# Global Stability of an Eco-Epidemiological Model with Time Delay and Saturation Incidence

Authors: Shuxue Mao; Rui Xu; Zhe Li; Yunfei Li
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

We investigate a delayed eco-epidemiological model with disease in predator and saturation incidence. First, by comparison arguments, the permanence of the model is discussed. Then, we study the local stability of each equilibrium of the model by analyzing the corresponding characteristic equations and find that Hopf bifurcation occurs when the delayτ passes through a sequence of critical values. Next, by means of an iteration technique, sufficient conditions are derived for the global stability of the disease-free planar equilibrium and the positive equilibrium. Numerical examples are carried out to illustrate the analytical results.


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## Body

## 1. Introduction

Recently, more attention has been paid to the eco-epidemiology model which considers both the ecological and epidemiological issues simultaneously due to the fact that most of the ecological populations suffer from various infectious diseases which have a significant role in regulating population sizes (see, e.g., [1–6]). Mukherjee [7] discussed a predator-prey model with disease in prey. The criteria were derived for both local stability and instability involving system parameters. In addition, considering the time required by the susceptible individuals to become infective after their interaction with the infectious individuals, Zhou et al. [8] formulated a delayed eco-epidemiology model and found that the Hopf bifurcation occurs when the delay passes through a sequence of critical values. They also gave an estimation of the length of the time delay to preserve stability. On the other hand, in the predator-prey system, the disease not only can spread in prey but also can spread in predator. Therefore, Zhang et al. [9] studied an eco-epidemiological model with disease in predator and showed that a Hopf bifurcation can occur as the delay increased. The above-mentioned works all used bilinear incidence to model disease transmission.Note that ecologically the assumption of standard incidence instead of the former bilinear mass action incidence is meaningful for large populations and a low number of infected individuals, a very good justification behind this assumption being found in [10]. Han et al. [11] proposed four modifications of a predator-prey model with standard incidence to include an SIS or SIR parasitic infection. Thresholds were identified, and global stability results were proved. When the disease persists in the prey population and the predators have a sufficient feeding efficiency to survive, the disease also persists in the predator population. Hethcote et al. [12] considered a predator-prey model including an SIS parasitic infection in the prey with infected prey being more vulnerable to predation. Thresholds were identified which determine when the predator population survives and when the disease remains endemic.However, there are a variety of factors that emphasize the need for a modification of the bilinear incidence and standard incidence. For example, the underlying assumption of homogeneous mixing may not always hold. Incidence rates that increase more gradually than linearly inI and S may arise from saturation effects. It has been strongly suggested by several authors that the disease transmission process may follow saturation incidence. After studying the cholera epidemic spread in Bari in 1973, Capasso and Serio [13] introduced a saturated incidence rate g(I)S into epidemic models with g(I)=βI/(1+αI). A general saturation incidence rate g(I)S=βIpS/(1+αIp) was proposed by Liu et al. [14] and used by a number of authors; see, for example, Ruan and Wang [15] (p=2), Bhattacharyya and Mukhopadhyay [16] (p=1), and so forth. βIp measures the infection force of the disease, and 1/(1+αIp) measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals. This incidence rate seems more reasonable than the bilinear incidence rate βSI, because it includes the behavioral change and crowding effect of the infective individuals and prevents the unboundedness of the contact rate by choosing suitable parameters.Motivated by the works of Zhang et al. [9] and Capasso and Serio [13], in this paper, we are concerned with the effect of disease in predator and saturated incidence on the dynamics of eco-epidemiological model. To this end, we consider the following delay differential equations: (1.1)ẋ(t)=rx(t)(1-x(t)K)-ax(t)S(t),Ṡ(t)=bx(t-τ)S(t-τ)-cS2(t)-βS(t)I(t)1+αI(t),İ(t)=βS(t)I(t)1+αI(t)-dI(t),
with initial conditions (1.2)x(θ)=ϕ1(θ),S(θ)=ϕ2(θ),I(θ)=ϕ3(θ),ϕi(θ)≥0,θ∈[-τ,0],ϕi(0)>0(i=1,2,3),
where (ϕ1(θ),ϕ2(θ),ϕ3(θ))∈C([-τ,0]), ℝ+3), the Banach space of continuous functions mapping the interval [-τ,0] into ℝ+03, here ℝ+03={(x1,x2,x3):xi≥0,i=1,2,3}.We make the following assumptions for our model (1.1).  (A1)
The prey population grows logistically with intrinsic growth rater and environmental carrying capacity  K.(A2)
There is a spread of disease in predators which are divided solely into susceptible and infectious population.a is the capturing rate of susceptible predators, b is the growth rate of susceptible predator due to predation of prey.(A3)
Susceptible predators become infected when they come in contact with infected predator, and this contact process is assumed to follow the saturation incidence rateβS(t)I(t)/(1+αI(t)), with β measuring the force of infection and α the inhibition effect.(A4)
c>0 models death rate due to overcrowding, and τ is the time required for the gestation of susceptible predator. d is the death rate of infected predator. All the above-mentioned parameters are assumed to be positive.The paper is organized as follows. In the next section, the positivity of solutions and the permanence of system are discussed. By analyzing the corresponding characteristic equations, we find conditions for local stability and bifurcation results in Section3. In Section 4, sufficient conditions are derived for the global stability of the disease-free planar equilibrium and the positive equilibrium of the system. Numerical examples are carried out to illustrate the validity of the main results. The paper ends with a conclusion in the last section.


## 2. Permanence

To prove the permanence of system (1.1), we need the following lemma, which is a direct application of Theorem 4.9.1 in the study by Kuang [17].Lemma 2.1.
Consider the following equation:(2.1)ẋ(t)=ax(t-τ)-bx(t)-cx2(t),
where a,b,c,τ>0 and x(t)>0 for all t∈[-τ,0].(1)
Ifa>b, then limsupt→+∞x(t)=(a-b)/c.(2)
Ifa<b, then limsupt→+∞x(t)=0.Theorem 2.2.
All the solutions of (1.1) with initial conditions (1.2) are all nonnegative.Proof.
Let(x(t),S(t),I(t)) be the solution of system (1.1) satisfying conditions (1.2). From the first and last equations of system (1.1), we have
(2.2)x(t)=x(0)e∫0t[r(1-x(ξ)/K)-aS(ξ)]dξ,I(t)=I(0)e∫0t[βS(ξ)/(1+αI(ξ))-d]dξ.
Hence, x(t) and I(t) are positive.
We now claim thatS(t)>0 for all t>0. Otherwise, there exists a t1>0 such that S(t1)=0 and S(t)>0 for all t∈[0,t1). Then Ṡ(t1)≤0. From the second equation of (1.1), we have
(2.3)Ṡ(t1)=bx(t1-τ)S(t1-τ)>0,
which is a contradiction.Theorem 2.3.
All the solutions of (1.1) with initial conditions (1.2) are ultimately bounded.Proof.
From the first equation of (1.1), we have
(2.4)ẋ(t)≤rx(t)(1-x(t)K).
Hence, we get
(2.5)limsup⁡t→+∞x(t)≤K=̇M1.
From the second equation of system (1.1), for t sufficiently large, we have
(2.6)Ṡ(t)=bx(t-τ)S(t-τ)-cS2(t)-βS(t)I(t)1+αI(t)≤bKS(t-τ)-cS2(t).
Hence, by Lemma 2.1, one can get
(2.7)lim⁡⁡sup⁡t→+∞S(t)≤bKc=̇M2.
It follows from the third equation of (1.1) and the above inequality, that for t sufficiently large, we have
(2.8)İ(t)=βS(t)I(t)1+αI(t)-dI(t)≤βM2I(t)1+αI(t)-dI(t).
Hence, one can see limsupt→+∞I(t)≤(1/(dα))|βM2-d|=̇M3.Now, we show that system (1.1) is permanent.Theorem 2.4.
Suppose that(H1)
(2.9)βm2>d,where m2 is defined in (2.13), then system (1.1) is permanent.Proof.
From the first equation of system (1.1), we have
(2.10)ẋ(t)≥rx(t)(1-x(t)K-aM2r).
It then follows that
(2.11)liminf⁡t→+∞x(t)≥K[1-aM2r]=̇m1.
Using the second equation of system (1.1), for t sufficiently large, we have
(2.12)Ṡ(t)≥bm1S(t-τ)-cS2(t)-βS(t)M31+αM3.
Hence, by Lemma 2.1 and (H1), one can derive that
(2.13)liminf⁡t→+∞S(t)≥1c[bm1-βM31+αM3]=̇m2.
From the third equation of system (1.1) and, above inequality, we have
(2.14)İ(t)≥βm2I(t)1+αI(t)-dI(t).
Since (H1) holds, then
(2.15)liminf⁡t→+∞I(t)≥1dα[βm2-d]=̇m3.
Therefore, the above calculations and Theorem2.2 imply that there exist Mi,mi(i=1,2,3) such that
(2.16)0<m1≤liminf⁡t→+∞x(t)≤limsup⁡t→+∞x(t)≤M1,0<m2≤liminf⁡t→+∞S(t)≤limsup⁡t→+∞S(t)≤M2,0<m3≤liminf⁡t→+∞I(t)≤limsup⁡t→+∞I(t)≤M3.


## 3. Local Stability

System (1.1) possesses the following equilibria.(1)
The trivial equilibriumE0(0,0,0).(2)
The axial equilibriumE1(K,0,0).(3)
The disease-free planar equilibriumE2(x2,S2,0), where(3.1)x2=KcrKab+rc,S2=KbrKab+rc.(4) The unique positive equilibriumE3(x3,S3,I3) exists if βS3>d, where (3.2)S3=Kbα-β+(Kbα-β)2+4dα(Kab/r+c)2α(Kab/r+c),I3=βS3-ddα,x3=K-KaS3r.In the following, we discuss the local stability of each equilibrium of system (1.1) by analyzing the corresponding characteristic equations, respectively.


### 3.1. Stability of EquilibriumE0

The characteristic equation of system (1.1) at the trivial equilibrium E0 is of the form (3.3)λ(λ-r)(λ+d)=0.
It is easy to see that (3.3) always has a positive root r. Hence, E0 is always unstable.


### 3.2. Stability of EquilibriumE1

The characteristic equation of system (1.1) at the axial equilibrium E1 is of the form (3.4)(λ+K)(λ-bKe-λτ)(λ+d)=0.
There are two characteristic roots λ1=-K, λ2=-d, and another characteristic root is given by the root of (3.5)λ=bKe-λτ.It is clear thatReλ>0. Hence, E1 is always unstable.


### 3.3. Stability of EquilibriumE2

Theorem 3.1.
The disease-free planar equilibriumE2 is locally asymptotically stable if  βS2<d, and the equilibrium  E2 is unstable if  βS2>d.Proof.
The characteristic equation of system (1.1) at the disease-free planar equilibrium E2 is of the form
(3.6)(λ+rx2K)(λ+2cS2-cS2e-λτ)(λ+d-βS2)=0.
Clearly,λ1=-rx2/K is a negative eigenvalue. The second eigenvalue is given by the root of
(3.7)λ2=cS2(e-λ2τ-2).
Suppose that Reλ2≥0, then Reλ2=cS2(e-Reλ2τcos(τImλ2)-2)<0. It is a contradiction, so Reλ2<0. The last eigenvalue is λ3=βS2-d. The equilibrium E2 is locally asymptotically stable if βS2<d, and the equilibrium E2 is unstable if βS2>d.


### 3.4. Stability of EquilibriumE3

The characteristic equation of system (1.1) at the positive equilibrium E3 is of the form (3.8)λ3+A1λ2+A2λ+A3+e-λτ(B1λ2+B2λ+B3)=0,
where (3.9)A1=rx3K+2cS3+βI31+αI3+dαI31+αI3,A2=(rx3K+dαI31+αI3)×(2cS3+βI31+αI3)+dαI31+αI3×rx3K+βI3(1+αI3)2,A3=rx3K×[dβI31+αI3+(2cS3+βI31+αI3)×dαI31+αI3],B1=-bx3,B2=bx3×(-dαI31+αI3-rx3K+aS3),B3=bx3×dαI31+αI3×(-rx3K+aS3).Forτ=0, the transcendental (3.8) reduces to the following equation: (3.10)λ3+(A1+B1)λ2+(A2+B2)λ+A3+B3=0.
We can easily get (3.11)A1+B1=rx3K+cS3+dαI31+αI3>0,A2+B2=(rx3K+dαI31+αI3)×cS3+dαI31+αI3×rx3K+βI3(1+αI3)2+bx3aS3>0,A3+B3=rx3K×[dβI31+αI3+cS3×dαI31+αI3]+bx3×dαI31+αI3×aS3>0,[A1+B1]×[A2+B2]-[A3+B3]>0.
Therefore, the Routh-Hurwitz criterion implies that all the roots of (3.8) have negative real parts and we can conclude that the positive equilibrium E3 is asymptotically stable in the absence of delay.Theorem 3.2.
For system (1.1), if the condition  (H2)A3<B3 holds, the positive equilibrium  E3 is conditionally stable.Proof.
Substitutingλ=iω into (3.8) and separating the real and imaginary parts, one can get
(3.12)A1ω2-A3=(B3-B1ω2)cos⁡(ωτ)+B2ωsin⁡(ωτ),ω3-A2ω=B2ωcos⁡(ωτ)-(B3-B1ω2)sin⁡(ωτ).
Squaring and adding (3.12) we get (3.13)ω6+D1ω4+D2ω2+D3=0,
where
(3.14)D1=A12-2A2-B12,D2=A22-B22-2A1A3+2B1B3,D3=A32-B32.We know that D3<0 provided that the condition (H2) holds. There is at least a positive ω0 satisfying (3.13), that is, the characteristic equation (3.8) has a pair of purely imaginary roots of the form ±iω0. From (3.12), we can get the corresponding τk>0 such that the characteristic (3.8) has a pair of purely imaginary roots
(3.15)τk=1ω0arccos[(A1ω02-A3)(B3-B1ω02)+(ω03-A2ω0)B2ω0(B3-B1ω02)2+(B2ω0)2]+2kπω0,(k=0,1,2,…).
Let λ(τ)=ν(τ)+iω(τ) be the roots of (3.8) such that τ=τk satisfying ν(τk)=0 and ω(τk)=ω0. Differentiating the two sides of (3.8) with respect to τ, we get
(3.16)(dλdτ)-1=2λ3+A1λ2-A2-λ2(λ3+A1λ2+A2λ+A3)+B1λ2-B3λ2(B1λ2+B2λ+B3)-τλ.Therefore,
(3.17)sign⁡[dReλdτ]τ=τk=sign⁡[Re(dλdτ)-1]λ=iω0=1ω02sign⁡[Re(A3+A1ω02+i2ω03A1ω02-A3+i(ω03-A2ω0)+B1ω02+B3-B1ω02+iB2ω0+B3)]=1ω02sign⁡[2ω06+(A12-2A2-B12)ω04+B32-A32(B3-B1ω02)2+(B2ω0)2].
If the conditions (H2) and (H3)A12-2A2>B12 hold, one can see
(3.18)sign⁡[dReλdτ]τ=τk>0.Therefore, the transversality condition holds, hence, the Hopf bifurcation occurs at ω=ω0 and τ=τk.Theorem 3.3.
Suppose that the conditions(H2) and  (H3) are satisfied.   (1)
The positive equilibriumE3 of system (1.1) is asymptotically stable for all τ∈[0,τ0) and unstable for τ>τ0.(2)
System (1.1) undergoes a Hopf Bifurcation at the positive equilibrium E3 when τ=τk(k=0,1,…).


## 3.1. Stability of EquilibriumE0

The characteristic equation of system (1.1) at the trivial equilibrium E0 is of the form (3.3)λ(λ-r)(λ+d)=0.
It is easy to see that (3.3) always has a positive root r. Hence, E0 is always unstable.


## 3.2. Stability of EquilibriumE1

The characteristic equation of system (1.1) at the axial equilibrium E1 is of the form (3.4)(λ+K)(λ-bKe-λτ)(λ+d)=0.
There are two characteristic roots λ1=-K, λ2=-d, and another characteristic root is given by the root of (3.5)λ=bKe-λτ.It is clear thatReλ>0. Hence, E1 is always unstable.


## 3.3. Stability of EquilibriumE2

Theorem 3.1.
The disease-free planar equilibriumE2 is locally asymptotically stable if  βS2<d, and the equilibrium  E2 is unstable if  βS2>d.Proof.
The characteristic equation of system (1.1) at the disease-free planar equilibrium E2 is of the form
(3.6)(λ+rx2K)(λ+2cS2-cS2e-λτ)(λ+d-βS2)=0.
Clearly,λ1=-rx2/K is a negative eigenvalue. The second eigenvalue is given by the root of
(3.7)λ2=cS2(e-λ2τ-2).
Suppose that Reλ2≥0, then Reλ2=cS2(e-Reλ2τcos(τImλ2)-2)<0. It is a contradiction, so Reλ2<0. The last eigenvalue is λ3=βS2-d. The equilibrium E2 is locally asymptotically stable if βS2<d, and the equilibrium E2 is unstable if βS2>d.


## 3.4. Stability of EquilibriumE3

The characteristic equation of system (1.1) at the positive equilibrium E3 is of the form (3.8)λ3+A1λ2+A2λ+A3+e-λτ(B1λ2+B2λ+B3)=0,
where (3.9)A1=rx3K+2cS3+βI31+αI3+dαI31+αI3,A2=(rx3K+dαI31+αI3)×(2cS3+βI31+αI3)+dαI31+αI3×rx3K+βI3(1+αI3)2,A3=rx3K×[dβI31+αI3+(2cS3+βI31+αI3)×dαI31+αI3],B1=-bx3,B2=bx3×(-dαI31+αI3-rx3K+aS3),B3=bx3×dαI31+αI3×(-rx3K+aS3).Forτ=0, the transcendental (3.8) reduces to the following equation: (3.10)λ3+(A1+B1)λ2+(A2+B2)λ+A3+B3=0.
We can easily get (3.11)A1+B1=rx3K+cS3+dαI31+αI3>0,A2+B2=(rx3K+dαI31+αI3)×cS3+dαI31+αI3×rx3K+βI3(1+αI3)2+bx3aS3>0,A3+B3=rx3K×[dβI31+αI3+cS3×dαI31+αI3]+bx3×dαI31+αI3×aS3>0,[A1+B1]×[A2+B2]-[A3+B3]>0.
Therefore, the Routh-Hurwitz criterion implies that all the roots of (3.8) have negative real parts and we can conclude that the positive equilibrium E3 is asymptotically stable in the absence of delay.Theorem 3.2.
For system (1.1), if the condition  (H2)A3<B3 holds, the positive equilibrium  E3 is conditionally stable.Proof.
Substitutingλ=iω into (3.8) and separating the real and imaginary parts, one can get
(3.12)A1ω2-A3=(B3-B1ω2)cos⁡(ωτ)+B2ωsin⁡(ωτ),ω3-A2ω=B2ωcos⁡(ωτ)-(B3-B1ω2)sin⁡(ωτ).
Squaring and adding (3.12) we get (3.13)ω6+D1ω4+D2ω2+D3=0,
where
(3.14)D1=A12-2A2-B12,D2=A22-B22-2A1A3+2B1B3,D3=A32-B32.We know that D3<0 provided that the condition (H2) holds. There is at least a positive ω0 satisfying (3.13), that is, the characteristic equation (3.8) has a pair of purely imaginary roots of the form ±iω0. From (3.12), we can get the corresponding τk>0 such that the characteristic (3.8) has a pair of purely imaginary roots
(3.15)τk=1ω0arccos[(A1ω02-A3)(B3-B1ω02)+(ω03-A2ω0)B2ω0(B3-B1ω02)2+(B2ω0)2]+2kπω0,(k=0,1,2,…).
Let λ(τ)=ν(τ)+iω(τ) be the roots of (3.8) such that τ=τk satisfying ν(τk)=0 and ω(τk)=ω0. Differentiating the two sides of (3.8) with respect to τ, we get
(3.16)(dλdτ)-1=2λ3+A1λ2-A2-λ2(λ3+A1λ2+A2λ+A3)+B1λ2-B3λ2(B1λ2+B2λ+B3)-τλ.Therefore,
(3.17)sign⁡[dReλdτ]τ=τk=sign⁡[Re(dλdτ)-1]λ=iω0=1ω02sign⁡[Re(A3+A1ω02+i2ω03A1ω02-A3+i(ω03-A2ω0)+B1ω02+B3-B1ω02+iB2ω0+B3)]=1ω02sign⁡[2ω06+(A12-2A2-B12)ω04+B32-A32(B3-B1ω02)2+(B2ω0)2].
If the conditions (H2) and (H3)A12-2A2>B12 hold, one can see
(3.18)sign⁡[dReλdτ]τ=τk>0.Therefore, the transversality condition holds, hence, the Hopf bifurcation occurs at ω=ω0 and τ=τk.Theorem 3.3.
Suppose that the conditions(H2) and  (H3) are satisfied.   (1)
The positive equilibriumE3 of system (1.1) is asymptotically stable for all τ∈[0,τ0) and unstable for τ>τ0.(2)
System (1.1) undergoes a Hopf Bifurcation at the positive equilibrium E3 when τ=τk(k=0,1,…).


## 4. Global Stability

In this section, we study the global stability of equilibriumsE2 and E3. The strategy of proofs is to use an iteration technique and comparison arguments, respectively.Theorem 4.1.
If(H4)
βbK<cd,Kab<rc  holds, then the disease-free planar equilibrium E2 is globally asymptotically stable.Proof.
Let(x(t),S(t),I(t)) be any positive solution of system (1.1) with initial conditions (1.2). Let the following hold:
(4.1)U1=limsup⁡t→+∞x(t),U2=limsup⁡t→+∞S(t),U3=limsup⁡t→+∞I(t),V1=liminf⁡t→+∞x(t),V2=liminf⁡t→+∞S(t),V3=liminf⁡t→+∞I(t).
In the following we shall claim that U1=V1=x2,U2=V2=S2,U3=V3=0.
It follows from the first equation of system (1.1) that
(4.2)ẋ(t)≤rx(t)(1-x(t)K).
By comparison, we obtain that
(4.3)U1=limsup⁡t→+∞x(t)≤K+ε.
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that U1≤M1x, where
(4.4)M1x=K.
Hence, for ε>0 sufficiently small, there is a T1>0 such that, if t>T1, x(t)≤M1x+ε.
We, therefore, derive from the second equation of system (1.1) that, for t>T1+τ,
(4.5)Ṡ(t)≤b(M1x+ε)S(t-τ)-cS2(t).
Hence, by Lemma 2.1, one can get
(4.6)U2=limsup⁡t→+∞S(t)≤b(M1x+ε)c=̇M1S.
Hence, for ε>0 sufficiently small, there is a T2>0 such that, if T2>T1+τ, S(t)≤M1S+ε.
It follows from the third equation of system (1.1) that, for t>T2,
(4.7)İ(t)≤β(M1S+ε)I(t)1+αI(t)-dI(t).
Since (H4) holds, one can see
(4.8)U3=limsup⁡t→+∞I(t)≤0.
According to Theorem 2.2, we can get limt→+∞I(t)=U3=V3=0.
We derive from the first equation of system (1.1) that, for t>T2+τ,
(4.9)ẋ(t)≥rx(t)(1-x(t)K-a(M1S+ε)r).
By comparison we derive that
(4.10)V1=liminf⁡t→+∞x(t)≥K[1-a(M1S+ε)r].
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that V1≥N1x, where
(4.11)N1x=K[1-aM1Sr].
Hence, for ε>0 sufficiently small, there is a T3>0 such that, if T3>T2+τ, x(t)≥N1x-ε.
We derive from the second equation of system (1.1) that, for t>T3,
(4.12)Ṡ(t)≥b(N1x-ε)S(t-τ)-cS2(t)-βS(t)ε1+αε.
Hence, by Lemma 2.1, one can get
(4.13)V2=liminf⁡t→+∞S(t)≥1c[b(N1x-ε)-βε1+αε].
Since this is true for arbitrary ε>0 sufficiently small, we conclude that V2≥N1S, where
(4.14)N1S=bN1xc.
Hence, for ε>0 sufficiently small, there is a T4>0 such that, if T4>T3+τ, S(t)≥N1S-ε.
Again, it follows from the first equation of system (1.2) that, for t>T4,
(4.15)ẋ(t)≤rx(t)(1-x(t)K-a(N1S-ε)r).
A comparison argument yields
(4.16)U1=limsup⁡t→+∞x(t)≤K[1-a(N1S-ε)r].
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that U1≤M2x, where
(4.17)M2x=K[1-aN1Sr].
Hence, for ε>0 sufficiently small, there is a T5>0 such that, if T5>T4+τ, x(t)≤M2x+ε.
It follows from the second equation of system (1.1) that, for t>T5,
(4.18)Ṡ(t)≤b(M2x+ε)S(t-τ)-cS2(t)-βS(t)ε1+αε.
By Lemma 2.1, one can derive that
(4.19)V2=limsup⁡t→+∞S(t)≤1c[b(M2x+ε)-βε1+αε].
Since this is true for arbitrary ε>0 sufficiently small, we conclude that U2≤M2S, where
(4.20)M2S=bM2xc.
Hence, for ε>0 sufficiently small, there is a T6>0 such that, if T6>T5+τ, S(t)≤M2S+ε.
We derive from the first equation of system (1.1) that, for t>T6,
(4.21)ẋ(t)≥rx(t)(1-x(t)K-a(M2S+ε)r).
By comparison it follows that
(4.22)V1=liminf⁡t→+∞x(t)≥K[1-a(M2S+ε)r].
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that V1≥N2x, where
(4.23)N2x=K[1-aM2Sr].
Hence, for ε>0 sufficiently small, there is a T7>0 such that, if T7>T6+τ, x(t)≥N2x-ε.
We derive from the second equation of system (1.1) that, for t>T7,
(4.24)Ṡ(t)≥b(N2x-ε)S(t-τ)-cS2(t)-βS(t)ε1+αε.
Hence, by Lemma 2.1, one can get
(4.25)V2=liminf⁡t→+∞S(t)≥1c[b(N2x-ε)-βε1+αε].
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that V2≥N2S, where
(4.26)N2S=bN2xc.
Hence, for ε>0 sufficiently small, there is a T7>0 such that, if T8>T7+τ, S(t)≥N2S-ε.
Continuing this process, we get four sequencesMnx,MnS,Nnx,NnS(n=1,2,…) such that, for n≥2,
(4.27)Mnx=K[1-aNn-1Sr],Nnx=K[1-aMnSr],MnS=bMnxc,NnS=bNnxc.
Clearly, we have
(4.28)Nnx≤V1≤U1≤Mnx,NnS≤V2≤U2≤MnS.
It follows from (4.27) that
(4.29)Mn+1x=K[1-Kabrc]+MnxK2a2b2r2c2.
Noting that Mnx≥S2 and Kab<rc, we derive from (4.29) that
(4.30)Mn+1x=K[1-Kabrc]+Mnx[Kabrc-1][Kabrc+1]≤K[1-Kabrc]+KcrKab+rc[Kabrc-1][Kabrc+1]=0.
Thus, the sequence Mnx is monotonically nonincreasing. Therefore, it follows that limn→+∞Mnx exists. Taking n→+∞, we obtain from (4.29) that
(4.31)lim⁡n→+∞Mn+1x=K[1-Kabrc]+lim⁡n→+∞MnxK2a2b2r2c2.
Noting that
(4.32)lim⁡n→+∞Mn+1x=lim⁡n→+∞Mnx,
it follows from (4.31) that
(4.33)lim⁡n→+∞Mn+1x=lim⁡n→+∞Mnx=x2.
We derive from (4.33) and the third equation of (4.27) that
(4.34)lim⁡n→+∞Mn+1S=lim⁡n→+∞MnS=S2.
Similarly, one can derive from (4.27) and (4.34) that
(4.35)lim⁡n→+∞Nnx=x2,lim⁡n→+∞NnS=S2.
It follows from (4.28), (4.33), and (4.35) that
(4.36)V1=U1=x2,V2=U2=S2.
We, therefore, have
(4.37)lim⁡t→+∞x(t)=x2,lim⁡t→+∞S(t)=S2,lim⁡t→+∞I(t)=0.
Hence, the disease-free planar equilibrium E2 is globally asymptotically stable. The proof is complete.Theorem 4.2.
If(H5)
βbK>cdandKab<rc,β>Kbα holds, then the positive equilibrium E3 is globally asymptotically stable.Proof.
Let(x(t),S(t),I(t)) be any positive solution of system (1.1) with initial conditions (1.2). Let the following hold:
(4.38)x¯=limsup⁡t→+∞x(t),S¯=limsup⁡t→+∞S(t),I¯=limsup⁡t→+∞I(t),x̲=liminf⁡t→+∞x(t),S̲=liminf⁡t→+∞S(t),I̲=liminf⁡t→+∞I(t).
In the following we claim that x¯=x̲=x3,S¯=S̲=S3,I¯=I̲=I3.
It follows from the first equation of system (1.1) that
(4.39)ẋ(t)≤rx(t)(1-x(t)K).
By comparison we obtain
(4.40)x¯=limsup⁡t→+∞x(t)≤K+ε.
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that x¯≤M1x, where
(4.41)M1x=K.
Hence, for ε>0 sufficiently small, there is a T1>0 such that, if t>T1, x(t)≤M1x+ε.We obtain from the second equation of system (1.1) that, for t>T1+τ,
(4.42)Ṡ(t)≤b(M1x+ε)S(t-τ)-cS2(t).
Hence, by Lemma 2.1, we derive that
(4.43)S¯=limsup⁡t→+∞S(t)≤b(M1x+ε)c.
Since it is true for arbitrary ε>0 sufficiently small, we conclude that S¯≤M1S, where
(4.44)M1S=bM1xc.
Hence, for ε>0 sufficiently small, there is a T2>0 such that, if T2>T1+τ, S(t)≤M1S+ε.
It follows from the third equation of system (1.1) that
(4.45)İ(t)≤β(M1S+ε)I(t)1+αI(t)-dI(t).
Since (H5) holds, one can see
(4.46)I¯=limsup⁡t→+∞I(t)≤β(M1S+ε)-ddα.
Since it is true for arbitrary ε>0 sufficiently small, we conclude that I¯≤M1I, where
(4.47)M1I=βM1S-ddα.
We derive from the first equation of system (1.1) that, for t>T2,
(4.48)ẋ(t)≥rx(t)(1-x(t)K-a(M1S+ε)r).
By comparison we derive that
(4.49)x̲=liminf⁡t→+∞x(t)≥K[1-a(M1S+ε)r].
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that x̲≥N1x, where
(4.50)N1x=K[1-aM1Sr].
Hence, for ε>0 sufficiently small, there is a T3>0 such that, if T3>T2+τ, x(t)≥N1x-ε.
We derive from the second equation of system (1.1) that, for t>T3,
(4.51)Ṡ(t)≥b(N1x-ε)S(t-τ)-cS2(t)-βS(t)M1I1+αM1I.
Hence, by Lemma 2.1 and (H5), one can get
(4.52)S̲=liminf⁡t→+∞S(t)≥1c[b(N1x-ε)-βM1I1+αM1I].
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that S̲≥N1S, where
(4.53)N1S=1c[bN1x-βM1I1+αM1I].
Hence, for ε>0 sufficiently small, we get S(t)≥N1S-ε.
It follows from the third equation of system (1.1) that
(4.54)İ(t)≥β(N1S-ε)I(t)1+αI(t)-dI(t).
Provided that βN1S>d, one can see
(4.55)I̲=liminf⁡t→+∞I(t)≥β(N1S-ε)-ddα.
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that I̲≥N1I, where
(4.56)N1I=βN1S-ddα.
It follows from the first equation of system (1.1) that
(4.57)ẋ(t)≤rx(t)(1-x(t)K-a(N1S-ε)r).
By comparison we derive that
(4.58)x¯=limsup⁡t→+∞x(t)≤K[1-a(N1S-ε)r].
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that x¯≤M2x, where
(4.59)M2x=K[1-aN1Sr].
Hence, for ε>0 sufficiently small, there is a T4>0 such that, if t>T4, x(t)≤M2x+ε.
We obtain from the second equation of system (1.1) that, for t>T4+τ,
(4.60)Ṡ(t)≤b(M2x+ε)S(t-τ)-cS2(t)-βS(t)N1I1+αN1I.
Hence, by Lemma 2.1, one can get
(4.61)S¯=limsup⁡t→+∞S(t)≤1c[b(M2x+ε)-βN1I1+αN1I].
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that S¯≤M2S, where
(4.62)M2S=1c[bM2x-βN1I1+αN1I].
Hence, for ε>0 sufficiently small, there is a T5>0 such that, if T5>T4+τ, S(t)≤M2S+ε.
It follows from the third equation of system (1.1) that
(4.63)İ(t)≤β(M2S+ε)I(t)1+αI(t)-dI(t).
Hence, by (H5), one can see
(4.64)I¯=limsup⁡t→+∞I(t)≤β(M2S+ε)-ddα.
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that I¯≤M2I, where
(4.65)M2I=βM2S-ddα.
We derive from the first equation of system (1.1) that, for t>T5,
(4.66)ẋ(t)≥rx(t)(1-x(t)K-a(M2S+ε)r).
By comparison we derive that
(4.67)x̲=liminf⁡t→+∞x(t)≥K[1-a(M2S+ε)r].
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that x̲≥N2x, where
(4.68)N2x=K[1-aM2Sr].
Hence, for ε>0 sufficiently small, there is a T6>0 such that, if T6>T5+τ, x(t)≥N2x-ε.
We derive from the second equation of system (1.1) that, for t>T6,
(4.69)Ṡ(t)≥b(N2x-ε)S(t-τ)-cS2(t)-βS(t)M2I1+αM2I.
By Lemma 2.1, one can get
(4.70)S̲=liminf⁡t→+∞S(t)≥1c[b(N2x-ε)-βM2I1+αM2I].
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that S̲≥N1S, where
(4.71)N2S=1c[bN2x-βM2I1+αM2I].
Hence, for ε>0 sufficiently small, we get S(t)≥N2S-ε.
It follows from the third equation of system (1.1) that
(4.72)İ(t)≥β(N2S-ε)I(t)1+αI(t)-dI(t).
Since (H5) holds, one can see
(4.73)I̲=liminf⁡t→+∞I(t)≥β(N2S-ε)-ddα.
Since this inequality holds true for arbitrary ε>0 sufficiently small, we conclude that I̲≥N2I, where
(4.74)N2I=βN2S-ddα.
Continuing this process, we obtain six sequencesMnx,MnS,MnI,Nnx,NnS,NnI(n=1,2,…) such that, for n≥2,
(4.75)Mnx=K[1-aNn-1Sr],MnS=1c[bMnx-βNn-1I1+αNn-1I],MnI=βMnS-ddα,Nnx=K[1-aMnSr],NnS=1c[bNnx-βMnI1+αMnI],NnI=βNnS-ddα.
A direct calculation shows that
(4.76)M2x-M1x=-KaN1Sr<0,M2S-M1S=1c[b(M2x-M1x)-βN1I1+αN1I]<0,M2I-M1I=β(M2S-M1S)dα<0,N2x-N1x=Ka(M1S-M2s)r>0,N2S-N1S=1c[b(N2x-N1x)+βM1I1+αM1I-βM2I1+αM2I]>0,N2I-N1I=β(N2S-N1S)dα>0.
By induction, we can show that Mn+1S<MnS, Nn+1S>NnS. Therefore, the sequence MnS is decreasing, and the sequence NnS is increasing. Clearly, we have
(4.77)Nnx≤x̲≤x¯≤Mnx,NnS≤S̲≤S¯≤MnS,NnI≤I̲≤I¯≤MnI.
Hence, the limits of the sequences MnS and NnS exist. Denote
(4.78)S¯=lim⁡t→+∞MnS,S̲=lim⁡t→+∞NnS.
We derive from (4.75) that
(4.79)[S¯-S̲][β-Kbα+Kbadr(S¯+S̲)]=0.
Since (H5) holds, β-Kbα+(Kba/dr)(S¯+S̲)>0. It, therefore, follows from (4.79) that S¯=S̲. Accordingly, we derive from (4.75) that
(4.80)S¯=1c[bK(1-aS¯r)-βα+dS¯].
By a simple calculation, we obtain
(4.81)S¯=S̲=S3.
It follows from (4.75) and (4.81) that I¯=I̲=I3, x¯=x̲=x3. Hence, the unique positive equilibrium E3 is globally asymptotically stable. The proof is complete.In the following we will present two examples to verify our results obtained earlier.Example 4.3.
In system (1.1), we let r=2,K=2,a=1,c=2,β=2,α=1,d=0.8,b=0.3,τ=1. It is easy to show that Kbβ-cd=-0.4<0,rc-Kab=3.4>0. By Theorem 4.1 we see that the equilibrium E2(1.7391,0.2609,0) of system (1.1) is globally stable (see Figure 1).Figure 1
The temporal solution found by numerical integration of system (1.1) with r=2,K=2,a=2,c=2,β=2,α=1,d=0.8,b=0.3,τ=1, and (x0,S0,I0)=(1,1,1).Example 4.4.
In system (1.1), we let r=2,K=2,a=1,c=2,β=2,α=1,d=0.4,b=0.3,τ=1. It is easy to show that Kbβ-cd=0.4>0,Kab-rc=-3.4<0,β-Kbα=1.400>0. By Theorem 4.2 we see that the equilibrium E3(1.7881,0.2119,0.0596) of system (1.1) is globally stable, as depicted in Figure 2.Figure 2
The temporal solution found by numerical integration of system (1.1) with r=2,K=2,a=1,c=2,β=2,α=1,d=0.4,b=0.3,τ=1, and (x0,S0,I0)=(1,1,1).


## 5. Conclusion

In this paper, we have incorporated the disease for the predator and the time delay into an eco-epidemiology model. A saturation incidence function was used to model the behavioral change of the susceptible predator when their number increases or due to the crowding effect of the infected predator. First, by comparison arguments, the permanence of system (1.1) was studied. Then, by analyzing the corresponding characteristic equations, sufficient conditions were derived for the local stability of each equilibrium of system (1.1). From Theorem 3.3, we showed that system (1.1) undergoes a Hopf bifurcation when the delay passes through a sequence of critical values. Next, by using the iteration technique and comparison arguments, we derived sufficient conditions for the global stability of the disease-free planer equilibrium and positive equilibrium of system (1.1). By Theorems 4.1 and 4.2, we showed that (1) if (H4) holds, the infected predator population becomes extinct and the disease will be eliminated; that is, only sound predator and prey coexist; (2) if (H5) holds, the prey, the sound predator and the infected predator coexist. The disease will not be eliminated, and the system is permanent.



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*Source: 730783-2011-11-30.xml*