# Global Stability of a Virus Infection Model with Time Delay and Absorption

Authors: Xiaohong Tian; Rui Xu
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

In this paper, a virus infection model with time delay and absorption is studied. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the model is established. By using comparison arguments, it is shown that the infection free equilibrium is globally asymptotically stable when the basic reproduction ratio is less than unity. When the basic reproduction ratio is greater than unity, sufficient conditions are derived for the global stability of the virus-infected equilibrium. Numerical simulations are carried out to illustrate the theoretical results.


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## Body

## 1. Introduction

Mathematical modelling has been proven to be valuable in studying the dynamics of virus infection, such as HIV/AIDS, HBV, and HCV. In recent years, great attention has been paid by many researchers to the pathogen infectious agent, or germ, which can cause disease or illness to its host (see, e.g., [1–3]). Based on the clinical experiment of chronic HBV carriers treated with various doses of lamivudine, Nowak and Bangham [4] and Bonhoeffer et al. [5] proposed a classical mathematical model describing the interaction between the susceptible host cells (x), infected host cells (y), and free virus particles (v), which is formulated by the following differential equations:  (1.1)ẋ(t)=λ-dx(t)-βx(t)v(t),ẏ(t)=βx(t)v(t)-ay(t),v̇(t)=ky(t)-uv(t),
where hepatocytes are produced at a rate λ, die at rate dx, and become infected at rate βxv; infected hepatocytes are produced at rate βxv and die at rate ay; free viruses are produced from infected cells at rate ky and are removed at rate uv. It is assumed that parameters a, d, k, u, λ, and β are positive constants. The basic reproductive ratio ℛ0 was found in [4]. Furthermore, in [6], by constructing suitable Lyapunov function, Korobeinikov showed that the infection-free equilibrium is globally asymptotically stable if ℛ0<1, and the virus-infected equilibrium is globally asymptotically stable if ℛ0>1.It is assumed in model (1.1) that the infection process follows the principle of mass action [7], namely, for each uninfected cell and free virus particle is assumed to be constant β between the rate of infection. However, studies of parasitic infections have shown that the relationship between the dose and rate of infection is clearly nonlinear. In [8], a more general saturated infection rate βxvp/(1+αvq) was suggested, where p, q, and α are positive constants.The virus life cycle plays a crucial role in disease progression. The binding of a viral particle to a receptor on a target cell initiates a cascade of events that can ultimately lead to the target cell becoming productively infected, that is, producing new virus. In model (1.1), it is assumed that as soon as virus contacts a target cell, the cell begins producing virus. This is not biologically sensible. In reality, there is a time delay between initial viral entry into a cell and subsequent viral production [9]. There have been some works on virus infection model in the literature (see, e.g., [9–14]). In [10], Herz et al. examined the effect of including a constant delay in the source term for productively infected T-cells. In [11], Li and Ma considered the following more general HIV-1 infection model with saturation incidence rate and time delay:  (1.2)ẋ(t)=λ-dx(t)-βx(t)v(t)1+v(t),ẏ(t)=βx(t-τ)v(t-τ)1+v(t-τ)-ay(t),v̇(t)=ky(t)-uv(t).
By analyzing the transcendental characteristic equations, sufficient conditions for the local asymptotic stability of the equilibria were studied, and by using Lyapunov-LaSalle invariance principal, the global asymptotic stability of the viral-free equilibrium was given.We note that in (1.1) and (1.2), the loss of pathogens due to the absorption into uninfected cells are ignored. In reality, when a pathogen enters an uninfected cell, the number of pathogens in the blood decreases by one. This is called the absorption effect [7]. In [15], considering the erythrocytic cycle in the absence of an immunological response by the host, Anderson et al. presented the following model for malaria infection:  (1.3)ẋ(t)=λ-dx(t)-βx(t)v(t),ẏ(t)=βx(t)v(t)-ay(t),v̇(t)=ky(t)-βx(t)v(t)-uv(t),
where the term -βxv in the third equation of (1.3) represents the absorption effect.Motivated by the works of Anderson et al.  [15], Li and Ma [11], and Song and Neumann [8], in this paper, we study the following virus infection model with a time delay and absorption:  (1.4)ẋ(t)=λ-dx(t)-βx(t)v(t)1+αv(t),ẏ(t)=βx(t-τ)v(t-τ)1+αv(t-τ)-ay(t),v̇(t)=ky(t)-βx(t)v(t)1+αv(t)-uv(t).
The initial conditions for system (1.4) take the form  (1.5)x(θ)=ϕ1(θ),y(θ)=ϕ2(θ),v(θ)=ϕ3(θ),ϕ1(θ)≥0,ϕ2(θ)≥0,ϕ3(θ)≥0,θ∈[-τ,0],ϕ1(0)>0,ϕ2(0)>0,ϕ3(0)>0,
where (ϕ1(θ),ϕ2(θ),ϕ3(θ))∈C([-τ,0],ℝ+03), here ℝ+03={(x1,x2,x3):xi≥0,i=1,2,3}.It is easy to show that all solutions of system (1.4) with initial condition (1.5) is defined on [0,+∞) and remain positive for all t≥0.The organization of this paper is as follows. In the next section, we introduce some notations and state several lemmas which will be essential to our proofs. In Section3, by analyzing the corresponding characteristic equations, the local stability of each of nonnegative equilibria of system (1.4) is discussed. In Section 4, by using an iteration technique, we study the global stability of the uninfected equilibrium of system (1.4). By comparison arguments, we discuss the global stability of the virus-infected equilibrium of system (1.4). Numerical simulations are carried out in Section 5 to illustrate the main theoretical results. The paper ends with conclusions in Section 6.


## 2. Preliminaries

In this section, based on the work developed by Xu and Ma [16], we introduce some notations and state several results which will be useful in next section.Letℝ+n be the cone of nonnegative vectors in ℝn. If x,y∈ℝn, we write x≤y(x<y) if xi≤yi(xi<yi) for 1≤i≤n. Let {e1,e2,…,en} denote the standard basis in ℝn. Suppose, r≥0 and let C=C([-r,0],ℝn) be the Banach space of continuous functions mapping the interval [-r,0] into ℝn with supremum norm. If ϕ,ψ∈C, we write ϕ≤ψ(ϕ<ψ) when the indicated inequality holds at each point of [-r,0]. Let C+={ϕ∈C:ϕ≥0}, and let ⋀ denote the inclusion ℝn→C([-r,0],ℝn) by x→x̂, x̂(θ)=x, θ∈[-r,0]. Denote the space of functions of bounded variation on [-r,0] by BV[-r,0]. If t0∈ℝ, A≥0 and x∈C([-t0-r,t0+A],ℝn), then for any t∈[t0,t0+A], we let xt∈C be defined by xt(θ)=x(t+θ), -r≤θ≤0.We now consider(2.1)ẋ(t)=f(t,xt).We assume throughout this section thatf:ℝ×C→ℝn is continuous, f(t,ϕ) is continuously differentiable in ϕ,f(t+T,ϕ)=f(t,ϕ) for all (t,ϕ)∈ℝ×C+, and some T>0. Then, by [17], there exists a unique solution of (2.1) through (t0,ϕ) for t0∈ℝ, ϕ∈C+. This solution will be denoted by x(t,t0,ϕ) if we consider the solution in ℝn, or by xt(t0,ϕ) if we work in the space C. Again, by [17], x(t,t0,ϕ)(xt(t0,ϕ)) is continuously differentiable in ϕ. In the following, the notation xt0=ϕ will be used as the condition of the initial data of (2.1), by which we mean that we consider the solution x(t) of (2.1) which satisfies x(t0+θ)=ϕ(θ), θ∈[-r,0].To proceed further, we need the following results. Letr=(r1,r2,…,rn)∈ℝ+n, |r|=maxi{ri}, and define (2.2)Cr=∏i=1nC([-ri,0],R).We writeϕ=(ϕ1,ϕ2,…,ϕn) for a generic point of Cr. Let Cr+={ϕ∈Cr:ϕ≥0}. Due to the ecological applications, we choose Cr+ as the state space of (2.1) in the following discussions.Fixϕ0∈Cr+ arbitrarily. Then, we set L(t,·)=Dϕ0f(t,ϕ0),Dϕ0f(t,ϕ0) denotes the Frechet derivation of f with respect to ϕ0. It is convenient to have the standard representation of L=(L1,L2,…,Ln) as (2.3)Li(t,ϕ)=∑j=1n∫-rj0ϕj(θ)dθηij(θ,t),1≤i≤n,
in which ηij:ℝ×ℝ→ℝ satisfies(2.4)ηij(θ,t)=ηij(0,t),θ≥0,ηij(θ,t)=0,θ≤-rj,ηij(⋅,t)∈BV[-rj,0],
where ηij(·,t) is continuous from the left in (-rj,0).We make the following assumptions for (2.1).(h0)
Ifϕ,ψ∈C+, ϕ≤ψ, and ϕi(0)=ψi(0) for some i, then fi(t,ϕ)≤fi(t,ψ).(h1)
For allϕ∈Cr+ with ϕi(0)=0, Li(t,ϕ)≥0 for t∈ℝ.(h2)
The matrixA(t) defined by
(2.5)A(t)=col(L(t,ê1),L(t,ê2),…,L(t,ên))=(ηij(0,t))
is irreducible for each t∈ℝ.(h3)
For eachj, for which rj>0, there exists i such that for all t∈ℝ and for positive constant ɛ sufficiently small, ηij(-rj+ɛ,t)>0.(h4)
Ifϕ=0, then x(t,t0,ϕ)≡0 for all t≥t0.The following result was established by Wang et al. [18].Lemma 2.1.
Let(h1)–(h4) hold. Then, the hypothesis (h0) is valid  (i)
Ifϕ and ψ are distinct elements of Cr+ with ϕ≤ψ and [t0,t0+σ) with n|r|<σ≤∞ is the intersection of the maximal intervals of existence of x(t,t0,ϕ) and x(t,t0,ψ), then
(2.6)0≤{x(t,t0,ϕ)≤x(t,t0,ψ)fort0≤t<t0+σ,x(t,t0,ϕ)<x(t,t0,ψ)fort0+n|r|≤t<t0+σ.(ii)
Ifϕ∈Cr+, ϕ≠0, t0∈ℝ and x(t,t0,ϕ) is defined on [t0,t0+σ) with σ>n|r|, then
(2.7)0<x(t,t0,ϕ)fort0+n|r|≤t<t0+σ.This lemma shows that if(h1)–(h4) hold, then the positivity of solutions of (2.1) follows.The following definition and results are useful in proving our main result in this section.Definition 2.2.
LetA=(aij)n×n be an n×n matrix, and let P1,…,Pn be distinct points of the complex plane. For each nonzero element aij of A, connect Pi to Pj with a directed line PiPj¯. The resulting figure in the complex plane is a directed graph for A. We say that a directed graph is strongly connected if, for each pair of nodes Pi,Pj with i≠j, there is a directed path
(2.8)PiPk1⃗,Pk1Pk2⃗,…,Pkr-1Pj⃗,
connecting Pi and Pj. Here, the path consists of r directed lines.Lemma 2.3 (see [19]).
A square matrix is irreducible if and only if its directed graph is strongly connected.Lemma 2.4 (see [20]).
If (2.1) is cooperative and irreducible in D, where D is an open subset of C, and the solutions with positive initial data is bounded, then the trajectory of (2.1) tends to some single equilibrium.We now consider the following delay differential system:(2.9)u̇1(t)=a1u2(t-τ)1+αu2(t-τ)-au1(t),u̇2(t)=ku1(t)-a2u2(t)1+αu2(t)-uu2(t)
with initial conditions  (2.10)ui(s)=ϕi(s)≥0,s∈[-τ,0),ϕi(0)>0,ϕi∈C([-τ,0),R+)(i=1,2).System (2.9) always has a trivial equilibrium A0(0,0). If ka1-aa2>au, then system (2.9) has a unique positive equilibrium A*(u1*,u2*), where (2.11)u1*=a1(ka1-aa2-au)αa(ka1-aa2),u2*=ka1-aa2-auαau.The characteristic equation of system (2.9) at the positive equilibrium A0 takes the form (2.12)λ2+p1λ+p0+q0e-λτ=0,
where (2.13)p0=a(a2+u),p1=a+a2+u,q0=-ka1.
Noting that (2.14)p1>0,p0+q0=au-(ka1-aa2),
if ka1-aa2<au, then the equilibrium A0 is locally stable when τ=0. If ka1-aa2>au, then A0 is unstable when τ=0.It is easy to show thatp12-2p0=a2+(a2+u)2>0. If ka1-aa2<au, then p02-q02>0. By Kuang [21], we see that the equilibrium A0 is locally asymptotically stable for all τ>0. If ka1-aa2>au, then A0 is unstable for all τ>0.The characteristic equation of system (2.9) at the positive equilibrium A* is of the form (2.15)λ2+g1λ+g0+h0e-λτ=0,
where (2.16)g0=a(u+a2(1+αu2*)2),g1=a+u+a2(1+αu2*)2,h0=-ka1(1+αu2*)2.
Note that (2.17)g1>0,g0+h0=au[au-(ka1-aa2)]ka1-aa2.
Hence, if ka1-aa2>au, then the positive equilibrium A* is locally stable when τ=0. If ka1-aa2<au, then A* is unstable when τ=0.It is easy to see that(2.18)g12-2g0=a2+(u+a2(1+αu2*)2)2>0.
If ka1-aa2>au, then g02-h02>0. By Kuang [21], we see that the positive equilibrium A* is locally asymptotically stable for all τ>0. If ka1-aa2<au, then A* is unstable for all τ>0.Lemma 2.5.
For system (2.9), one hase the following.  (i)
Ifka1-aa2>au, then the positive equilibrium A*(u1*,u2*) is globally stable.(ii)
Ifka1-aa2<au, then the equilibrium A0(0,0) is globally stable.Proof.
We represent the right-hand side of (2.9) by f(t,xt)=(f1(t,xt),f2(t,xt)), and set
(2.19)L(t,⋅)=Dϕf(t,ϕ).
By direct calculation, we have
(2.20)L1(t,h)=a1(1+αϕ2(-τ))2h2(-τ)-ah1(0),L2(t,h)=kh1(0)-a2(1+αϕ2(0))2h2(0)-uh2(0).
We now claim that the hypotheses (h1)–(h4) hold for system (2.9). It is easily seen that (h1) and (h4) hold for system (2.9). We need only to verify that (h2) and (h3) hold.
The matrixA(t) takes the form(2.21)(-aa1(1+αϕ2(-τ))2k-a2(1+αϕ2(0))2-u).
Clearly, the matrix A(t) is irreducible for each t∈ℝ.
From the definition ofA(t) and ηij, it is readily seen that η12(θ,t)=η12(0,t)=a1/(1+αϕ2(-τ))2, η21(θ,t)=η21(0,t)=k for θ≥0, ηij(θ,t)=0, i≠j for θ≤-τ, and ηij(·,t)∈BV[-τ,0], where ηij is a positive Borel measure on [-τ,0]. Therefore, ηij(·,t)>0. Thus, for each j, there is i≠j such that ηij(-rj+ɛ,t)=ηij(-τ+ɛ,t)>0 for all t∈ℝ and for ɛ>0 sufficiently small, i=1,2. Hence, (h3) holds.
Thus, the conditions of Lemma2.1 are satisfied. Therefore, the positivity of solutions of system (2.9) follows. It is easy to see that system (2.9) is cooperative. By Lemma 2.3, we see that any solution starting from D=Cτ+ converges to some single equilibrium. However, system (2.9) has only two equilibria: A0 and A*. Note that if ka1-aa2>au, then the positive equilibrium A* is locally stable, and the equilibrium A0 is unstable. Hence, any solution starting from D converges to A*(u1*,u2*) if ka1-aa2>au. Using a similar argument one can show the global stability of the equilibrium A0 when ka1-aa2<au. This completes the proof.


## 3. Local Stability

In this section, we discuss the local stability of equilibria of system (1.4) by analyzing the corresponding characteristic equations.System (1.4) always has an infection free equilibrium E0(λ/d,0,0).Let(3.1)R0=λβ(k-a)aud.
ℛ0 is called the basic reproduction ratio of system (1.4). It is easy to show that if ℛ0>1, system (1.4) has a virus infected equilibrium E*(x*,y*,v*), where (3.2)x*=au(1+αv*)β(k-a),y*=uk-av*,v*=λβ(k-a)-audau(αd+β).The characteristic equation of system (1.4) at the infection free equilibrium E0 is of the form  (3.3)(s+d)(s2+p1s+p0+q0e-sτ)=0,
where (3.4)p0=a(u+λβd),p1=a+u+λβd,q0=kλβd.
Obviously, (3.3) always has a negative real root s=-d. All other roots of (3.3) are determined by  (3.5)s2+p1s+p0+q0e-sτ=0.
It is easy to show that p1>0, p0+q0>0, then the infection free equilibrium E0 of system (1.4) is locally asymptotically stable when τ=0.Ifiω(ω>0) is a solution of (3.3), by calculating, we have the following:  (3.6)ω4+(p12-2p0)ω2+p02-q02=0.
Note that (3.7)p12-2p0=a2+(u+λβd)>0,p02-q02=1d2[aud+λβ(k+a)][aud+λβ(a-k)].
If ℛ0<1, then p02-q02>0. Therefore, (3.6) has no positive roots. Accordingly, if ℛ0<1, the infection free equilibrium E0 of system (1.4) is locally asymptotically stable for all τ>0; if ℛ0>1, (3.6) has at least a positive real root. Accordingly, E0 is unstable.The characteristic equation of system (1.4) at the virus infected equilibrium E*(x*,y*,v*) takes the form  (3.8)s3+g2s2+g1s+g0+(h1s+h0)e-sτ=0,
where (3.9)g0=a(du+dβx*(1+αv*)2+uβv*1+αv*),g1=au+du+ad+(a+d)βx*(1+αv*)2+(a+u)βv*1+αv*,g2=a+u+d+βx*(1+αv*)2+βv*1+αv*,h0=-kdβx*(1+αv*)2,h1=-kβx*(1+αv*)2.
When τ=0, (3.8) becomes  (3.10)s3+g2s2+(g1+h1)s+g0+h0=0.
By direct calculation, we have (3.11)g0+h0=au(αd+β)v*1+αv*>0,g2(g1+h1)-(g0+h0)=a2uαv*1+αv*+a2λx*+kdβx*(1+αv*)2+[u+λx*+βx*(1+αv*)2]×[auαv*1+αv*+(u+a)λx*+dβx*(1+αv*)2]>0.
Clearly, all roots of (3.10) have only negative real parts.Ifiω(ω>0) is a solution of (3.8), separating real and imaginary parts, it follows that  (3.12)ω3-g1ω=h1ωcos⁡ωτ-g0sinωτ,g2ω2-g0=h1ωsinωτ+h0cos⁡ωτ.
Squaring and adding the two equations of (3.12), we derive that  (3.13)ω6+C1ω4+C2ω2+C3=0,
where (3.14)C1=g22-2g1,C2=g12-2g0g2-h12,C3=g02-h02.
Clearly, C3=g02-h02>0. It is easy to show that (3.15)C1=a2+u2+d2+βx*1+αv*[2(d+u1+αv*)+(1+11+αv*)2]>0,C2=(a2+u2)(d+βv*1+αv*)2+auαv*1+αv*[au+(k+a)βx*(1+αv*)2]+2(a2+du)β2x*v*(1+αv*)3+d2βx*(1+αv*)2[2u+βx*(1+αv*)2]>0.
Hence, (3.13) has no positive roots. Accordingly, if ℛ0>1, the virus-infected equilibrium E* of system (1.4) exists and is locally asymptotically stable for all τ>0.Based on the discussions above, we have the following result.Theorem 3.1.
For system (1.4), one has the following.  (i)
Ifℛ0<1, the infection free equilibrium E0(λ/d,0,0) is locally asymptotically stable. If ℛ0>1, then E0(λ/d,0,0) is unstable.(ii)
Ifℛ0>1, the virus infected equilibrium E*(x*,y*,v*) is locally asymptotically stable.


## 4. Global Stability

In this section, we discuss the global stability of the uninfected equilibrium and the virus infected equilibrium of system (1.4), respectively. The technique of proofs is to use a comparison argument and an iteration scheme [22].Theorem 4.1.
Letℛ0>1. The virus infected equilibrium E*(x*,y*,v*) of system (1.4) is globally asymptotically stable provided that  (H1)
αd(k-a)>aβ,(H2)
0<(k+a)[λα(k+a)-au]<4kλaα.Proof.
Let(x(t),y(t),v(t)) be any positive solution of system (1.4) with initial condition (1.5). Let
(4.1)U1=lim⁡sup⁡⁡t→+∞x(t),V1=lim⁡inf⁡t→+∞x(t),U2=lim⁡sup⁡⁡t→+∞y(t),V2=lim⁡inf⁡t→+∞y(t),U3=lim⁡sup⁡⁡t→+∞v(t),V3=lim⁡inf⁡⁡t→+∞v(t).
Now, we claim that U1=V1=x*, U2=V2=y*, U3=V3=v*.
It follows from the first equation of system (1.4) that(4.2)ẋ(t)≤λ-dx(t).
By comparison, we derive that
(4.3)U1=limsup⁡t→+∞x(t)≤λd:=M1x.
Hence, for ɛ>0 sufficiently small, there exists a T1>0 such that if t>T1, x(t)≤M1x+ɛ. We therefore, derive from the second and the third equations of system (1.4) that for t>T1+τ,
(4.4)ẏ(t)≤β(M1x+ɛ)v(t-τ)1+αv(t-τ)-ay(t),v̇(t)≤ky(t)-uv(t).
Consider the following auxiliary equations:
(4.5)u̇1(t)=β(M1x+ɛ)u2(t-τ)1+αu2(t-τ)-au1(t),u̇2(t)=ku1(t)-uu2(t).
Since ℛ0>1, by Lemma 2.5, it follows from (4.5) that
(4.6)lim⁡t→+∞u1(t)=kβ(M1x+ɛ)-aukaα,lim⁡t→+∞u2(t)=kβ(M1x+ɛ)-auauα.
By comparison, we obtain that
(4.7)U2=lim⁡sup⁡⁡t→+∞y(t)≤kβ(M1x+ɛ)-aukaα,U3=lim⁡sup⁡⁡t→+∞v(t)≤kβ(M1x+ɛ)-auauα.
Since these inequalities are true for arbitrary ɛ>0, it follows that U2≤M1y, U3≤M1v, where
(4.8)M1y=kβM1x-aukaα,M1v=kβM1x-auauα.
Hence, for ɛ>0 sufficiently small, there is a T2≥T1+τ such that if t>T2, y(t)≤M1y+ɛ,v(t)≤M1v+ɛ.
Forɛ>0 sufficiently small, we derive from the first equation of system (1.4) that for t>T2,(4.9)ẋ(t)≥λ-dx(t)-βx(t)(M1v+ɛ)1+α(M1v+ɛ).
A comparison argument shows that
(4.10)V1=lim⁡inf⁡t→+∞x(t)≥λ(1+α(M1v+ɛ))d+(αd+β)(M1v+ɛ).
Since these inequalities are true for arbitrary ɛ>0, it follows that V1≥N1x, where
(4.11)N1x=λ(1+αM1v)d+(αd+β)M1v.
Hence, for ɛ>0 sufficiently small, there is a T3≥T2 such that if t>T3, x(t)≥N1x-ɛ.
Forɛ>0 sufficiently small, it follows from the second and the third equations of system (1.4) that for t>T3+τ,(4.12)ẏ(t)≥β(N1x-ɛ)v(t-τ)1+αv(t-τ)-ay(t),v̇(t)≥ky(t)-β(M1x+ɛ)v(t)1+αv(t)-uv(t).
Consider the following auxiliary equations:
(4.13)u̇1(t)=β(N1x-ɛ)u2(t-τ)1+αu2(t-τ)-au1(t),u̇2(t)=ku1(t)-β(M1x+ɛ)u2(t)1+αu2(t)-uu2(t).
Since (H1) holds, by Lemma 2.5, it follows from (4.13) that
(4.14)lim⁡t→+∞u1(t)=β(N1x-ɛ)[kβ(N1x-ɛ)-aβ(M1x+ɛ)-au]aα[kβ(N1x-ɛ)-aβ(M1x+ɛ)],lim⁡t→+∞u2(t)=kβ(N1x-ɛ)-aβ(M1x+ɛ)-auauα.
By comparison, we derive that
(4.15)V2=lim⁡inf⁡t→+∞y(t)≥β(N1x-ɛ)[kβ(N1x-ɛ)-aβ(M1x+ɛ)-au]aα[kβ(N1x-ɛ)-aβ(M1x+ɛ)],V3=lim⁡inf⁡⁡t→+∞v(t)≥kβ(N1x-ɛ)-aβ(M1x+ɛ)-auauα.
Since these two inequalities hold for arbitrary ɛ>0 sufficiently small, we conclude that V2≥N1y, V3≥N1v, where
(4.16)N1y=βN1x(kβN1x-aβM1x-au)aα(kβN1x-aβM1x),N1v=kβN1x-aβM1x-auauα.
Therefore, for ɛ>0 sufficiently small, there exists a T4≥T3+τ such that if t>T4, y(t)≥N1y-ɛ,v(t)≥N1v-ɛ.
Forɛ>0 sufficiently small, it follows from the first equation of system (1.4) that for t>T4,(4.17)ẋ(t)≤λ-dx(t)-βx(t)(N1v-ɛ)1+α(N1v-ɛ).
By comparison, we derive that
(4.18)U1=limsup⁡t→+∞x(t)≤λ(1+α(N1v-ɛ))d+(αd+β)(N1v-ɛ).
Since this is true for arbitrary ɛ>0, it follows that U1≤M2x, where
(4.19)M2x=λ(1+αN1v)d+(αd+β)N1v.
Hence, for ɛ>0 sufficiently small, there is a T5≥T4 such that if t>T5, x(t)≤M2x+ɛ. It therefore, follows from the second and the third equations of system (1.4) that for t>T5+τ,
(4.20)ẏ(t)≤β(M2x+ɛ)v(t-τ)1+αv(t-τ)-ay(t),v̇(t)≤ky(t)-β(N2x-ɛ)v(t-τ)1+αv(t-τ)-uv(t).
By Lemma 2.5 and a comparison argument, we derive from (4.20) that
(4.21)U2=limsup⁡t→+∞y(t)≤β(M2x+ɛ)[kβ(M2x+ɛ)-aβ(N1x-ɛ)-au]aα[kβ(M2x+ɛ)-aβ(N1x-ɛ)],U3=lim⁡sup⁡⁡t→+∞v(t)≤kβ(M2x+ɛ)-aβ(N1x-ɛ)-auauα.
Since these inequalities are true for arbitrary ɛ>0, it follows that U2≤M2y, U3≤M2v, where
(4.22)M2y=βM2x(kβM2x-aβN1x-au)aα(kβM2x-aβN1x),M2v=kβM2x-aβN1x-auauα.
Hence, for ɛ>0 sufficiently small, there is a T6≥T5+τ such that if t>T6, y(t)≤M2y+ɛ, v(t)≤M2v+ɛ.
Again, forɛ>0 sufficiently small, we derive from the first equation of system (1.4) that for t>T6,(4.23)ẋ(t)≥λ-dx(t)-βx(t)(M2v+ɛ)1+α(M2v+ɛ).
A comparison argument shows that
(4.24)V1=lim⁡inf⁡t→+∞x(t)≥λ(1+α(M2v+ɛ))d+(αd+β)(M2v+ɛ).
Since this is true for arbitrary ɛ>0, it follows that V1≥N2x, where
(4.25)N2x=λ(1+αM2v)d+(αd+β)M2v.
Hence, for ɛ>0 sufficiently small, there is a T7≥T6 such that if t>T7, x(t)≥N2x-ɛ.
Forɛ>0 sufficiently small, it follows from the second and the third equations of system (1.4) that for t>T7+τ,(4.26)ẏ(t)≥β(N2x-ɛ)v(t-τ)1+αv(t-τ)-ay(t),v̇(t)≥ky(t)-β(M2x+ɛ)v(t)1+αv(t)-uv(t).
Since (H1) holds, by Lemma 2.5 and a comparison argument, it follows from (4.26) that
(4.27)V2=lim⁡inf⁡⁡t→+∞y(t)≥β(N2x-ɛ)[kβ(N2x-ɛ)-aβ(M2x+ɛ)-au]aα[kβ(N2x-ɛ)-aβ(M2x+ɛ)],V3=lim⁡inf⁡⁡t→+∞v(t)≥kβ(N2x-ɛ)-aβ(M2x+ɛ)-auauα.
Since these two inequalities hold for arbitrary ɛ>0 sufficiently small, we conclude that V2≥N2y, V3≥N2v, where
(4.28)N2y=βN2x(kβN2x-aβM2x-au)aα(kβN2x-aβM2x),N2v=kβN2x-aβM2x-auauα.
Therefore, for ɛ>0 sufficiently small, there exists a T8≥T7+τ such that if t>T8, y(t)≥N2y-ɛ, v(t)≥N2v-ɛ.
Continuing this process, we derive six sequencesMnx, Mny, Mnv, Nnx, Nny, Nnv  (n=1,2,…) such that for n≥2,(4.29)Mnx=λ(1+αNn-1v)d+(αd+β)Nn-1v,Mny=βMnx(kβMnx-aβNn-1x-au)aα(kβMnx-aβNn-1x),Mnv=kβMnx-aβNn-1x-auauα,Nnx=λ(1+αMnv)d+(αd+β)Mnv,Nny=βNnx(kβNnx-aβMnx-au)aα(kβNnx-aβMnx),Nnv=kβNnx-aβMnx-auauα.
Clearly, we have
(4.30)Nnx≤V1≤U1≤Mnx,Nny≤V2≤U2≤Mny,Nnv≤V3≤U3≤Mnv.
It is easy to show that the sequences Mnx, Mny, and Mnv are nonincreasing, and the sequences Nnx, Nny, Nnv are nondecreasing. Hence, the limit of each sequence in Mnx, Mny, Mnv, Nnx, Nny, and Nnv exists. Denote
(4.31)x¯=lim⁡n→+∞Mnx,x̲=lim⁡n→+∞Nnx,y¯=lim⁡n→+∞Mny,y̲=lim⁡n→+∞Nny,v¯=lim⁡n→+∞Mnv,v̲=lim⁡n→+∞Nnv.
We, therefore, obtain from (4.29) and (4.31) that
(4.32)a(αd+β)x¯2-k(αd+β)x¯x̲=a(λα-u)x¯-kλαx̲,(4.33)a(αd+β)x̲2-k(αd+β)x¯x̲=a(λα-u)x̲-kλαx¯.
(4.32) minus (4.33),
(4.34)a(αd+β)(x¯2-x̲2)=(λaα+λkα-au)(x¯-x̲).
Assume that x¯≠x̲. Then, we derive from (4.34) that
(4.35)x¯+x̲=λaα+λkα-aua(αd+β).
(4.32) plus (4.33),
(4.36)a(αd+β)(x¯+x̲)2-2(k+a)(αd+β)x¯x̲=(λaα-λkα-au)(x¯+x̲).
On substituting (4.35) into (4.36), it follows that
(4.37)x¯x̲=λkαa(k+a)(αd+β)2(λaα+λkα-au).
Note that x¯>0, x̲>0. Let (H2) hold. It follows from (4.35) and (4.37) that
(4.38)(x¯+x̲)2-4x¯x̲=(λaα+λkα-au)a2(αd+β)2[(λaα+λkα-au)-4kλaαk+a].
Hence, we have (x¯+x̲)2-4x¯x̲<0. This is a contradiction. Accordingly, we have x¯=x̲. We, therefore, derive from (4.31) that y¯=y̲, v¯=v̲. Note that if (H1) and (H2) hold, by Theorem 3.1, the virus-infected equilibrium E* is locally stable, we conclude that E* is globally stable. The proof is complete.Theorem 4.2.
Ifℛ0<1 holds, the infection-free equilibrium E0(λ/d,0,0) of system (1.4) is globally asymptotically stable.Proof.
Let(x(t),y(t),v(t)) be any positive solution of system (1.4) with initial condition (1.5).
Ifℛ0<1, choose ɛ>0 sufficiently small satisfying(4.39)kβ(λd+ɛ)<au.
It follows from the first equation of system (1.4) that
(4.40)ẋ(t)≤λ-dx(t).
By comparison, we derive that
(4.41)limsup⁡t→+∞x(t)≤λd.
Hence, for ɛ>0 sufficiently small satisfying (4.39), there exists a T1>0 such that if t>T1, x(t)≤(λ/d)+ɛ. We, therefore, derive from the second and the third equations of system (1.4) that for t>T1+τ,
(4.42)ẏ(t)≤β((λ/d)+ɛ)v(t-τ)1+αv(t-τ)-ay(t),v̇(t)≤ky(t)-uv(t).
Consider the following auxiliary equation:
(4.43)u̇1(t)=β((λ/d)+ɛ)u2(t-τ)1+αu2(t-τ)-au1(t),u̇2(t)=ku1(t)-uu2(t).
If ℛ0<1, then by Lemma 2.5, it follows from (4.41) and (4.43) that
(4.44)lim⁡t→+∞u1(t)=0,lim⁡t→+∞u1(t)=0.
By comparison, we obtain that
(4.45)lim⁡t→+∞y(t)=0,lim⁡t→+∞v(t)=0.
Therefore, for ɛ>0 sufficiently small, there is a T2>T1+τ such that if t>T2, y(t)<ɛ, v(t)<ɛ.
It follows from the first equation of system (1.4) that for t>T2,(4.46)ẋ(t)≥λ-dx(t)-βx(t)ɛ1+αɛ.
By comparison, we derive that
(4.47)lim⁡inf⁡t→+∞x(t)≥λ(1+αɛ)d+(αd+β)(1+αɛ).
Letting ɛ→0, it follows that
(4.48)lim⁡inf⁡t→+∞x(t)≥λd.
Noting that (4.41) holds, we conclude that
(4.49)lim⁡t→+∞x(t)=λd.
This completes the proof.


## 5. Numerical Examples

In this section, we give two examples to illustrate the main theoretical results above.Example 5.1.
In system (1.4), let α=2, β=2, λ=4, a=1, d=1.5, k=2, u=3, and τ=1. By calculation, we have ℛ0=16/9>1, and system (1.4) has a virus-infected equilibrium E*(11/5,7/10,7/30). Clearly, (H1) and (H2) hold. By Theorem 4.1, we see that E* is globally asymptotically stable. Numerical simulation illustrates the above result (see Figure 1).Figure 1
The numerical solution of system (1.4) when α=2, β=2, λ=4, a=1, d=1.5, k=2, u=3, τ=1, and (ϕ1,ϕ2,ϕ3)=(10,4,6).Example 5.2.
In system (1.4), let α=1, β=1, λ=2.5, a=2, d=1, k=1, u=2, and τ=3. Noting that ℛ0=-5/8<1, system (1.4) always has an infection-free equilibrium E0(5/2,0,0). By Theorem 4.2, we see that E0 is globally asymptotically stable. Numerical simulation illustrates this fact (see Figure 2).Figure 2
The numerical solution of system (1.4) when α=1, β=1, λ=2.5, a=2, d=1, k=1, u=2, τ=3, and (ϕ1,ϕ2,ϕ3)=(10,8,6).


## 6. Conclusions

In this paper, we have discussed a virus infection model with time delay, absorption, and saturation incidence. The basic reproduction ratioℛ0 was found. We investigated the global asymptotic stability of the infection-free equilibrium and the virus-infected equilibrium of system (1.4), respectively. When the basic reproduction ratio is greater than unity, by using the iteration scheme, we have established sufficient conditions for the global stability of the virus-infected equilibrium of system (1.4). By Theorem 4.1, we see that when ℛ0>1 and (H1) and (H2) hold, the virus-infected equilibrium is globally stable. Biologically, these indicate that when the death rate of infected cells and the production rate of free viruses from infected cells are sufficiently large, then the solutions of system (1.4) tend to the virus infected equilibrium which means that the virus persists in the host. On the other hand, by Theorem 4.2, we see that if the basic reproduction ratio is less than unity, the infection free equilibrium is globally asymptotically stable. Biologically, if the rate at which new uninfected cells are generated and the average number of effective contacts of one infective individual per unit time are small enough and the death rates of uninfected cells, infected cells and pathogens are large enough such that ℛ0<1, then the virus is cleared. We would like to point out here that Theorem 4.1 has room for improvement, we leave this for future work.



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*Source: 152415-2011-06-23.xml*