891minus120591]

(119905)120597119867 (119905 + 120591)

120597119906120591

= 0 (25)

That is

1198604119906 + 120594[0119905119891minus120591]

(119905) (120582+

3minus 120582+

1) 119878 = 0 (26)

By the bounds in U of the control it is easy to rewrite 119906lowast inthe form (23)

4 Numerical Simulation

In this section we give a numerical method to solve the opti-mality system which is a two-point boundary value problemwith separated boundary conditions at times 119905

0= 0 and 119905

119891

Let there exist a step size ℎ gt 0 and (119899119898) isin N2 with120591 = 119898ℎ and 119905

119891minus 1199050= 119899ℎ For reasons of programming we

0 30 60 90 120 150 1800

05

1

15

2

25

3

Time (days)

With controlWithout control

Susc

eptib

les

S(t)

times107

Figure 1 The function 119878 with and without control

consider 119898 knots to left of 1199050and right of 119905

119891 and we obtain

the following partition

Δ = (119905minus119898= minus120591 lt sdot sdot sdot lt 119905

minus1lt 1199050= 0 lt 119905

1lt sdot sdot sdot lt 119905

119899

= 119905119891lt 119905119899+1

lt sdot sdot sdot lt 119905119899+119898

)

(27)

Then we have 119905119894= 1199050+ 119894ℎ (minus119898 le 119894 le 119899 + 119898) Next we

define the state and adjoint variables 119878(119905) 119868(119905) 119877(119905) 1205821 1205822

1205823 and the control 119906 in terms of nodal points 119878

119894 119868119894 119877119894 1205821198941 1205821198942

120582119894

3 and 119906

119894 Now using combination of forward and backward

difference approximations we obtain the Algorithm 1The numerical simulations were carried out using data

regarding the course of the influenza A (H1N1) in MoroccoThe initial conditions and parameters of the system (2) aretaken from [15 16] while the time delay value is taken from[17]

120573 = 03095 Λ = 117417

119889 = 39139 times 10minus5 120574 = 02

120576 = 00063 120591 = 10

(28)

the initial conditions for the ordinary differential systemwere

119878 (0) = 30 times 106 119868 (0) = 30 119877 (0) = 28 (29)

and the transversality conditions for the ordinary differentialsystem were

120582119894(119905119891) = 0 (119894 = 1 3) (30)

Figure 1 indicates that the number of susceptible indi-viduals (119878) decreases more rapidly in the case with controlIt reaches 7399 times 105 at the end of the vaccination periodagainst 6732 times 106 in case without control that is a reduc-tion of 5992 times 106


Step 1 For 119894 = minus119898 0 do 119878119894= 1198780 119868119894= 1198680 119877119894= 1198770 119906119894= 0

End forFor 119894 = 119899 119899 + 119898 do 120582119894

1= 0 120582119894

2= 0 120582119894

3= 0

End forStep 2 For 119894 = 0 119899 minus 1 do

119878119894+1=119878119894+ ℎ(Λ minus 119906

119894minus119898119878119894minus119898)

1 + ℎ(120573(119868119894119873) + 119889)

119868119894+1=

119868119894

1 + ℎ(120574 + 119889 + 120576 minus 120573(119878119894+1119873))

119877119894+1=119877119894+ ℎ(120574119868

119894+1+ 119906119894minus119898119878119894minus119898)

1 + ℎ119889

120582119899minus119894minus1

1=

120582119899minus119894

1+ ℎ(119860

1+ 120573(119868

119894+1119873)120582

119899minus119894

2+ (120582119899minus119894+119898

3minus 120582119899minus119894minus1+119898

1)120594[0119905119891minus120591]

(119905119899minus119894)119906119894)

1 + ℎ(119889 + 120573(119868119894+1119873))

120582119899minus119894minus1

2=120582119899minus119894

2+ ℎ(119860

2+ 120582119899minus119894

3120574 minus 120582119899minus119894minus1

1120573(119878119894+1119873))

1 + ℎ(120574 + 119889 + 120576 minus 120573(119878119894+1119873))

120582119899minus119894minus1

3=120582119899minus119894

3minus 1198603ℎ

1 + 119889ℎ

119879119894+1=(120582119899minus119894minus1+119898


3)

1198604

120594[0119905119891minus120591]

(119905119894+1)119878119894+1

119906119894+1= min (119887max (0 119879119894+1))

Step 3 For 119894 = 0 n write119878lowast(119905119894) = 119878119894 119868lowast(119905119894) = 119868119894 119877lowast(119905

119894) = 119877119894 119906lowast(119905

119894) = 119906119894

End for

Algorithm 1

0

2

0 30 60 90 120 150 1800

5

Time (days)

times104 times10

6

With

out c

ontro

lI(t)

With

cont

rol

I(t)


0 30 60 90 120 150 1800

05

1

15

2

Time (days)

times104

Infe

cted

I(t)

Figure 3 The function 119868 with control

Figures 2 3 and 4 represent the number of infectedindividuals (119868) with control (solid curve) and without control

0 30 60 90 120 150 1800

051

152

253

354

455

Time (days)

times106

Infe

cted

I(t)

Figure 4 The function 119868 without control

(dashed curve) It shows that in the presence of a controlthe number of infected individuals (119868) decreases greatly Themaximum number of infected individuals in the case withcontrol is 1787 times 104 and is 4404 times 106 in the case withoutcontrol then the efficiency of our strategy in reducing thespread of infection is nearly 9952

Figure 5 shows that the number of people removed beginsto grow notably from 24th day instead of 48th day inthe absence of control Moreover the number of recoveredindividuals at the end of the vaccination period is 2926 times107 instead of 2342 times 107 which represents an increase of

5840 times 106 cases

Figures 6 and 7 represent respectively the optimal con-trol and optimal value of the costThe curves start to increaseduring the first month because of the high infection level


0 30 60 90 120 150 1800

05

1

15

2

25

3

Time (days)

times107

Rem

oved

R(t)



0 30 60 90 120 150 1800

001

002

003

004

005

006

Time (days)

Con

trol

u(t)

Figure 6 The optimal control 119906lowast

0 30 60 90 120 150 1800

05

1

15

Time (days)

Cos

t

times10minus3

Figure 7 The optimal value of the cost

then they drop off steadily which is because of the constantand steady eradication of the infection

5 Conclusion

The work in this paper contributes to a growing literatureon applying optimal control techniques to epidemiology Weproposed amore realistic controlledmodel by including timedelay which represents the needed time for the migrationfrom the susceptible class to the recovered class after vac-cination The optimal control theory has been applied inthe context of an SIR model with time delay in state andcontrol variables and that includes a control 119906 that representsthe percentage of susceptible individuals being vaccinatedper time unit By using Pontryaginrsquos maximum principlethe explicit expression of the optimal controls was obtainedSimulation results indicate that the proposed control strategyis effective in reducing the number of susceptible and infectedindividuals and maximizing the recovered individuals

Acknowledgment

The authors would like to thank the anonymous referee forhisher valuable comments on the first version of the paperwhich have led to an improvement in this paper Researchreported in this paper was supported by the MoroccanSystemsTheory Network

References

[1] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press Oxford UK1992

[2] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ology of Infectious Diseases John Wiley amp Sons Chisteter UK2000

[3] F Brauer and C C CastilloMathematical Models in PopulationBiology and Epidemiology Springer New York NY USA 2000

[4] P Van Den Driessche ldquoSome epidemiological models withdelaysrdquo inDifferential Equations andApplications to Biology andto Industry (Claremont CA 1994) pp 507ndash520 World Scien-tific River Edge NJ USA 1996

[5] Q J A Khan and E V Krishnan ldquoAn epidemic model with atime delay in transmissionrdquo Applications of Mathematics vol48 no 3 pp 193ndash203 2003

[6] N Yoshida and T Hara ldquoGlobal stability of a delayed SIRepidemic model with density dependent birth and death ratesrdquoJournal of Computational and AppliedMathematics vol 201 no2 pp 339ndash347 2007

[7] R Xu Z Ma and Z Wang ldquoGlobal stability of a delayed SIRSepidemic model with saturation incidence and temporaryimmunityrdquo Computers and Mathematics with Applications vol59 no 9 pp 3211ndash3221 2010

[8] XMeng L Chen and BWu ldquoA delay SIR epidemicmodel withpulse vaccination and incubation timesrdquo Nonlinear AnalysisReal World Applications vol 11 no 1 pp 88ndash98 2010

[9] G Zaman Y H Kang and I H Jung ldquoOptimal treatment of anSIR epidemic model with time delayrdquo BioSystems vol 98 no 1pp 43ndash50 2009

[10] K Hattaf and N Yousfi ldquoOptimal control of a delayed HIVinfection model with immune response using an efficientnumerical methodrdquo ISRN Biomathematics vol 2012 Article ID215124 7 pages 2012


[11] G Birkhoff and G C Rota Ordinary Differential EquationsJohn Wiley amp Sons New YorkNY USA 4th edition 1989

[12] W H Fleming and R W Rishel Deterministic and StochasticOptimal Control Springer New York NY USA 1975

[13] D L Lukes Differential Equations Classical to Controlled Vol162 of Mathematics in Science and Engineering Academic PressNew York NY USA 1982

[14] L Gollmann D Kern and H Maurer ldquoOptimal control prob-lems with delays in state and control variables subject to mixedcontrol-state constraintsrdquo Optimal Control Applications andMethods vol 30 no 4 pp 341ndash365 2009

[15] K Hattaf and N Yousfi ldquoMathematical Model of the InfluenzaA(H1N1) Infectionrdquo Advanced Studies in Biology vol 1 no 8pp 383ndash390 2009

[16] M El hia O Balatif J Bouyaghroumni E Labriji and MRachik ldquoOptimal control applied to the spread of influenzaA(H1N1)rdquo Applied Mathematical Sciences vol 6 no 82 pp4057ndash4065 2012

[17] httpwwwpoumoncadiseases-maladiesa-zswineflu-grippe-porcinevaccines-vaccins fphp

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Section 3 In Section 4 we will give a numerical appropriatemethod and the corresponding simulation results Finally theconclusions are summarized in Section 5

2 Mathematical Model

We consider the SIR epidemic model with constant totalpopulation sizeThe population is divided into three disease-state compartments susceptible individuals (119878) people whocan catch the disease infectious (infective) individuals (119868)people who have the disease and can transmit the diseaserecovered individuals (119877) people who have recovered fromthe disease We assume that an individual can be infectedonly through contacts with infectious individuals and thatimmunity is permanent The transitions between differentstates are described by the following parameters

(i) Λ is the recruitment rate of susceptibles

(ii) 120573 is the effective contact rate

(iii) 119889 is the natural mortality rate

(iv) 120574 is the recovery rate

(v) 120598 is the disease induced death rate

The population dynamics is given by the following system ofordinary differential equations subject to nonnegative initialconditions

119889119878 (119905)

119889119905= Λ minus 120573119878 (119905)

119868 (119905)

119873 (119905)minus 119889119878 (119905)

119889119868 (119905)

119889119905= 120573119878 (119905)

119868 (119905)

119873 (119905)minus (120574 + 119889 + 120598) 119868 (119905)

119889119877 (119905)

119889119905= 120574119868 (119905) minus 119889119877 (119905)

(1)

where 119878(0) = 1198780 119868(0) = 119868

0119877(0) = 119877

0 and119873(119905) = 119878(119905)+119868(119905)+

119877(119905) is the total population number at time 119905The strategy of the control we adopt consists of a vaccina-

tion program our goal is to minimize the level of susceptibleand infected individuals and to maximize the recoveredindividuals Into the model (1) we include a control 119906 thatrepresents the percentage of susceptible individuals beingvaccinated per time unit In order to have a realisticmodel weneed to take into account that themovement of the vaccinatedsusceptible individuals from the class of susceptibles into therecovered class is subject to delay Thus the time delay isintroduced in the system as follows at time 119905 only a per-centage of susceptible individuals that have been vaccinated120591 time unit ago that is at time 119905 minus 120591 are removed from thesusceptible class and added to the recovered class So themathematical system with time delay in state and control

variables is given by the nonlinear retarded differentialequations

119889119878 (119905)

119889119905= Λ minus 120573119878 (119905)

119868 (119905)

119873 (119905)minus 119889119878 (119905) minus 119906 (119905 minus 120591) 119878 (119905 minus 120591)

119889119868 (119905)

119889119905= 120573119878 (119905)

119868 (119905)

119873 (119905)minus (120574 + 119889 + 120598) 119868 (119905)

119889119877 (119905)

119889119905= 120574119868 (119905) minus 119889119877 (119905) + 119906 (119905 minus 120591) 119878 (119905 minus 120591)

(2)

In addition for biological reasons we assume for 120579 isin [minus120591 0]that 119878(120579) 119868(120579) and 119877(120579) are nonnegative continuous func-tions and 119906(120579) = 0 Note that the control 119906 is assumed to beintegrable in the sense of Lebesgue bounded with 0 le 119906 le119887 lt 1 and 119887 is a given constant

To show the existence of solutions for the control system(2) we first prove that the system (2) is dissipative that is allsolutions are uniformly bounded in a proper subset Ω isin R3

+

Let (119878 119868 119877) isin R3+be any solution with nonnegative initial

conditions Adding equations of (2) we get

119889119873

119889119905= Λ minus 119889119873 minus 120598119868 lt Λ minus 119889119873 (3)

After integration using the constant variation formula wehave

119873(119905) leΛ

119889+ 119873 (0) 119890

minus119889119905 (4)

It then follows that

0 le 119873 (119905) leΛ

119889as 119905 997888rarr infin (5)

Therefore all feasible solutions of the system (2) enter into theregion

Ω = (119878 119868 119877) isin R3

+ 119873 le

Λ

119889 (6)

Then we can rewrite (2) in the following form

119889119883

119889119905= 119860119883 + 119865 (119883119883

120591) = 119866 (119883119883

120591) (7)

where

119883 (119905) = [

[

119878 (119905)

119868 (119905)

119877 (119905)

]

]

119860 = [

[

minus119889 0 0

0 minus (119889 + 120574 + 120576) 0

0 120574 minus119889

]

]

119865 (119883 (119905) 119883120591(119905)) =

[[[[[[

[

Λ minus120573119878 (119905) 119868 (119905)

119873 (119905)minus 119906120591(119905) 119878120591(119905)

120573119878 (119905) 119868 (119905)

119873 (119905)

119906120591 (119905) 119878120591 (119905)

]]]]]]

]

119906120591(119905) = 119906 (119905 minus 120591) 119883

120591(119905) = 119883 (119905 minus 120591)

(8)



1003816100381610038161003816

le 1198721

10038161003816100381610038161198831 (119905) minus 1198832 (119905)1003816100381610038161003816

+ 1198722

10038161003816100381610038161198831120591 (119905) minus 1198832120591 (119905)1003816100381610038161003816

(9)

where1198721and119872



1003816100381610038161003816 =10038161003816100381610038161198781 (119905) minus 1198782 (119905)

1003816100381610038161003816

+10038161003816100381610038161198681 (119905) minus 1198682 (119905)

1003816100381610038161003816 +10038161003816100381610038161198771 (119905) minus 1198772 (119905)

1003816100381610038161003816

10038161003816100381610038161198831120591 (119905) minus 1198832120591 (119905)1003816100381610038161003816 =

10038161003816100381610038161198781120591 (119905) minus 1198782120591 (119905)1003816100381610038161003816

+10038161003816100381610038161198681120591 (119905) minus 1198682120591 (119905)

1003816100381610038161003816 +10038161003816100381610038161198771120591 (119905) minus 1198772120591 (119905)

1003816100381610038161003816

(10)

Here

119878119894120591 (119905) = 119878119894 (119905 minus 120591) 119868

119894120591 (119905) = 119868119894 (119905 minus 120591)

119877119894120591(119905) = 119877

119894(119905 minus 120591) for 119894 = 1 2

(11)


1003816100381610038161003816 le 119872 (10038161003816100381610038161198831 minus 1198832

1003816100381610038161003816 +10038161003816100381610038161198831120591 minus 1198832120591

1003816100381610038161003816)

(12)

where

119872 = max (1198721+ 119860 1198722) lt infin (13)






119869 (119906) = int

119905119891

0

1198601119878 (119905) + 119860

2119868 (119905) minus 119860

3119877 (119905) +

1198604

21199062(119905) 119889119905

(14)



119869 (119906lowast) = min 119869 (119906) 119906 isin U (15)


U = 119906 0 le 119906 le 119887 lt 1 119905 isin [0 119905119891]




119869 (119906lowast) = min119906isinU

119869 (119906) (17)








119871 (119878 119868 119877 119906) ge 1198882+ 1198881(|119906|2)1205882

(18)



119871 (119878 119868 119877 119906) ge 1198882+ 1198881(|119906|2)1205882

(19)


119867 = 1198601119878 (119905) + 119860

2119868 (119905) minus 119860

3119877 (119905) +

1198604

21199062(119905) +

3

sum

119894=1

120582119894119891119894

(20)


state variable



1 1205822 and 120582

3satisfying

1= minus 119860

1+ 1205821119889 + (120582

1minus 1205822) 120573

119868lowast

119873lowast

+ 120594[0119905119891minus120591]

(119905) (120582+

1minus 120582+

3) 119906lowast

2= minus119860

2+ (1205821minus 1205822) 120573

119878lowast

119873lowast+ (119889 + 120574 + 120598) 120582

2minus 1205741205823

3= 1198603+ 1205823119889

(21)


1205821(119905119891) = 1205822(119905119891) = 1205823(119905119891) = 0 (22)


119906lowast(119905) = min(119887max(0

(120582+

1minus 120582+

3)

1198604

120594[0119905119891minus120591]

(119905) 119878lowast)) (23)

where 120582+119894(119905) = 120582

119894(119905 + 120591) for 119894 = 1 3


1= minus

120597119867 (119905)

120597119878 (119905)minus 120594[0119905119891minus120591]

(119905)120597119867 (119905 + 120591)

120597119878 (119905 minus 120591) 120582

1(119905119891) = 0

2= minus

120597119867

120597119868 120582

2(119905119891) = 0

3= minus

120597119867

120597119877 120582

3(119905119891) = 0

(24)


120597119867

120597119906+ 120594[0119905119891minus120591]

(119905)120597119867 (119905 + 120591)

120597119906120591

= 0 (25)

That is

1198604119906 + 120594[0119905119891minus120591]

(119905) (120582+

3minus 120582+

1) 119878 = 0 (26)




0= 0 and 119905

119891



0 30 60 90 120 150 1800

05

1

15

2

25

3

Time (days)


Susc

eptib

les

S(t)

times107






minus1lt 1199050= 0 lt 119905


119899

= 119905119891lt 119905119899+1


)

(27)




119894 119868119894 119877119894 1205821198941 1205821198942

120582119894

3 and 119906




120573 = 03095 Λ = 117417

119889 = 39139 times 10minus5 120574 = 02

120576 = 00063 120591 = 10

(28)


119878 (0) = 30 times 106 119868 (0) = 30 119877 (0) = 28 (29)


120582119894(119905119891) = 0 (119894 = 1 3) (30)



Step 1 For 119894 = minus119898 0 do 119878119894= 1198780 119868119894= 1198680 119877119894= 1198770 119906119894= 0

End forFor 119894 = 119899 119899 + 119898 do 120582119894

1= 0 120582119894

2= 0 120582119894

3= 0


119878119894+1=119878119894+ ℎ(Λ minus 119906

119894minus119898119878119894minus119898)

1 + ℎ(120573(119868119894119873) + 119889)

119868119894+1=

119868119894

1 + ℎ(120574 + 119889 + 120576 minus 120573(119878119894+1119873))

119877119894+1=119877119894+ ℎ(120574119868

119894+1+ 119906119894minus119898119878119894minus119898)

1 + ℎ119889

120582119899minus119894minus1

1=

120582119899minus119894

1+ ℎ(119860

1+ 120573(119868

119894+1119873)120582

119899minus119894

2+ (120582119899minus119894+119898


1)120594[0119905119891minus120591]

(119905119899minus119894)119906119894)

1 + ℎ(119889 + 120573(119868119894+1119873))

120582119899minus119894minus1

2=120582119899minus119894

2+ ℎ(119860

2+ 120582119899minus119894


1120573(119878119894+1119873))

1 + ℎ(120574 + 119889 + 120576 minus 120573(119878119894+1119873))

120582119899minus119894minus1

3=120582119899minus119894

3minus 1198603ℎ

1 + 119889ℎ

119879119894+1=(120582119899minus119894minus1+119898


3)

1198604

120594[0119905119891minus120591]

(119905119894+1)119878119894+1

119906119894+1= min (119887max (0 119879119894+1))


119894) = 119877119894 119906lowast(119905

119894) = 119906119894

End for

Algorithm 1

0

2

0 30 60 90 120 150 1800

5

Time (days)

times104 times10

6

With

out c

ontro

lI(t)

With

cont

rol

I(t)


0 30 60 90 120 150 1800

05

1

15

2

Time (days)

times104

Infe

cted

I(t)



0 30 60 90 120 150 1800

051

152

253

354

455

Time (days)

times106

Infe

cted

I(t)







0 30 60 90 120 150 1800

05

1

15

2

25

3

Time (days)

times107

Rem

oved

R(t)



0 30 60 90 120 150 1800

001

002

003

004

005

006

Time (days)

Con

trol

u(t)


0 30 60 90 120 150 1800

05

1

15

Time (days)

Cos

t

times10minus3



5 Conclusion


Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003816100381610038161003816

le 1198721

10038161003816100381610038161198831 (119905) minus 1198832 (119905)1003816100381610038161003816

+ 1198722

10038161003816100381610038161198831120591 (119905) minus 1198832120591 (119905)1003816100381610038161003816

(9)

where1198721and119872



1003816100381610038161003816 =10038161003816100381610038161198781 (119905) minus 1198782 (119905)

1003816100381610038161003816

+10038161003816100381610038161198681 (119905) minus 1198682 (119905)

1003816100381610038161003816 +10038161003816100381610038161198771 (119905) minus 1198772 (119905)

1003816100381610038161003816

10038161003816100381610038161198831120591 (119905) minus 1198832120591 (119905)1003816100381610038161003816 =

10038161003816100381610038161198781120591 (119905) minus 1198782120591 (119905)1003816100381610038161003816

+10038161003816100381610038161198681120591 (119905) minus 1198682120591 (119905)

1003816100381610038161003816 +10038161003816100381610038161198771120591 (119905) minus 1198772120591 (119905)

1003816100381610038161003816

(10)

Here

119878119894120591 (119905) = 119878119894 (119905 minus 120591) 119868

119894120591 (119905) = 119868119894 (119905 minus 120591)

119877119894120591(119905) = 119877

119894(119905 minus 120591) for 119894 = 1 2

(11)


1003816100381610038161003816 le 119872 (10038161003816100381610038161198831 minus 1198832

1003816100381610038161003816 +10038161003816100381610038161198831120591 minus 1198832120591

1003816100381610038161003816)

(12)

where

119872 = max (1198721+ 119860 1198722) lt infin (13)






119869 (119906) = int

119905119891

0

1198601119878 (119905) + 119860

2119868 (119905) minus 119860

3119877 (119905) +

1198604

21199062(119905) 119889119905

(14)



119869 (119906lowast) = min 119869 (119906) 119906 isin U (15)


U = 119906 0 le 119906 le 119887 lt 1 119905 isin [0 119905119891]




119869 (119906lowast) = min119906isinU

119869 (119906) (17)








119871 (119878 119868 119877 119906) ge 1198882+ 1198881(|119906|2)1205882

(18)



119871 (119878 119868 119877 119906) ge 1198882+ 1198881(|119906|2)1205882

(19)


119867 = 1198601119878 (119905) + 119860

2119868 (119905) minus 119860

3119877 (119905) +

1198604

21199062(119905) +

3

sum

119894=1

120582119894119891119894

(20)


state variable



1 1205822 and 120582

3satisfying

1= minus 119860

1+ 1205821119889 + (120582

1minus 1205822) 120573

119868lowast

119873lowast

+ 120594[0119905119891minus120591]

(119905) (120582+

1minus 120582+

3) 119906lowast

2= minus119860

2+ (1205821minus 1205822) 120573

119878lowast

119873lowast+ (119889 + 120574 + 120598) 120582

2minus 1205741205823

3= 1198603+ 1205823119889

(21)


1205821(119905119891) = 1205822(119905119891) = 1205823(119905119891) = 0 (22)


119906lowast(119905) = min(119887max(0

(120582+

1minus 120582+

3)

1198604

120594[0119905119891minus120591]

(119905) 119878lowast)) (23)

where 120582+119894(119905) = 120582

119894(119905 + 120591) for 119894 = 1 3


1= minus

120597119867 (119905)

120597119878 (119905)minus 120594[0119905119891minus120591]

(119905)120597119867 (119905 + 120591)

120597119878 (119905 minus 120591) 120582

1(119905119891) = 0

2= minus

120597119867

120597119868 120582

2(119905119891) = 0

3= minus

120597119867

120597119877 120582

3(119905119891) = 0

(24)


120597119867

120597119906+ 120594[0119905119891minus120591]

(119905)120597119867 (119905 + 120591)

120597119906120591

= 0 (25)

That is

1198604119906 + 120594[0119905119891minus120591]

(119905) (120582+

3minus 120582+

1) 119878 = 0 (26)




0= 0 and 119905

119891



0 30 60 90 120 150 1800

05

1

15

2

25

3

Time (days)


Susc

eptib

les

S(t)

times107






minus1lt 1199050= 0 lt 119905


119899

= 119905119891lt 119905119899+1


)

(27)




119894 119868119894 119877119894 1205821198941 1205821198942

120582119894

3 and 119906




120573 = 03095 Λ = 117417

119889 = 39139 times 10minus5 120574 = 02

120576 = 00063 120591 = 10

(28)


119878 (0) = 30 times 106 119868 (0) = 30 119877 (0) = 28 (29)


120582119894(119905119891) = 0 (119894 = 1 3) (30)



Step 1 For 119894 = minus119898 0 do 119878119894= 1198780 119868119894= 1198680 119877119894= 1198770 119906119894= 0

End forFor 119894 = 119899 119899 + 119898 do 120582119894

1= 0 120582119894

2= 0 120582119894

3= 0


119878119894+1=119878119894+ ℎ(Λ minus 119906

119894minus119898119878119894minus119898)

1 + ℎ(120573(119868119894119873) + 119889)

119868119894+1=

119868119894

1 + ℎ(120574 + 119889 + 120576 minus 120573(119878119894+1119873))

119877119894+1=119877119894+ ℎ(120574119868

119894+1+ 119906119894minus119898119878119894minus119898)

1 + ℎ119889

120582119899minus119894minus1

1=

120582119899minus119894

1+ ℎ(119860

1+ 120573(119868

119894+1119873)120582

119899minus119894

2+ (120582119899minus119894+119898


1)120594[0119905119891minus120591]

(119905119899minus119894)119906119894)

1 + ℎ(119889 + 120573(119868119894+1119873))

120582119899minus119894minus1

2=120582119899minus119894

2+ ℎ(119860

2+ 120582119899minus119894


1120573(119878119894+1119873))

1 + ℎ(120574 + 119889 + 120576 minus 120573(119878119894+1119873))

120582119899minus119894minus1

3=120582119899minus119894

3minus 1198603ℎ

1 + 119889ℎ

119879119894+1=(120582119899minus119894minus1+119898


3)

1198604

120594[0119905119891minus120591]

(119905119894+1)119878119894+1

119906119894+1= min (119887max (0 119879119894+1))


119894) = 119877119894 119906lowast(119905

119894) = 119906119894

End for

Algorithm 1

0

2

0 30 60 90 120 150 1800

5

Time (days)

times104 times10

6

With

out c

ontro

lI(t)

With

cont

rol

I(t)


0 30 60 90 120 150 1800

05

1

15

2

Time (days)

times104

Infe

cted

I(t)



0 30 60 90 120 150 1800

051

152

253

354

455

Time (days)

times106

Infe

cted

I(t)







0 30 60 90 120 150 1800

05

1

15

2

25

3

Time (days)

times107

Rem

oved

R(t)



0 30 60 90 120 150 1800

001

002

003

004

005

006

Time (days)

Con

trol

u(t)


0 30 60 90 120 150 1800

05

1

15

Time (days)

Cos

t

times10minus3



5 Conclusion


Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1 1205822 and 120582

3satisfying

1= minus 119860

1+ 1205821119889 + (120582

1minus 1205822) 120573

119868lowast

119873lowast

+ 120594[0119905119891minus120591]

(119905) (120582+

1minus 120582+

3) 119906lowast

2= minus119860

2+ (1205821minus 1205822) 120573

119878lowast

119873lowast+ (119889 + 120574 + 120598) 120582

2minus 1205741205823

3= 1198603+ 1205823119889

(21)


1205821(119905119891) = 1205822(119905119891) = 1205823(119905119891) = 0 (22)


119906lowast(119905) = min(119887max(0

(120582+

1minus 120582+

3)

1198604

120594[0119905119891minus120591]

(119905) 119878lowast)) (23)

where 120582+119894(119905) = 120582

119894(119905 + 120591) for 119894 = 1 3


1= minus

120597119867 (119905)

120597119878 (119905)minus 120594[0119905119891minus120591]

(119905)120597119867 (119905 + 120591)

120597119878 (119905 minus 120591) 120582

1(119905119891) = 0

2= minus

120597119867

120597119868 120582

2(119905119891) = 0

3= minus

120597119867

120597119877 120582

3(119905119891) = 0

(24)


120597119867

120597119906+ 120594[0119905119891minus120591]

(119905)120597119867 (119905 + 120591)

120597119906120591

= 0 (25)

That is

1198604119906 + 120594[0119905119891minus120591]

(119905) (120582+

3minus 120582+

1) 119878 = 0 (26)




0= 0 and 119905

119891



0 30 60 90 120 150 1800

05

1

15

2

25

3

Time (days)


Susc

eptib

les

S(t)

times107






minus1lt 1199050= 0 lt 119905


119899

= 119905119891lt 119905119899+1


)

(27)




119894 119868119894 119877119894 1205821198941 1205821198942

120582119894

3 and 119906




120573 = 03095 Λ = 117417

119889 = 39139 times 10minus5 120574 = 02

120576 = 00063 120591 = 10

(28)


119878 (0) = 30 times 106 119868 (0) = 30 119877 (0) = 28 (29)


120582119894(119905119891) = 0 (119894 = 1 3) (30)



Step 1 For 119894 = minus119898 0 do 119878119894= 1198780 119868119894= 1198680 119877119894= 1198770 119906119894= 0

End forFor 119894 = 119899 119899 + 119898 do 120582119894

1= 0 120582119894

2= 0 120582119894

3= 0


119878119894+1=119878119894+ ℎ(Λ minus 119906

119894minus119898119878119894minus119898)

1 + ℎ(120573(119868119894119873) + 119889)

119868119894+1=

119868119894

1 + ℎ(120574 + 119889 + 120576 minus 120573(119878119894+1119873))

119877119894+1=119877119894+ ℎ(120574119868

119894+1+ 119906119894minus119898119878119894minus119898)

1 + ℎ119889

120582119899minus119894minus1

1=

120582119899minus119894

1+ ℎ(119860

1+ 120573(119868

119894+1119873)120582

119899minus119894

2+ (120582119899minus119894+119898


1)120594[0119905119891minus120591]

(119905119899minus119894)119906119894)

1 + ℎ(119889 + 120573(119868119894+1119873))

120582119899minus119894minus1

2=120582119899minus119894

2+ ℎ(119860

2+ 120582119899minus119894


1120573(119878119894+1119873))

1 + ℎ(120574 + 119889 + 120576 minus 120573(119878119894+1119873))

120582119899minus119894minus1

3=120582119899minus119894

3minus 1198603ℎ

1 + 119889ℎ

119879119894+1=(120582119899minus119894minus1+119898


3)

1198604

120594[0119905119891minus120591]

(119905119894+1)119878119894+1

119906119894+1= min (119887max (0 119879119894+1))


119894) = 119877119894 119906lowast(119905

119894) = 119906119894

End for

Algorithm 1

0

2

0 30 60 90 120 150 1800

5

Time (days)

times104 times10

6

With

out c

ontro

lI(t)

With

cont

rol

I(t)


0 30 60 90 120 150 1800

05

1

15

2

Time (days)

times104

Infe

cted

I(t)



0 30 60 90 120 150 1800

051

152

253

354

455

Time (days)

times106

Infe

cted

I(t)







0 30 60 90 120 150 1800

05

1

15

2

25

3

Time (days)

times107

Rem

oved

R(t)



0 30 60 90 120 150 1800

001

002

003

004

005

006

Time (days)

Con

trol

u(t)


0 30 60 90 120 150 1800

05

1

15

Time (days)

Cos

t

times10minus3



5 Conclusion


Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Step 1 For 119894 = minus119898 0 do 119878119894= 1198780 119868119894= 1198680 119877119894= 1198770 119906119894= 0

End forFor 119894 = 119899 119899 + 119898 do 120582119894

1= 0 120582119894

2= 0 120582119894

3= 0


119878119894+1=119878119894+ ℎ(Λ minus 119906

119894minus119898119878119894minus119898)

1 + ℎ(120573(119868119894119873) + 119889)

119868119894+1=

119868119894

1 + ℎ(120574 + 119889 + 120576 minus 120573(119878119894+1119873))

119877119894+1=119877119894+ ℎ(120574119868

119894+1+ 119906119894minus119898119878119894minus119898)

1 + ℎ119889

120582119899minus119894minus1

1=

120582119899minus119894

1+ ℎ(119860

1+ 120573(119868

119894+1119873)120582

119899minus119894

2+ (120582119899minus119894+119898


1)120594[0119905119891minus120591]

(119905119899minus119894)119906119894)

1 + ℎ(119889 + 120573(119868119894+1119873))

120582119899minus119894minus1

2=120582119899minus119894

2+ ℎ(119860

2+ 120582119899minus119894


1120573(119878119894+1119873))

1 + ℎ(120574 + 119889 + 120576 minus 120573(119878119894+1119873))

120582119899minus119894minus1

3=120582119899minus119894

3minus 1198603ℎ

1 + 119889ℎ

119879119894+1=(120582119899minus119894minus1+119898


3)

1198604

120594[0119905119891minus120591]

(119905119894+1)119878119894+1

119906119894+1= min (119887max (0 119879119894+1))


119894) = 119877119894 119906lowast(119905

119894) = 119906119894

End for

Algorithm 1

0

2

0 30 60 90 120 150 1800

5

Time (days)

times104 times10

6

With

out c

ontro

lI(t)

With

cont

rol

I(t)


0 30 60 90 120 150 1800

05

1

15

2

Time (days)

times104

Infe

cted

I(t)



0 30 60 90 120 150 1800

051

152

253

354

455

Time (days)

times106

Infe

cted

I(t)







0 30 60 90 120 150 1800

05

1

15

2

25

3

Time (days)

times107

Rem

oved

R(t)



0 30 60 90 120 150 1800

001

002

003

004

005

006

Time (days)

Con

trol

u(t)


0 30 60 90 120 150 1800

05

1

15

Time (days)

Cos

t

times10minus3



5 Conclusion


Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 30 60 90 120 150 1800

05

1

15

2

25

3

Time (days)

times107

Rem

oved

R(t)



0 30 60 90 120 150 1800

001

002

003

004

005

006

Time (days)

Con

trol

u(t)


0 30 60 90 120 150 1800

05

1

15

Time (days)

Cos

t

times10minus3



5 Conclusion


Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Optimal Control of an SIR Model with ...downloads.hindawi.com/archive/2013/403549.pdf · Hindawi Publishing Corporation ISRN Biomathematics Volume , Article ID, pages