Dynamic Behavior of Leptospirosis Disease with Saturated ...Muhammad Altaf Khan1 · Syed Farasat Saddiq2 ... Abdul Wali Khan, University Mardan, Mardan, Khyber Pakhtunkhwa, Pakistan

Int. J. Appl. Comput. Math (2016) 2:435–452DOI 10.1007/s40819-015-0102-2

INVITED PAPER

Dynamic Behavior of Leptospirosis Disease withSaturated Incidence Rate

Muhammad Altaf Khan1 · Syed Farasat Saddiq2 ·Saeed Islam1 · Ilyas Khan3 · Sharidan Shafie4

Published online: 25 November 2015© Springer India Pvt. Ltd. 2015

Abstract Leptospirosis is a tropical disease and found almost in globe. Human as well asmammals are mostly infected from this disease. This work presents a mathematical studyof the leptospirosis disease with saturated incidence rate. Initially, we present the modelformulation and their fundamental properties. Then we find the local stability of the diseasefree and endemic equilibrium. The disease free equilibrium is stable both locally and globallywhen R0 < 1. Further, we find that endemic equilibrium is stable locally and globally ifR0 > 1. The numerical results are shown for analytical results.

Keywords Leptospirosis · Saturated incidence · Basic reproduction number · Globalstability · Numerical results

Mathematics Subject Classification 92D25 · 49J15 · 93D20

Introduction

Leptospirosis is a worldwide zoonotic disease which is commonly occurs in tropical andsubtropical regions. Themagnitude of the problem in the tropics and subtropicsmaybe largelynot only because of their climatic and environmental conditions, but also likely relationship

B Saeed [email protected]

Muhammad Altaf [email protected]

1 Department of Mathematics, Abdul Wali Khan, University Mardan, Mardan, Khyber Pakhtunkhwa,Pakistan

2 Department of Mathematics, Islamia College University Peshawar, Peshawar, KhyberPakhtunkhwa, Pakistan

3 College of Engineering Majmaah University, Majmaah, Kingdom of Saudi Arabia

4 Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Skudai,Johor, Malaysia

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436 Int. J. Appl. Comput. Math (2016) 2:435–452

with the caused leptospira contaminated environment, such as local agricultural practicesand poor housing and disposal, all what will be many sources of infection [1,2]. In tropicalcountries, people are always risky, especially if their contact with potentially infected areas,rodents and pets in a wet and hot climate. In template climate, the cattle, pits and dogs are thecarriers for important source of leptospirosis [3]. Valuable information about the risk factorof leptospirosis disease can be found in [4].

Tobetter understand the epidemiologyof an infectious disease,mathematicalmodelinghasplayed an important role [5–8]. These models provide us the quantitative descriptions of thecomplicated, non linear process of disease transmission and help us to obtain inside into thedynamics of the disease and we are able to make such decision for public health policy. Manymathematical models [9–11] have been proposed to represent the compartmental dynamicsof both susceptible, infected and recovered human and vector population.

Various mathematical studies have been proposed on leptospirosis disease. Here, we givean overview. Triampo et al. presented a mathematical model for the transmission of lep-tospirosis disease in [12]. In their work, they considered a number of leptospirosis diseasein Thailand and shown their numerical simulations. Zaman [13] considered the real datapresented in [12] to studied the dynamical behavior and role of optimal control theory of thisdisease. Pongsumpun et al. [14] developed mathematical model to study the behavior of lep-tospirosis disease. In their work, they represent the rate of change for both the vector (rats)and human population. They divided the human population further into two main groupsJuveniles and adults. Moreover, we refer the reader to see [15–17] and the references therein.

In epidemic models a variety of non linear incidence rate have been used [18–20]. Liu etal. [21] showed that the case where hosts can exhibit prolonged immunity to infection is notlinear incidence rate of infection could greatly expand the breadth dynamics caused by thisdisease. The saturated incidence was first time used by Capasso and Serio [20] in choleraepidemic model.

In this work, we present analysis of nonlinear incidence in leptospirosis epidemic model.Here, we study the model presented in [22]. In [22], the authors used the standard optimalcontrol technique Pontryagin’s Maximum Principle and used three control variables for pos-sible eradication of the infection in the community. We study the model [22], to obtain itsstability analysis on behalf of reproduction number. First, we find the threshold/ basic repro-duction number for the model and then show that the Disease free equilibrium is locally aswell as globally asymptotically stable for R0 < 1 and unstable for R0 > 1. Further, we findthat an endemic equilibrium is locally as well as globally asymptotically stable for R0 > 1.The global stability of an endemic equilibrium is shown by the geometric approach method.We solve, the proposed model numerically and the results are presented in the form of plotsfor justification purpose. Finally, the conclusion and references are presented.

This paper is organized as follows: section “Mathematical Model” is devoted to the math-ematical formulation of the model. We discuss the equilibria and local stability of both thedisease-free and endemic equilibrium in section “Equilibria and Local Stability”. In section“Global Stability”, we show the global stability of both the disease-free and endemic equilib-rium. In section “Numerical Simulations” the theoretical results are demonstrated in the formof numerical graphics. Finally, we conclude our work by conclusion in section “Conclusion”.

Mathematical Model

In this section, we present the model formulation of the leptospirosis disease with theirinteraction with human population. The human population denoted by Nh(t), is subdivided

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Int. J. Appl. Comput. Math (2016) 2:435–452 437

into three compartments; Sh(t)-Susceptible individuals, Ih(t)-infected individuals (thosewhohave been infected) and Rh(t)-recovered individuals(those who recovered from infection orremoved), so

Nh(t) = Sh(t) + Ih(t) + Rh(t). (1)

The population of susceptible is increased by recruitment of the individuals (which is assumedsusceptible), at the rate of bh . The susceptible population decreased by following the effectivecontact with the infected individuals and vectors (in the Ih and Iv) at a rate λ(t), where

λ(t) =(

β2 Iv1 + α1 Iv

+ β1 Ih1 + α2 Ih

). (2)

In (2),β2 andβ1 are the effective contact rate (capable of leading to infection). The populationof susceptible is further decreased with the natural deathμh , while increased with the numberof individuals, which are susceptible again at a rate of λh . So, we can write, the rate of changeof the susceptible population given by

dShdt

= bh − μh Sh − Shλ(t) + λh Rh . (3)

The population of infected individuals is generated by the infection of susceptible individuals(at a rate λ(t)). The population of infected individuals is decreased by the natural mortalityrate μh , disease induced death rate δh and from the recovery rate of infection γh . So, we canwrite their rate of change for infected population given by

d Ihdt

= Shλ(t) − (μh + δh + γh)Ih . (4)

The population of recovered individuals is generated by the rate of recovery γh (from infectedclass), while decreased by the natural death rateμh and λh (those individuals who susceptibleagain). Thus, their rate of change can be expressed as follows:

dRh

dt= γh Ih − (μh + λh)Rh . (5)

The vector population is denoted by Nv(t), is subdivided into two classes, Sv(t)-susceptiblevector and Iv(t)-infected vector, so

Nv(t) = Sv(t) + Iv(t). (6)

The population of susceptible vector is generated by the recruitment of the vector at a rateof bv . The population of susceptible is decreased by following the effective contact withinfected individuals Ih (at a rate of λ(t)), where λ(t) = β3 Ih/1+α2 Ih and β3 is the effectivecontact rate that leads to infection. Also, the population of susceptible vector is decreased bythe natural death rate at a rate γv . Thus, the rate of change for the susceptible vector is givenby

dSv

dt= bv − γvSv − λ(t)Sv. (7)

The population of infected vector is generated by the infection of susceptible vector at rateof λ(t), while decreased by the natural death γv and disease related death rate δv . Their rateof change can be written as:

d Ivdt

= λ(t)Sv − γv Iv − δv Iv. (8)

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Human Popula�on Vector Popula�on

hb h hSμ vb

v vSγ

2

11h v

v

S II

βα+

1

21h h

h

S II

βα+

3

21v h

h

S II

βα+

h hIμv vIγ

v vSγ

h hIδ

h hRλ h hIγ v vIδ

h hRμ

Sh

Ih Iv

Sv

Rh

Fig. 1 The flow diagram of human and vector interaction

The complete transfer flow diagram of human and vector population is depicted in Fig. 1.Thus, the system for the transmission dynamics of the leptospirosis disease is given by thefollowing nonlinear system of differential equations:

dShdt

= bh − μh Sh − Shλ(t) + λh Rh, Sh(0) = S0 ≥ 0,

d Ihdt

= Shλ(t) − (μh + δh + γh)Ih, Ih(0) = I0 ≥ 0,

dRh

dt= γh Ih − (μh + λh)Rh, Rh(0) = R0 ≥ 0,

dSv

dt= bv − γvSv − λ(t)Sv, Sv(0) = S0 ≥ 0,

d Ivdt

= λ(t)Sv − γv Iv − δv Iv, Iv(0) = I0 ≥ 0,

(9)

Subject to nonnegative initial conditions.One infected human will shed leptospirosis in their urine during and after the illness of

the period, and thus may pose a risk of infection to others but only in a certain ways. Chiefsocial interaction is perfectly safe, as airborne bacteria. Saliva is not consider high risky,because the bacteria cannot be tolerate the acidity of the human mouth for a long time, soeven we advise against the risk of sharing food, cups or cutlery very small. Items that candry out between uses, such as towels, also very low risk once they dry, but handle very bloodsoaked clothing, wet bed or similar may present risks, for detail see the reference [23]. LetN (h) = Sh + Ih + Rh , shows the total dynamics of human population at time t , is given by

dNh

dt= bh − μh Nh − δh Ih ≤ bh − μh Nh,

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N (v) = Sv + Iv , represent the total population of vector at time t , given by

dNv

dt= bv − Nvγv − δv Iv ≤ bv − Nv.

The feasible region for the system (9) is

� =(

(Sh, Ih, Rh, Sv, Iv) ∈ R5+,

(Nh ≤ bh

μh, Nv ≤ bv

γv

)).

It is easy to prove that the set � is positively invariant with respect to system (9).

Lemma 2.1 Let the initial data Sh(0) > 0, Ih(0) > 0, Rh(0) > 0, Sv(0) > 0 and Iv(0) > 0.Then the solution of Sh(t), Ih(t), Rh(t), Sv(t) and Iv(t) corresponding to model (9), are non-negative ∀ time t > 0.

Proposition 2.1 Let (Sh, Ih, Rh, Sv, Iv), be the solution of the system (9) with non-negativeinitial conditions and closed set� = ((Sh, Ih, Rh, Sv, Iv) ∈ R5+, Nh ≤ bh

μh, Nv ≤ bv

γv), then � is positively invariant and

attracting under the flow described by system (9).

Proof Consider the Lyapunov function

M(t) = (Nh(t), Nv(t)) = (Sh + Ih + Rh, Sv + Iv). (10)

The time derivative of Eq. (10) is

dM

dt= (bh − μh Nh − δh Ih, bv − γvNv − δv Iv). (11)

Now it is easy to prove that

dNh

dt≤ bh − μh Nh ≤ 0 f or Nh ≥ bh

μh,

dNv

dt≤ bv − γvNv ≤ 0 f or Nv ≥ bv

γv

. (12)

Thus, it follows that dMdt ≤ 0 which implies that � is positively invariant set. Also a standard

comparison theorem [24] is used to show that

0 ≤ (Nh, Nv) ≤ ((Nh(0)e−μh t + bh

μh(1 − e−μh t ), Nv(0)e

−γv t + bv

γv

(1 − e−γv t )).

Thus as t → ∞, 0 ≤ (Nh, Nv) ≤ (bhμh

, bv

γv) and so � is an attracting and positive invariant

set. �

Equilibria and Local Stability

To obtain the endemic equilibria of the system (9), setting the left side of the system (9) equalto zero, we get ⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

S∗h = P2(1+α2 I ∗

h )(P1(γv(1+α2 I ∗h )+β3 I ∗

h )+α1β3bv I ∗h )

β2β3bv(1+α2 I ∗h )+β1(P1(γv(1+α2 I ∗

h )+β3 I ∗h )+α1β3bv I ∗

h ),

R∗h = γh I ∗

h(μh+δh)

,

S∗v = bv(1+α2 I ∗

h )

(γv(1+α2 I ∗h )+β3 I ∗

h )

I ∗v = β3bv I ∗

h(γv+δv)(γv(1+α2 I ∗

h )+β3 I ∗h )

,

123


where P1 = (γv + δv), P2 = (μh + δh + γh), P3 = (μh + λh).The disease free equilibrium of the system (9) at E0 is

S0h = bhμh

, S0v = bv

γv

.

The basic reproduction number R0 for system (9) is

R0 = bhβ2β3bv + β1(γv + δv)bhγv

μh(γv + δv)(μh + δh + γh)γv

.

Disease Free Local Stability

Theorem 3.1 For R0 < 1, the DFE of the model (9) at E0 is stable locally asymptoticallyif ((μh + δh + γh) − β1bh/μh) > 0, otherwise unstable.

Proof To show the local stability of the disease free equilibrium, we setting the left side ofthe system (9), equating to zero, we get the following jacobian matrix Jo

Jo =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−μh −β1Soh λh 0 −β2Soh

0 β1Soh − (μh + δh + γh) 0 0 β2Soh

0 γh −(μh + λh) 0 0

0 −β3Sov 0 −γv 0

0 β3Sov 0 0 −(γv + δv)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

The characteristics equation to the Jacobian matrix Jo is follows as

(μh + λ)(γv + λ)(λ3 + a1λ2 + a2λ + a3) = 0, (13)

where,

a1 = (μh + λh) + (γv + δv) + (μh + δh + γh) − β1bhμh

,

a2 = (μh + λh)((μh + δh + γh) − β1bh

μh

)+ (μh + λh)(γv + δv)

+(μh + δh + γh)(γv + δv)(1 − R0),

a3 = (μh + λh)(μh + δh + γh)(γv + δv)(1 − R0). (14)

Clearly, the two roots of the characteristics equation (14) are−μh and−γv have negative realparts. The other roots can be determined from the cubic terms of the characteristics equation(14). Direct calculations by using (15), we get

a1a2 − a3 =[(μh + λh) + (γv + δv) + (μh + δh + γh) − β1bh

μh

]

×[(μh + λh)

{(γv + δv) +

((μh + δh + γh) − β1bh

μh

)}]

+(γv + δv)((μh + δh + γh) − β1bh

μh

)(γv + δv)(μh + δh + γh)(1 − R0) > 0.

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It follows from Routh-Hurwitz crietria, ai > 0 for i = 1, 2, 3 and a1a2 − a3 > 0. Thus,all the eigenvalues of the system (9) have negative real parts if R0 < 1 and ((μh + δh +γh) − β1bh/μh) > 0. Therefore, the system (9) at the disease free equilibrium E0 is locallyasymptotically stable, if R0 < 1 and ((μh + δh + γh) − β1bh/μh) > 0. �

The stability of the Disease free equilibrium of the system (9) for R0 < 1 means that thedisease dies out from the community. In this case the endemic equilibrium does not existsbut we are interesting to find the properties of the model about the endemic equilibrium pointE1, when R0 > 1. In the following, we show that the endemic equilibrium point E1 of themodel (9) is locally asymptotically stable, when R0 > 1.

Local Stability of Endemic Equilibrium

In this subsection, we discuss the local stability of the model (9) at E1, by setting the lefthand side of the system (9), equal to zero, we get the Jacobian matrix J ∗ in the followingtheorem.

Theorem 3.2 For R0 > 1, the endemic equilibrium point E1 of the system (9) is locallyasymptotically stable if condition (18) and the terms under braces are positive, otherwiseunstable.

Proof The Jacobian matrix J ∗ of the system (9) at E1 is given by

J ∗ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−μh − β2 I ∗v

1+α1 I ∗v

− β1 I ∗h

1+α2 I ∗h

− β1S∗h

(1+α2 I ∗h )2

λh 0 − β2S∗h

(1+α1 I ∗v )2

β2 I ∗v

1+α1 I ∗v

+ β1 I ∗h

1+α2 I ∗h

β1S∗h

(1+α2 I ∗h )2

− (μh + δh + γh) 0 0β2S∗

h(1+α1 I ∗

v )2

0 γh −(μh + λh) 0 0

0 − β3S∗v

(1+α2 I ∗h )2

0 −γv − β3 I ∗h

1+α2 I ∗h

0

0 β3S∗v

(1+α2 I ∗h )2

0β3 I ∗

h1+α2 I ∗

h−(γv + δv)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(15)The characteristics equation of (15) is given by

λ5 + A1λ4 + A2λ

3 + A3λ2 + A4λ + A5 = 0, (16)

where

A1 = B4 + (μh + λh) − (B1 + B3)︸︷︷︸+(γv + δv),

A2 = B4(γv + δv) + B5B6 + ((μh + λh) − (B1 + B3)︸︷︷︸)(B4 + (γv + δv))

+[(B2B7 − B1B3) − (B3 + B1)(μh + λh)]︸︷︷︸,A3 = (μh + λh)[B2B7 − B1B3] − B2γhλh︸︷︷︸+[(B2B7 − B1B3) − (B3 + B1)(μh + λh)]︸︷︷︸

×[B4 + (γv + δv)] + [(μh+λh) − (B1+B3)︸︷︷︸]B4(γv+δv)+[(2μh+λh)+γv]B5B6,

123


A4 = {(μh + λh)[B2B7 − B1B3] − B2γhλh}︸︷︷︸[B4 + (γv + δv)]

+B4(γv + δv) [(B2B7 − B1B3) − (B3 + B1)(μh + λh)]︸︷︷︸+[(2μh + λh)γv + μh(μh + λh)]B5B6,

A5 = B4(γv + δv) {(μh + λh)[B2B7 − B1B3] − B2γhλh}︸︷︷︸+B5B6μh(μh + λh)γv, (17)

and

B1 = μh + β1 Ih1 + α2 Ih

+ β2 Iv1 + α1 Iv

, B2 = β1 Ih1 + α2 Ih

+ β2 Iv1 + α1 Iv

,

B3 = β1S∗h

(1 + α2 Ih)2− (μh + δh + γh),

B4 = (γv + β3 Ih1 + α2 Ih

), B5 = β3Sv

(1 + α2 Ih)2, B6 = β2Sh

(1 + α1 Iv)2,

B7 = β1Sh(1 + α2 Ih)2

, B8 = β3 Ih(1 + α2 Ih)

The Routh-Hurwitz criteria for (16) is follows as

H5 =

∣∣∣∣∣∣∣∣∣∣

A1 A3 A5 0 01 A2 A4 0 00 A1 A3 A5 00 1 A2 A4 00 0 A1 A3 A5

∣∣∣∣∣∣∣∣∣∣= A1A4A5(A2A3 − A1A4) − A1A

22A

25

+A1A4A25 − A4A5(A

23 − A1A5) + A2A3A

25 − A2

5 > 0 (18)

The eigenvalues of the characteristics equations (16) have negative real parts if Ai > 0 fori = 1, 2, 3, 4, 5, H5 > 0 and R0 > 1 and the terms under braces in positive. Thus, it followsfromRouth-Hurtwiz criteria that the system (9) at the endemic equilibrium point E1 is locallyasymptotically stable, if R0 > 1 and the terms under braces are positive and the conditions(18) are satisfied. �

Global Stability

In this section, we investigate the global stability of the system (9) at the disease free equi-librium E0 and the endemic equilibrium at E2. In the following theorem, we first show thedisease free global stability for the case (λh = 0).

Theorem 4.1 The system (9) at E0 is globally asymptotically stable if R0 < 1, otherwiseunstable.

Proof In order to show this result, we construct the following lyapanove function:

L(t) = (γv + δv)(Sh − S0h − S0h log

ShS0h

)+ (γv + δv)Ih + β2bh

μh

(Sv − S0v − S0v log

Sv

S0v

)

+β2bhμh

Iv. (19)

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Taking the time derivative of (19) along the solutions of system (9), we obtain

L ′(t) = (γv + δv)( Sh − S0h

Sh

)[bh − μh Sh − (

β1 Ih1 + α2 Ih

+ β2 Iv1 + α1 Iv

)Sh]

+(γv + δv)[( β1 Ih1 + α2 Ih

+ β2 Iv1 + α1 Iv

)Sh − (μh + δh + γh)Ih]

+β2bhμh

( Sv − S0vSv

)[bv − γvSv − β3Sv Ih

1 + α2 Ih] + β2bh

μh[ β3Sv Ih1 + α2 Ih

−(γv + δv)Iv]. (20)

Making use of S0h = bhμh

and S0v = bv

γvin equation (20), and taking some arrangements, we

get

L ′(t) = −μh(γv + δv)(Sh − S0h )

2

Sh− γv

β2bhμh

(Sv − S0v )2

Sv

−(γv + δv)(μh + δh + γh)α2Ih2

1 + α2 Ih

− (γv + δv)β2bhα1

μh

Iv2

1 + α1 Iv− Ih

1 + α2 Ih(γ + δv)(μh + δh + γh)(1 − R0). (21)

L ′(t) is negative if R0 < 1 and L ′(t) = 0 if Sh = S0h , Sv = S0v , Ih = Iv = 0.Hence, the largestcompact invariant set (Sh, Ih, Rh, Sv, Iv) ∈ � : L ′(t) = 0 is the singleton set E0, where E0

is the disease free equilibrium. Thus, by Principle [25], E0 is globally asymptotically stablein �. �

To prove that the endemic equilibrium point E1 is globally asymptotically stable, wereduce the system (9), by using Rh = Nh − Sh − Ih in first equation of the system (9)and Sv = bv−(γv+δv)Iv

γvin fifth equation of the system (9), we obtain the following reduced

system:

dShdt

= bh − μh Sh − Sh

(β2 Iv

1 + α1 Iv+ β1 Ih

1 + α2 Ih

)+ λh(Nh − Sh − Ih),

d Ihdt

= Sh

(β2 Iv

1 + α1 Iv+ β1 Ih

1 + α2 Ih

)− (μh + δh + γh)Ih,

d Ivdt

= β3 Ih(bv − (γv + δv)Iv)

γv(1 + α2 Ih)− (γv + δv)Iv, (22)

subject to nonnegative initial conditions

Sh = Sh(0) ≥ 0, Ih = Ih(0) ≥ 0, Iv = Iv(0) ≥ 0.

The endemic equilibrium of the system (22) is denoted by E2.To prove the global stability of the endemic equilibrium, we first give the following lemma.

Lemma If the model dxdt = g(x), where g : D −→ Rn, posses a unique equilibrium x∗

and also a compact absorbing set exists for x∗, then x∗ is stable globally asymptoticallygiven that the functionP(x) and a Lozinskii measure � exist such that q = limitt−→∞ supsupx

1t

∫ t0 �(B(x(s, x)))ds < 0 [26]. The symbols P, � and B will stated in the following

result.

Theorem 4.2 If R0 > 1, then the system (22) is globally asymptotically stable at E2.

123


Proof At E2, the Jacobian matrix J (E2) is given by

J (E2) =

⎛⎜⎜⎜⎜⎝

−μh − β2 Iv1+α1 Iv

− β1 Ih1+α2 Ih

− λh − β1Sh(1+α2 Ih )2

− λh − β2Sh(1+α1 Iv)2

β2 Iv1+α1 Iv

+ β1 Ih1+α2 Ih

β1Sh(1+α2 Ih )2

− (μh + δh + γh)β2Sh

(1+α1 Iv)2

0 β3(bv−(γv+δv)Iv)

γv(1+α2 Ih )2− β3 Ih (γv+δv)

γv(1+α2 Ih )− (γv + δv)

⎞⎟⎟⎟⎟⎠ .

The second additive compound matrix of J (E2) is denoted by J [2] given by

J [2] =⎛⎜⎝

A11β2Sh

(1+α1 Iv)2β2Sh

(1+α1 Iv)2

β3(bv−(γv+δv)Iv)

γv(1+α2 Ih)2A22 − β1Sh

(1+α2 Ih)2− λh

0 β2 Iv1+α1 Iv

+ β1 Ih1+α2 Ih

A33

⎞⎟⎠ ,

where

A11 = −μh − λh − β2 Iv1 + α1 Iv

− β1 Ih1 + α2 Ih

+ β1Sh(1 + α2 Ih)2

− (μh + δh + γh),

A22 = −μh − λh − β2 Iv1 + α1 Iv

− β1 Ih1 + α2 Ih

− β3(γv + δv)Ihγv(1 + α2 Ih)

− (γv + δv),

A33 = β1Sh(1 + α2 Ih)2

− P2 − β3(γv + δv)Ihγv(1 + α2 Ih)

− (γv + δv), (23)

Consider the function

P = P(Sh, Ih, Iv) = diag

(1,

IhIv

,IhIv

)(24)

with P−1 is

P−1 = diag

(1,

IvIh

,IvIh

).

And

Pf = diag

(0,

Iv I ′h − I ′

v Ih

Iv2,Iv I ′

h − I ′v Ih

Iv2

).

Pf P−1 is

Pf P−1 = diag

(0,

I ′h

Ih− I ′

v

Iv,I ′h

Ih− I ′

v

Iv

),

And Pf J [2]P−1 is

P J [2]P−1 =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

A11β2Sh

(1+α1 Iv)2IvIh

β2Sh(1+α1 Iv)2

IvIh

β3(bv−(γv+δv)Iv)

γv(1+α2 Ih)2IhIv

A22 − β1Sh(1+α2 Ih)2

− λh

0 β2 Iv1+α1 Iv

+ β1 Ih1+α2 Ih

A33

⎞⎟⎟⎟⎟⎟⎟⎟⎠

.

So, we write

B = Pf P−1 + P J [2]P−1 =

(B11 B12

B21 B22

),

123


where

B11 = −μh − λh − β2 Iv1 + α1 Iv

− β1 Ih1 + α2 Ih

+ β1Sh(1 + α2 Ih)2

− (μh + δh + γh),

B12 = max

{β2Sh

(1 + α1 Iv)2IvIh

,β2Sh

(1 + α1 Iv)2IvIh

},

B21 =(IhIv

β3(bv − (γv + δv)Iv)

γv(1 + α2 Ih)2, 0

)T

,

B22 =(T11 T12T21 T22

),

where

T11 = I ′h

Ih− I ′

v

Iv− μh − λh − β2 Iv

1 + α1 Iv− β1 Ih

1 + α2 Ih− β3(γv + δv)Ih

γv(1 + α2 Ih)− (γv + δv),

T12 = − β1Sh(1 + α2 Ih)2

− λh,

T21 = β2 Iv1 + α1 Iv

+ β1 Ih1 + α2 Ih

,

T22 = I ′h

Ih− I ′

v

Iv+ β1Sh

(1 + α2 Ih)2− β3(γv + δv)Ih

γv(1 + α2 Ih)− (γv + δv) − (μh + δh + γh).

Suppose the vector (u, v, w) in R3 and its norm ||.|| will be defined as||(u, v, w)|| = max{|u|, |v| + |w|}.

Suppose μB represent Lozinski measure with the above defined norm. So as described in[26], we choose

μ(B) ≤ sup(g1, g2),

where

g1 = μ(B11) + |B12|, g2 = |B21| + μ(B22),

|B21| and |B12| are the matrix norm associated with vector � and μ represent the Lozinskimeasure with respect to � norm, then

μ(B11) = −μh − λh − β2 Iv1 + α1 Iv

− β1 Ih1 + α2 Ih

+ β1Sh(1 + α2 Ih)2

− (μh + δh + γh),

|B12| = max

{β2Sh

(1 + α1 Iv)2IvIh

,β2Sh

(1 + α1 Iv)2IvIh

}, (25)

Therefore,

g1 = μ(B11) + |B12|

= −μh − λh − β2 Iv1+α1 Iv

− β1 Ih1+α2 Ih

+ β1Sh(1+α2 Ih)2

+ β2Sh(1+α1 Iv)2

IvIh

− (μh+δh+γh),

≤ −μh − λh − β2 Iv1+α1 Iv

− β1 Ih1+α2 Ih

+ β1Sh(1+α2 Ih)

+ β2Sh(1+α1 Iv)

IvIh

− (μh + δh + γh),

≤ I ′h

Ih− μh − λh − β2 Iv

1 + α1 Iv− β1 Ih

1 + α2 Ih,

123


using system (22),

I ′h

Ih= Iv

Ih

β2Sh(1 + α1 Iv)

+ β1Sh1 + α2 Ih

− (μh + δh + γh),

we get

g1 ≤ I ′h

Ih− μh − λh − β2 Iv

1 + α1 Iv− β1 Ih

1 + α2 Ih.

Again

|B21| = IhIv


γv(1 + α2 Ih)2,

μ(B22) = Sup{ I ′

h

Ih− I ′

v

Iv− μh − λh − β2 Iv

1 + α1 Iv− β1 Ih

1 + α2 Ih− β3(γv + δv)Ih

γv(1 + α2 Ih)

−(γv + δv) + β2 Iv1 + α1 Iv

+ β1 Ih1 + α2 Ih

,I ′h

Ih− I ′

v

Iv+ β1Sh

(1+α2 Ih)2− β3(γv+δv)Ih

γv(1+α2 Ih)

−(γv + δv) − (μh + δh + γh) − β1Sh(1 + α2 Ih)2

− λh

},

= I ′h

Ih− I ′

v

Iv− μh − λh − (γv + δv) − β3(γv + δv)Ih

γv(1 + α2 Ih)− (μh + δh + γh) − λh .

So

g2 = μ(B22) + |B21|,= I ′

h

Ih− I ′

v


γv(1 + α2 Ih)

−(μh + δh + γh) − λh + IhIv


γv(1 + α2 Ih)2,

≤ I ′h

Ih− I ′

v


γv(1 + α2 Ih)

−(μh + δh + γh) − λh + IhIv


γv(1 + α2 Ih),

≤ I ′h

Ih− μh − λh − β3(γv + δv)Ih

γv(1 + α2 Ih)− (μh + δh + γh) − λh,

we used in above g2, the third equation of the system (22),

I ′v

Iv= Ih

Iv


γv(1 + α2 Ih)2− (γv + δv).

So,

μB ≤ sup(g1, g2) ≤ I ′h

Ih− μ

then,

q = 1

t

∫ t

0μBds ≤ 1

t

∫ t

0

(I ′h

Ih− μ

)ds = 1

tln

Ih(t)

Ih(0)− μ.

123


Table 1 Parameter values used in numerical simulation

Notation Parameter description Value References

bh Recruitment rate for humanpopulation

1.2 Assumed

β1 Transmission rate for humanpopulation

0.04 Assumed

β2 Transmission rate for vectorpopulation

0.04 Assumed

β3 Transmission rate between Sv and Ih 0.04 Assumed

μh Natural mortality rate of humanpopulation

4.6 × 10−5 [12]

λh Proportionality constant 2.85 × 10−3 [12]

δh Disease death rate for humanpopulation

1.0 × 10−3 [28]

γv Natural mortality rate of vectorpopulation

1.8 × 10−3 [12]

δv Disease death rate for vectorpopulation

0.04 Assumed

α1 The rate at which the infection forcesaturates effect

0.83 Assumed

α2 The rate at which the infection forcesaturates

0.83 Assumed

bv Recruitment rate for vectorpopulation

1.3 Assumed

γh Recovery rate of the infection 2.7 × 10−3 [28]

Implies that q ≤ −μ2 < 0. Thus the result [27], implies that the positive equilibrium point

E2 of the system (22) is globally asymptotically stable. �

Numerical Simulations

In this section, we present the numerical solution of the system (9). The parameter values usedin the numerical solution are shown in Table 1. Figures 2 and 3 shows the population behaviorof human and vector population respectively. Figures 4, 5, 6, 7 and 8 represents Susceptiblehuman, infected human, recovered human, susceptible vector and infected vector with theparameters α1 and α2. Figure 2 represent the human population, the dashed line representthe behavior of susceptible human, the bold line shows the population behavior of infectedindividuals, the dot dashed line shows the population behavior of recovered human. Figure 3shows the population of behavior of vector, the susceptible vector is represented by bold lineand the dashed line shows the population of infected vector. The plot shows the susceptiblehuman population with saturation effect for the values of α1 and α2 in Fig. 4. The graphof susceptible human decreases with the saturated effect, for the values of α1 = 0.83 andα2 = 0.83. The dotted line shows the population of susceptible human without saturationeffect and the dashed line shows the saturation effect of human and the saturation effect ofvector is represented by a dot dashed line. Due the effect of α1 and α2 the population of

123


0 2 4 6 8 10 12 14 16 18 204

6

8

10

12

14

16

18

Time(day)

Hum

an p

opul

atio

n

Population behavior of human population

Sh

Ih

Rh

Fig. 2 Population behavior of human

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Time(day)

Vec

tor

Pop

ulat

ion

Population behavior of vector population

Sv

Iv

Fig. 3 Population behavior of vector

susceptible individuals change sharply. The plot shows the infected human population withthe saturated effect of the values α1 and α2 in Fig. 5. The graph of infected human decreaseswith the saturated effect, for the values of α1 = 0.83 and α2 = 0.83. The dotted line showsthe population of infected human without saturated effect and the dashed line shows thesaturated effect. Due the effect of α1 and α2 the population of infected individuals changes.The plot shows the recovered human population with saturated effect for the values of α1

and α2 in Fig. 6. The graph of recovered human decreases with the saturated effect, for thevalues for α1 = 0.83 and α2 = 0.83. The dotted line shows the population of recoveredhuman without saturated effect and the dashed line shows the saturated effect of human andthe saturated effect of vector is represented by a dot dashed line. Due the effect of α1 andα2 the population of recovered individuals increases. The plot shows the vector populationwith saturated effect for the values of α1 and α2 in Fig. 7. The graph of susceptible vector

123


0 2 4 6 8 10 12 14 16 18 204

6

8

10

12

14

16

18

20

22

Time(day)

Sus

cept

ible

hum

an p

opul

atio

n


Sh

α1=0.83

α2=0.83

Fig. 4 Represents the graph of susceptible humanwith the effect of α1 and α2 with the parametersμh = 0.31,bh = 099, α1 = 0.83, α2 = 0.083, δv = 0.32, β1 = 0.91, bv = 290, γh = 0.71, γv = 0.8, β3 = 0.22,δh = 0.61, λh = 0.71, β2 = 0.92

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

Time(day)

Infe

cted

hum

an p

opul

atio

n


Ih

α1=0.83

α2=0.83

Fig. 5 Represents the graph of infected human with the effect of α1 and α2 with the parameters μh = 0.31,bh = 099, β2 = 0.92, α1 = 0.83, α2 = 0.083, β1 = 0.91, δv = 0.32, bv = 290, γh = 0.71, γv = 0.8,β3 = 0.22, δh = 0.61, λh = 0.71

decreases with the saturated infection rate, for the values for α1 = 0.83 and α2 = 0.83. Thedotted line shows the population of susceptible humanwithout saturated effect and the dashedline shows the saturated effect of human and the saturated effect of vector is represented bya dot dashed line. Due the effect of α1 and α2 the population of susceptible individualsdecreases. The plot shows the infected vector population with the effect of α1 and α2 inFig. 8. The graph of infected vector decreases with the with the saturation factor α1 = 0.83and α2 = 0.83. The dotted line shows the population of infected vector without saturationeffect and the dashed line shows the saturation effect of human and the saturated parametereffect of vector is represented by a dot dashed line. Due the effect of α1 and α2 the population

123


0 2 4 6 8 10 12 14 16 18 204

6

8

10

12

14

16

18

20

Time(day)

Rec

over

ed h

uman

pop

ulat

ion


Rh

α1=0.83

α2=0.83

Fig. 6 Represents the graph of recovered human with the effect of α1 and α2 with the parameters μh = 0.31,bh = 099, β1 = 0.91, α1 = 0.83, α22 = 0.083, δv = 0.32, bv = 290, γh = 0.71, β3 = 0.22, γv = 0.8,β2 = 0.92, δh = 0.61, λh = 0.71

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Time(day)

Vec

tor

Pop

ulat

ion


Sv

α1=0.83

α2=0.83

Fig. 7 Represents the graph of susceptible vector with the effect of α1 and α2 with the parametersμh = 0.31,bh = 099, α1 = 0.83, β3 = 0.22, α2 = 0.083, δv = 0.32, β1 = 0.91, bv = 290, γh = 0.71, γv = 0.8,β2 = 0.92, δh = 0.61, λh = 0.71

of infected vector decreases. The suggested parameters α1 = 0.83 and α2 = 0.83 can reducethe infection in the leptospirosis disease interaction with human.

Conclusion

The dynamic behavior of leptospirosis disease with saturated incidence has been presentedsuccessfully. The mathematical results obtained with their mathematical interpretation. Thestability of the disease free and endemic equilibrium is completely described by the basic

123


0 2 4 6 8 10 12 14 16 18 200

50

100

150

Time(day)

Vec

tor

Pop

ulat

ion


Iv

α1=0.83

α2=0.83

Fig. 8 Represents the graph of susceptible vector with the effect of α1 and α2 with the parametersμh = 0.31,bh = 099,α1 = 0.83, β3 = 0.22, α2 = 0.083, δv = 0.32, bv = 290, γh = 0.71, γv = 0.8, β1 = 0.91,δh = 0.61, λh = 0.71, β2 = 0.92

reproduction number R0. We found, when R0 < 1, the disease free equilibrium at E0 isstable locally as well as globally. Further, the endemic equilibrium is obtained. We obtainedthat, when R0 > 1, the endemic equilibrium is stable both locally and globally. Moreover,the numerical results for the model is obtained and briefly discussed. The parameters α1 andα2 effect have been discussed in numerical solution.

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Documents

Dynamic Behavior of Leptospirosis Disease with Saturated ...Muhammad Altaf Khan1 · Syed Farasat Saddiq2 ... Abdul Wali Khan, University Mardan, Mardan, Khyber Pakhtunkhwa, Pakistan