# Stability Analysis of a Delayed SIR Epidemic Model with Stage Structure and Nonlinear Incidence

Authors: Xiaohong Tian; Rui Xu
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

We investigate the stability of an SIR epidemic model with stage structure and time delay. By analyzing the eigenvalues of the corresponding characteristic equation, the local stability of each feasible equilibrium of the model is
established. By using comparison arguments, it is proved when the basic reproduction number is less than unity, the disease free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, sufficient
conditions are derived for the global stability of an endemic equilibrium of the model. Numerical simulations are carried out to illustrate the theoretical results.


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## Body

## 1. Introduction

LetS(t) denote the number of members of a population susceptible to a disease, I(t) the number of infective members, and R(t) the number of members who have been removed from the possibility of infection through full immunity, a standard SIR compartmental model is of the form [1](1.1)Ṡ(t)=A-μS(t)-βS(t)I(t),İ(t)=βS(t)I(t)-(μ+γ+ε)I(t),Ṙ(t)=γI(t)-μR(t),
where the parameters A,β,γ,μ,ε are positive constants in which A is the recruitment rate of susceptible population, μ represents the natural death rate of the population, ε is the disease-induced death rate of the infectives, and γ is the recovery rate from the infected compartment. It is assumed further that susceptibles become infectious by contact with infectious individuals. Later they may recover and join the group of immune (or dead) individuals. Based on the previous idea, different types of SIR epidemic models have been investigated (see, e.g., [2–6]). We note that most of the previous works assume that each species has the same contact and recovery rates ignoring the effect of stage structure. In the real world, any species has a process of growth and development, such as from immature to mature, and growth at various stages of life history showed differences in physiology. In the recent years, there have been a fair amount of work on epidemiological models with stage structure (see, e.g., [7–10]). In fact, the spread of disease is related to the species stage structure. Some diseases, such as measles, mumps, chickenpox and scarlet fever, only spread or have more opportunities to spread in children, and some other diseases, such as diphtheria, leptospirosis, a variety of sexually transmitted diseases, may spread in adult. By assuming that the mature population  does not contract the disease and the immature population is susceptible to the infection in [9], Xiao et al. proposed an SIR disease transmission model with stage structure and bilinear incidence rate as follows:(1.2)ẋ1(t)=a(1-γ)y(t)-r1x1(t)-βx1(t)x2(t),ẋ2(t)=βx1(t)x2(t)-bx2(t)-r2x2(t),ẋ3(t)=bx2(t)+aγy(t)-aγe-r3τy(t-τ)-r3x3(t),ẏ(t)=aγe-r3τy(t-τ)-ry2(t),
where x1(t),x2(t),x3(t) denote the densities of the immature population that are susceptible, infectious population and recovered population with immunity, respectively; y(t) denotes the density of the mature population which does not contract the disease. The parameters a,b,β,γ,r1,r2,r3,r are positive constants. a is the birth rate of the immature population. It is assumed that newborn individuals are the recovered population with immunity with probability γ(0<γ<1) and are susceptible population with probability 1-γ. β is the rate that the susceptible population become infective, and b is the rate that the infective population becomes recovered with immunity. r1,r2,r3 are the death rates of the susceptible, infective, recovered population, respectively, and r1≤r2 is reasonable for biological meaning. r is the death rate of the mature population. Finally, it is assumed that those immatures born at time t-τ that survive to time t exit from the immature population and enter the mature population. Xiao et al. [9] proved that if the basic reproduction number is less than unity, the disease-free equilibrium of system (1.2) is globally asymptotically stable; if the basic reproduction number is greater than unity, sufficient conditions were derived for the global stability of an endemic equilibrium.Incidence rate plays a very important role in the research of epidemiological models; it should generally be written asβU(N)S/N, where N is the total population size (see [1]). In classical epidemic models, bilinear incidence rate βSI and standard incidence rate βSI/N are frequently used. The bilinear incidence rate is based on the law of mass action. This contact law is more appropriate for communicable diseases such as influenza., but not for sexually transmitted diseases. For standard incidence rate, it may be a good approximation if the number of available partners is large enough and everybody could not make more contacts than is practically feasible [11]. After a study of the cholera epidemic spread in Bari in 1973, Capasso and Serio [12] introduced a saturated incidence rate g(I)S into epidemic models, where g(I) tends to a saturation level when I gets large, that is,(1.3)g(I)=βI1+αI,
where βI measures the force of infection of the disease, and 1/(1+αI) measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the susceptible individuals. This incidence rate seems more reasonable than the bilinear incidence rate g(I)S=βIS, 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 [13].Motivated by the work of Capasso and Serio [12] and Xiao et al. [9], in this paper, we are concerned with the effect of stage structure and saturation incidence on the dynamic of an SIR epidemic model. To this end, we study the following delayed differential system(1.4)ẋ1(t)=a(1-γ)y(t)-r1x1(t)-βx1(t)x2(t)1+αx2(t),ẋ2(t)=βx1(t)x2(t)1+αx2(t)-bx2(t)-r2x2(t),ẋ3(t)=bx2(t)+aγy(t)-aγe-r3τy(t-τ)-r3x3(t),ẏ(t)=aγe-r3τy(t-τ)-ry2(t).
The initial conditions for system (1.4) take the form(1.5)xi(θ)=ϕ1(θ),x2(θ)=ϕ2(θ),x3(θ)=ϕ3(θ),y(θ)=ψ(θ),ϕ1(θ)≥0,ϕ2(θ)≥0,ϕ3(θ)≥0,ψ(θ)≥0,θ∈[-τ,0],ϕ1(0)>0,ϕ2(0)>0,ϕ3(0)>0,ψ(0)>0,
where(1.6)ϕi,ψ∈C([-τ,0],ℝ+04),ϕi(0)>0(i=1,2,3),ψ(0)>0,
here ℝ+04={(x1,x2,x3,x4):xi≥0,i=1,2,3,4}.For continuity of initial conditions, we require(1.7)ϕ3(0)=∫-τ0aγψ(s)er3sds.It is easy to show that all solutions of system (1.4) with initial conditions (1.5) and (1.7) are defined on [0,+∞) and remain positive for all t≥0.The organization of this paper is as follows. In the next section, by analyzing the corresponding characteristic equations, the local stability of each of nonnegative equilibria of system (1.4) is discussed. In Section 3, we study the global stability of the disease-free equilibrium and the endemic equilibrium of system (1.4), respectively. Numerical simulations are carried out in Section 4 to illustrate the main theoretical results. A brief discussion is given in Section 5 to conclude this work.


## 2. Local Stability

In this section, we discuss the local stability of each of nonnegative equilibria of system (1.4) by analyzing the eigenvalues of the corresponding characteristic equations, respectively.System (1.4) always has a trivial equilibrium E0(0,0,0,0), and a disease free equilibrium E1(x10,0,x30,y0), where(2.1)x10=a2γ(1-γ)e-r3τrr1,x30=a2γ2e-r3τ(1-e-r3τ)rr3,y0=aγe-r3τr.
The basic reproduction number is given as(2.2)ℛ0=a2βγ(1-γ)e-r3τrr1(b+r2).
It is easy to prove that if ℛ0>1, system (1.4) has an endemic equilibrium E*(x1*,x2*,x3*,y*), where(2.3)x1*=aα(1-γ)y*+b+r2αr1+β,x2*=aβ(1-γ)y*-r1(b+r2)(b+r2)(αr1+β),x3*=bx2*+aγ(1-e-r3τ)y*r3,y*=aγe-r3τr.The characteristic equation of system (1.4) at the equilibrium E0(0,0,0,0) is of the form(2.4)(λ+r1)(λ+b+r2)(λ+r3)(λ-aγe-(λ+r3)τ)=0.
Obviously,  (2.4) always has three negative real roots λ=-r1,λ=-b2-r2, and λ=-r3. Noting that y=λ and y=aγe-τ(λ+r3) must intersect at a positive value of λ, hence, the equation λ-aγe-(λ+r3)τ=0 has a positive real root. Accordingly, E0 is unstable.The characteristic equation of system (1.4) at the equilibrium E1(x10,0,x30,y0) takes the form(2.5)(λ+r1)(λ+r3)[λ-a2βγ(1-γ)e-r3τ-rr1(b+r2)rr1][λ+2aγe-r3τ-aγe-(λ+r3)τ]=0.
Obviously, (2.5) always has three real roots λ1=-r1<0,λ2=-r3<0, and λ3=[a2βγ(1-γ)e-r3τ-rr1(b+r2)]/(rr1). Clearly, if ℛ0<1,λ3<0. Other roots are given by the roots of equation(2.6)λ+2aγe-r3τ-aγe-(λ+r3)τ=0.Letf(λ)=λ+2aγe-r3τ-aγe-(λ+r3)τ. Now, we claim that the roots of f(λ)=0 have only negative real parts. Suppose that Reλ≥0, then it follows from (2.6) that(2.7)Reλ=αγe-r3τ[e-τReλcos(τImλ)-2]≤-αγe-r3τ<0,
which leads to a contradiction. Hence, we have Reλ<0. Therefore, if ℛ0<1, the disease-free equilibrium E1(x10,0,x30,y0) is locally asymptotically stable. If ℛ0>1, (2.5) has a positive root, then the disease-free equilibrium E1 is unstable.The characteristic equation of system (1.4) at the endemic equilibrium E*(x1*,x2*,x3*,y*) takes the form(2.8)(λ+r3)(λ2+pλ+q)(λ+2aγe-r3τ-aγe-(λ+r3)τ)=0,
where(2.9)p=r1+[α(r1+b+r2)+β]x2*1+αx2*>0,q=(b+r2)(αr1+β)x2*1+αx2*>0.
Clearly, (2.8) always has a negative real root λ=-r3. Noting that p>0,q>0, roots of equation λ2+pλ+q=0 have only negative real parts. In addition, from the discussion above, we see that roots of the (2.6) have only negative real parts. By the general theory on characteristic equations of delay differential equations from [14], we see that if ℛ0>1, the endemic equilibrium E* is locally asymptotically stable.Based on the discussions above, we have the following result.Theorem 2.1.
For system (1.4), one has the following:  (i)
ifℛ0>1, the endemic equilibrium E*(x1*,x2*,x3*,y*) is locally asymptotically stable,(ii)
ifℛ0<1, the disease-free equilibrium E1(x10,0,x30,y0) is locally asymptotically stable.


## 3. Global Stability

In this section, we discuss the global stability of the disease-free equilibrium and the endemic equilibrium of system (1.4), respectively. The technique of proofs is to use a comparison argument and an iteration scheme.We first consider the subsystem of (1.4)(3.1)ẋ1(t)=a(1-γ)y(t)-r1x1(t)-βx1(t)x2(t)1+αx2(t),ẋ2(t)=βx1(t)x2(t)1+αx2(t)-bx2(t)-r2x2(t).
Letting z(t)=x1(t)+x2(t),x(t)=x2(t), system (3.1) becomes(3.2)ż(t)=a(1-γ)y(t)-r1z(t)+(r1-r2-b)x(t),ẋ(t)=x(t)[βz(t)1+αx(t)-βx(t)1+αx(t)-(b+r2)x(t)].
The initial conditions for system (3.2) take the form(3.3)z(θ)=φ1(θ),x(θ)=φ2(θ),φi(θ)≥0,φi(0)>0,i=1,2.
Clearly, system (3.2) has a nonnegative equilibrium A1(z0,0), where z0=a2γ(1-γ)e-r3τ/(rr1); when ℛ0>1, system (3.2) has a positive equilibrium A*(z*,x*), where(3.4)z*=a2γ(1-γ)e-r3τrr1+(r1-r2-b)x*r1,x*=a2βγ(1-γ)e-r3τ-rr1(b+r2)r(b+r2)(αr1+β).
Moreover, from Theorem 2.1, we see that A1 is locally asymptotically stable if ℛ0<1, and A* is locally asymptotically stable if ℛ0>1.To study the global dynamics of system (1.4), we need only to discuss the global behavior of solutions of system (3.2). In the following, we investigate the global asymptotic stability of the equilibria A1 and A* by using the comparison arguments and the iteration scheme [15], respectively. To this end, we need the following result developed by Song and Chen in [16].Lemma 3.1.
Consider the following equation:(3.5)ẋ(t)=ax(t-τ)-bx(t)-cx2(t),
where a,b,c,τ>0,x(t)>0 for t∈[-τ,0]. One has the following:  (i)
ifa>b, then limt→+∞x(t)=(a-b)/c;(ii)
ifa<b, then limt→+∞x(t)=0.Lemma 3.2.
Letℛ0>1. If αr1>β, then A*(z*,x*) is globally asymptotically stable.Proof.
Let(z(t),x(t)) be any positive solution of system (3.2) with initial condition (3.3). Let
(3.6)U1=lim supt→+∞z(t),V1=lim inft→+∞z(t),U2=lim supt→+∞x(t),V2=lim inft→+∞x(t).
Now we claim that U1=V1=z*,U2=V2=x*.
From Lemma3.1, it is easy to show that
(3.7)limt→∞y(t)=y0=y*=αβe-r3τr.
Hence, we know that for ε>0, there exists a T0>0 such that, if t>T0,
(3.8)y*-ε<y(t)<y*+ε.
We derive from the first equation of system (3.2) that
(3.9)ż(t)≤a(1-γ)(y*+ε)-r1z.
By comparison, we have
(3.10)lim supt→+∞z(t)≤a(1-γ)(y*+ε)r1.
Since this is true for arbitrary ε>0 sufficiently small, it follows that U1≤M1z, where
(3.11)M1z=a(1-γ)y*r1.
Hence, for ε>0 sufficiently small, there is a T1>T0 such that, if t>T1, z(t)≤M1z+ε.
Forε>0 sufficiently small, we derive from the second equation of system (3.2) that, for t>T1,
(3.12)ẋ(t)≤x(t)1+αx(t)[(β(M1z+ε)-(b+r2))-(β+α(b+r2))x(t)].
Consider the following auxiliary system:
(3.13)u̇(t)=u(t)[(β(M1z+ε)-(b+r2))-(β+α(b+r2))u(t)].
By Lemma 3.1 it follows from (3.13) that
(3.14)limt→+∞u(t)=β(M1z+ε)-(b+r2)β+α(b+r2).
By comparison, we obtain that
(3.15)lim supt→+∞x(t)≤β(M1z+ε)-(b+r2)β+α(b+r2).
Since the inequality is true for arbitrary ε>0 sufficiently small, it follows that U2≤M1x, where
(3.16)M1x=βM1z-(b+r2)β+α(b+r2).
Hence, for ε>0 sufficiently small, there is a T2>T1 such that, if t>T2,x(t)≤M1x+ε.
Forε>0 sufficiently small, we derive from the first equation of system (3.2) that, for t>T2,
(3.17)ż(t)≥a(1-γ)(y*-ε)-r1z(t)+(r1-r2-b)(M1x+ε).
By comparison and by Lemma 3.1, we have
(3.18)lim inft→+∞z(t)≥a(1-γ)(y*-ε)+(r1-r2-b)(M1x+ε)r1.
Since the inequality is true for arbitrary ε>0 sufficiently small, it follows that V1≥N1z, where
(3.19)N1z=a(1-γ)y*+(r1-r2-b)M1xr1.
Hence, for ε>0 sufficiently small, there is a T3>T2 such that, if t>T3, z(t)≥N1z-ε.
Forε>0 sufficiently small, we derive from the second equation of system (3.2) that, for t>T3,
(3.20)ẋ(t)≥x(t)1+αx(t)[(β(N1z-ε)-(b+r2))-(β+α(b+r2))x(t)].
By comparison and by Lemma 3.1, we have
(3.21)lim inft→+∞x(t)≥β(N1z-ε)-(b+r2)β+α(b+r2).
Since the inequality holds for arbitrary ε>0 sufficiently small, it follows that V2≥N1x, where
(3.22)N1x=βN1z-(b+r2)β+α(b+r2).
Therefore, for ε>0 sufficiently small, there is a T4>T3 such that if t>T4,x(t)≥N1x-ε.
Forε>0 sufficiently small, we derive from the first equation of system (3.2) that, for t>T4,
(3.23)ż(t)≤a(1-γ)(y*+ε)-r1z(t)+(r1-r2-b)(N1x-ε).
By comparison and by Lemma 3.1, we have
(3.24)lim supt→+∞z(t)≤a(1-γ)(y*+ε)+(r1-r2-b)(N1x-ε)r1.
Since the inequality holds for arbitrary ε>0 sufficiently small, it follows that U1≤M2z, where
(3.25)M2z=a(1-γ)y*+(r1-r2-b)N1xr1.
Hence, for ε>0 sufficiently small, there is a T5>T4 such that, if t>T5,z(t)≤M2z+ε.
Forε>0 sufficiently small, we derive from the second equation of system (3.2) that, for t>T5,
(3.26)ẋ(t)≤x(t)1+αx(t)[(β(M2z+ε)-(b+r2))-(β+α(b+r2))x(t)].
By comparison and by Lemma 3.1, we have
(3.27)lim supt→+∞x(t)≤β(M2z+ε)-(b+r2)β+α(b+r2).
Since the inequality holds for arbitrary ε>0 sufficiently small, we conclude that U2≤M2x, where
(3.28)M2x=βM2z-(b+r2)β+α(b+r2).
Therefore, for ε>0 sufficiently small, there is a T6>T5 such that, if t>T6,x(t)≤M2x+ε.
Forε>0 sufficiently small, we derive from the first equation of system (3.2) that, for t>T6,
(3.29)ż(t)≥a(1-γ)(y*-ε)-r1z(t)+(r1-r2-b)(M2x+ε).
By comparison and by Lemma 3.1 it follows that
(3.30)lim inft→+∞z(t)≥a(1-γ)(y*-ε)+(r1-r2-b)(M2x+ε)r1.
Since this is true for arbitrary ε>0 sufficiently small, we conclude that V1≥N2z, where
(3.31)N2z=a(1-γ)y*+(r1-r2-b)M2xr1.
Hence, for ε>0 sufficiently small, there is a T7>T6 such that, if t>T7,z(t)≥N2z-ε.
Forε>0 sufficiently small, we derive from the second equation of system (3.2) that, for t>T7,
(3.32)ẋ(t)≥x(t)1+αx(t)[(β(N2z-ε)-(b+r2))-(β+α(b+r2))x(t)].
By comparison and by Lemma 3.1 it follows that
(3.33)lim inft→+∞x(t)≥β(N2z-ε)-(b+r2)β+α(b+r2).
Since this is true for arbitrary ε>0 sufficiently small, we conclude that V2≥N2x, where
(3.34)N2x=βN2z-(b+r2)β+α(b+r2).
Hence, for ε>0 sufficiently small, there is a T8>T7 such that, if t>T8,x(t)≥N2x-ε.
Continuing this process, we derive four sequencesMnz,Mnx,Nnz,Nnx(n=1,2,…) such that for n≥2,
(3.35)Mnz=a(1-γ)y*+(r1-r2-b)Nn-1xr1,Nnz=a(1-γ)y*+(r1-r2-b)Mnxr1,Mnx=βMnz-(b+r2)β+α(b+r2),Nnx=βNnz-(b+r2)β+α(b+r2).
Clearly, we have
(3.36)Nnx≤V2≤U2≤Mnx,Nnz≤V1≤U1≤Mnz.
It follows from (3.36) that
(3.37)Mn+1x=aβ(1-γ)y*-r1(b+r2)r1[β+α(b+r2)][1+β(r1-r2-b)r1(β+α(b+r2))]+β2(r1-r2-b)2r12[β+α(b+r2)]2Mnx.
Noting that Mnx≥x* and αr1>β, we derive from (3.37) that
(3.38)Mn+1x-Mnx=aβ(1-γ)y*-r1(b+r2)r1[β+α(b+r2)][1+β(r1-r2-b)r1(β+α(b+r2))]+[β2(r1-r2-b)2r12[β+α(b+r2)]2-1]Mnx≤aβ(1-γ)y*-r1(b+r2)r1[β+α(b+r2)][1+β(r1-r2-b)r1(β+α(b+r2))]+[β2(r1-r2-b)2r12[β+α(b+r2)]2-1]x*=0.
Hence, the sequence Mnx is monotonically nonincreasing. Therefore, limn→+∞Mnx exists. Taking n→+∞, it follows from (3.37) that
(3.39)limn→+∞Mnx=a2βγ(1-γ)e-r3τ-rr1(b+r2)r(b+r2)(r1α+β)=x*.
We therefore obtain from (3.35) and (3.39) that
(3.40)limn→+∞Nnx=x*,limn→+∞Mnz=z*,limn→+∞Nnz=z*.
It follows from (3.36), (3.39), and (3.40) that
(3.41)U1=V1=z*,U2=V2=x*.
We therefore have
(3.42)limt→+∞z(t)=z*,limt→+∞x(t)=x*.
Noting that if ℛ0>1 and αr1>β hold, the positive equilibrium A* is locally asymptotically stable, we conclude that A* is globally asymptotically stable. The proof is complete.Theorem 3.3.
Ifℛ0>1 and αr1>β hold, then the endemic equilibrium E*(x1*,x2*,x3*,y*) of system (1.4) is globally asymptotically stable; that is, the disease remains endemic.Proof.
From (3.7), we know that limt→+∞y(t)=y*=y0=aγe-r3τ/r. According to the results of Lemma 3.2, we prove that
(3.43)limt→+∞x1(t)=x1*,limt→+∞x2(t)=x2*.
In the following, we show the existence oflimt→+∞x3(t).
By Lemma3.2, it follows from (3.43) that for ε>0 sufficiently small, there exists a T>0, such that, if t>T,
(3.44)y*-ε<y(t)<y*+ε,x1*-ε<x1(t)<x1*+ε,x2*-ε<x2(t)<x2*+ε.
Therefore, we derive from the third equation of system (1.4) that, for t>T+τ,
(3.45)ẋ3(t)≤b(x2*+ε)+aγ(y*+ε)-aγe-r3τ(y*-ε)-r3x3(t).
By comparison, we have
(3.46)lim supt→+∞x3(t)≤b(x2*+ε)+aγ(y*+ε)-aγe-r3τ(y*-ε)r3.
Since the inequality holds for arbitrary ε>0 sufficiently small, we have lim supt→+∞x3(t)≤Mx3, where
(3.47)Mx3=bx2*+aγ(1-e-r3τ)y*r3.
Hence, for ε>0 sufficiently small, there is a T1>T such that, if t>T1,
(3.48)x3(t)≤Mx3+ε.
Again, forε>0 sufficiently small, it follows from the third equation of system (1.4) that, for t>T1+τ,
(3.49)ẋ3(t)≥b(x2*-ε)+αγ(y*-ε)-αγe-r3τ(y*+ε)-r3x3(t).
By comparison, we have
(3.50)lim inft→+∞x3(t)≥b(x2*-ε)+aγ(y*-ε)-aγe-r3τ(y*+ε)r3.
Since the inequality holds for arbitrary ε>0 sufficiently small, we conclude that lim inft→+∞x3(t)≥Nx3, where
(3.51)Nx3=bx2*+aγ(1-e-r3τ)y*r3.
Hence, for ε>0 sufficiently small, there is a T2>T1 such that, if t>T2,
(3.52)x3(t)≥Nx3-ε.
It follows from (3.48) and (3.52) that
(3.53)limt→+∞x3(t)=bx2*+aγ(1-e-r3τ)y*r3=x3*.
Noting that if ℛ0>1 and αr1>β hold, the endemic equilibrium E* is locally asymptotically stable, we see that E* is globally asymptotically stable. This completes the proof.Theorem 3.4.
Ifℛ0<1 holds, the disease-free equilibrium E1(x10,0,x30,y0) of system (1.4) is globally asymptotically stable; that is, the disease fades out.Proof.
Chooseε>0 sufficiently small satisfying
(3.54)β(a(1-γ)y0r1+ε)<b+r2.
From (3.7), we know that for ε>0 sufficiently small, there exists a t0>0 such that if t>t0,(3.55)y0-ε<y(t)<y0+ε.
We derive from the first equation of system (3.2) that
(3.56)ż(t)≤a(1-γ)(y0+ε)-r1z.
By Lemma 3.1 and by a comparison argument, we get
(3.57)lim supt→+∞z(t)≤a(1-γ)(y0+ε)r1.
Since this inequality holds for arbitrary ε>0 sufficiently small, we conclude that
(3.58)lim supt→+∞z(t)≤a(1-γ)y0r1.
Hence, for ε>0 sufficiently small, there is a t1>t0 such that, for t>t1,
(3.59)z(t)≤a(1-γ)y0r1+ε.
For ε>0 sufficiently small satisfying (3.54), it follows from (3.59) and the second equation of system (3.2) that
(3.60)ẋ(t)≤x(t)1+αx(t)[β(a(1-γ)y0r1+ε)-(b+r2)-(β+α(b+r2))x(t)].
Noting that (3.54) holds, we conclude that
(3.61)limt→+∞x(t)=0.
Hence, for ε>0 sufficiently small satisfying (3.54), there is a t2>t1 such that, if t>t2,x(t)<ε.
On the other hand, we derive from the first equation of system (3.2) that, for t>t2,
(3.62)ż(t)≥a(1-γ)(y0-ε)-r1z(t)+(r1-r2-b)ε.
By Lemma 3.1 and by a comparison argument, we have
(3.63)lim inft→+∞z(t)≥a(1-γ)(y0-ε)+(r1-r2-b)εr1.
Since this inequality is true for arbitrary ε>0 sufficiently small, we conclude that
(3.64)lim inft→+∞z(t)≥a(1-γ)y0r1,
which, together with (3.58), yields
(3.65)limt→+∞z(t)=a(1-γ)y0r1=z0.
According to (3.61) and (3.65), we can easily prove that
(3.66)limt→+∞x1(t)=x10,limt→+∞x2(t)=0.
Using a similar argument as in the proof of Theorem 3.3, we can show that if ℛ0<1, then
(3.67)limt→+∞x3(t)=aγ(1-e-r3τ)y0r3=x30.
Noting that if ℛ0<1, the disease-free equilibrium E1 is locally stable, we conclude that E1 is globally asymptotically stable. This completes the proof.


## 4. Numerical Examples

In this section, we give two examples to illustrate the main theoretical results above.Example 4.1.
In system (1.4), let α=1,β=2,γ=0.5,r1=3,r2=1,r3=0.5,r=1,a=6,b=2,τ=1. Computation gives the following value for the basic reproduction number ℛ0=2e-1/2>1, and system (1.4) has a unique endemic equilibrium E*((9e-1/2+3)/5,(4e-1/2-3)/5,-3e-1+31e-1/2/5-12/5,3e-1/2). Clearly, αr1-β=1>0. By Theorem 3.3, we see that the endemic equilibrium E* of system (1.4) is globally asymptotically stable. Numerical simulation illustrates the previous result (see Figure 1).Figure 1
The numerical solution of system (1.4) with α=1,β=2,γ=0.5,r1=3,r2=1,r3=0.5,r=1,a=6,b=2,τ=1; (ϕ1,ϕ2,ϕ3,ψ)=(15,3,4,10).Example 4.2.
In system (1.4), let α=1,β=1,γ=0.5,r1=0.5,r2=1,r3=1,r=0.5,a=2,b=2,τ=1. Computation gives the following value for the basic reproduction number ℛ0=4e-1/3<1, system (1.4) has only a disease-free equilibrium E1(4e-1,0,2e-1(1-e-1),2e-1). By Theorem 3.4, we see that the disease-free equilibrium E1 of system (1.4) is globally asymptotically stable. Numerical simulation illustrates this fact (see Figure 2).Figure 2
The numerical solution of system (1.4) with α=1,β=1,γ=0.5,r1=0.5,r2=1,r3=1,r=0.5,a=2,b=2,τ=1;(ϕ1,ϕ2,ϕ3,ψ)=(20,2,6,15).


## 5. Discussion

In this paper, we have discussed the effect of stage structure and saturation incidence rate on an SIR epidemic model with time delay. The basic reproduction numberℛ0 was found. The local stability of each of feasible equilibria of system (1.4) was investigated. When the basic reproduction number is greater than unity, by using the iteration scheme, we have established sufficient conditions for the global stability of the endemic equilibrium of system (1.4). By Theorem 3.3, we see that when ℛ0>1 and αr1>β, the endemic equilibrium is globally stable. Biologically, these indicate that when the proportionality (infection) constant and /or the birth rate of the immature population is sufficiently large and the death rates of susceptible population and the mature population are sufficiently small such that ℛ0>1, then the disease remains endemic. On the other hand, by Theorem 3.4, we see that, if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. Biologically, if the proportionality (infection) constant and /or the birth rate of the immature population is small enough and the death rates of susceptible population and the mature population are large enough such that ℛ0<1, then the disease fades out. We would like to point out here that Theorem 3.3 has room for improvement, we leave this for future work.



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*Source: 979217-2009-10-29.xml*