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

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.

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: 𝑥̇(𝑦𝑡)̇(𝑣𝑡=)̇(=𝑡𝜆)−=𝛽𝑥𝑑𝑘(𝑥𝑦𝑡)((𝑡𝑣𝑡))(−−𝑡)𝛽𝑢−𝑥𝑣𝑎((𝑦𝑡𝑡))(,𝑣𝑡)(,𝑡),(1.1) 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 𝛽𝑥𝑣^𝑝/(1+𝛼𝑣^𝑞) 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: 𝑥̇(𝑡)=𝜆−𝑑𝑥(𝑡)−𝛽𝑥(𝑡)𝑣(𝑡)𝑦̇(𝑡)=𝛽𝑥(𝑡−𝜏)𝑣(𝑡−1𝜏+)𝑣(𝑡),𝑣̇(𝑡)1=+𝑘𝑣𝑦((𝑡𝑡−)−𝜏)𝑢𝑣(𝑡−).𝑎𝑦(𝑡),(1.2) 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: 𝑣̇(𝑥𝑡̇)(𝑦𝑡=)̇(𝑡=𝑘)𝑦=𝜆(𝑡−𝛽)𝑥−𝑑(𝑥𝑡𝛽)(𝑥𝑡𝑣)((−𝑡𝑡))𝑣𝛽−𝑥(𝑡𝑎)(𝑦𝑡−)(𝑣𝑡𝑢)(𝑣,𝑡)(𝑡,),(1.3) 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: 𝑥̇(𝑡)=𝜆−𝑑𝑥(𝑡)−𝛽𝑥(𝑡)𝑣(𝑡)𝑦̇(𝑡)=𝛽𝑥(𝑡−𝜏)𝑣(𝑡−1𝜏+)𝛼𝑣(𝑡),𝑣̇(𝑡)=𝑘𝑦1(𝑡+)−𝛼𝑣𝛽(𝑥𝑡(−𝑡)𝜏𝑣)(𝑡)−𝑎𝑦(𝑡),1+𝛼𝑣(𝑡)−𝑢𝑣(𝑡).(1.4) The initial conditions for system (1.4) take the form 𝑥𝜙(𝜃)=𝜙₁(𝜃),𝑦(𝜃)=𝜙₂(𝜃),𝑣(𝜃)=𝜙₃(𝜃),₁(𝜃)𝜙≥0,𝜙₂(𝜃)≥0,𝜙₃(𝜃)≥0,𝜃∈[−𝜏,0₁],(0)>0,𝜙₂(0)>0,𝜙₃(0)>0,(1.5) where (𝜙₁(𝜃),𝜙₂(𝜃),𝜙₃(𝜃))∈𝐶([−𝜏,0],ℝ³⁺⁰), here ℝ³⁺⁰={(𝑥₁,𝑥₂,𝑥₃)∶𝑥_𝑖≥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 Section 3, 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.

Let ℝ^𝑛+ be the cone of nonnegative vectors in ℝ^𝑛. If 𝑥,𝑦∈ℝ^𝑛, we write 𝑥≤𝑦(𝑥<𝑦) if 𝑥_𝑖≤𝑦_𝑖(𝑥_𝑖<𝑦_𝑖) for 1≤𝑖≤𝑛. Let {𝑒₁,𝑒₂,…,𝑒_𝑛} denote the standard basis in ℝ^𝑛. Suppose, 𝑟≥0 and let 𝐶=𝐶([−𝑟,0],ℝ^𝑛) be the Banach space of continuous functions mapping the interval [−𝑟,0] into ℝ^𝑛 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 ℝ^𝑛→𝐶([−𝑟,0],ℝ^𝑛) 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],ℝ^𝑛), then for any 𝑡∈[𝑡₀,𝑡₀+𝐴], we let 𝑥_𝑡∈𝐶 be defined by 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃), −𝑟≤𝜃≤0.

We now consider 𝑥̇(𝑡)=𝑓𝑡,𝑥_𝑡.(2.1)

We assume throughout this section that 𝑓∶ℝ×𝐶→ℝ^𝑛 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 ℝ^𝑛, or by 𝑥_𝑡(𝑡₀,𝜙) if we work in the space 𝐶. Again, by [17], 𝑥(𝑡,𝑡₀,𝜙)(𝑥_𝑡(𝑡₀,𝜙)) is continuously differentiable in 𝜙. In the following, the notation 𝑥_𝑡0=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. Let 𝑟=(𝑟₁,𝑟₂,…,𝑟_𝑛)∈ℝ^𝑛+, |𝑟|=max_𝑖{𝑟_𝑖}, and define 𝐶_𝑟=^𝑛_𝑖=1𝐶−𝑟_𝑖,0,ℝ.(2.2)

We write 𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛) for a generic point of 𝐶_𝑟. Let 𝐶^+𝑟={𝜙∈𝐶_𝑟∶𝜙≥0}. Due to the ecological applications, we choose 𝐶^+𝑟 as the state space of (2.1) in the following discussions.

Fix 𝜙₀∈𝐶^+𝑟 arbitrarily. Then, we set L(𝑡,⋅)=𝐷_𝜙0𝑓(𝑡,𝜙₀),𝐷_𝜙0𝑓(𝑡,𝜙₀) denotes the Frechet derivation of 𝑓 with respect to 𝜙₀. It is convenient to have the standard representation of 𝐿=(𝐿₁,𝐿₂,…,𝐿_𝑛) as 𝐿_𝑖(𝑡,𝜙)=^𝑛_𝑗=1^0−𝑟_𝑗𝜙_𝑗(𝜃)𝑑_𝜃𝜂_𝑖𝑗(𝜃,𝑡),1≤𝑖≤𝑛,(2.3) in which 𝜂_𝑖𝑗∶ℝ×ℝ→ℝ satisfies𝜂𝜂_𝑖𝑗(𝜃,𝑡)=𝜂_𝑖𝑗(0,𝑡),𝜃≥0,_𝑖𝑗𝜂_𝑖𝑗(𝜃,𝑡)=0,𝜃≤−𝑟_𝑗,(⋅,𝑡)∈BV−𝑟_𝑗,0,(2.4) where 𝜂_𝑖𝑗(⋅,𝑡) is continuous from the left in (−𝑟_𝑗,0).

We make the following assumptions for (2.1).(h0)If 𝜙,𝜓∈𝐶⁺, 𝜙≤𝜓, and 𝜙_𝑖(0)=𝜓_𝑖(0) for some 𝑖, then 𝑓_𝑖(𝑡,𝜙)≤𝑓_𝑖(𝑡,𝜓). (h1)For all 𝜙∈𝐶^+𝑟 with 𝜙_𝑖(0)=0, 𝐿_𝑖(𝑡,𝜙)≥0 for 𝑡∈ℝ. (h2)The matrix 𝐴(𝑡) defined by 𝐴(𝑡)=col𝐿𝑡,𝑒₁̂,𝐿𝑡,𝑒₂̂,…,𝐿𝑡,𝑒_𝑛̂=𝜂_𝑖𝑗(0,𝑡)(2.5) is irreducible for each 𝑡∈ℝ. (h3)For each 𝑗, for which 𝑟_𝑗>0, there exists 𝑖 such that for all 𝑡∈ℝ and for positive constant 𝜀 sufficiently small, 𝜂_𝑖𝑗(−𝑟_𝑗+𝜀,𝑡)>0.(h4)If 𝜙=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)If 𝜙 and 𝜓 are distinct elements of 𝐶^+𝑟 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 0≤𝑥𝑡,𝑡⁰𝑥𝑡,𝑡₀,𝜙≤𝑥𝑡,𝑡₀,𝜓for𝑡₀≤𝑡<𝑡₀+𝜎,,𝜙<𝑥𝑡,𝑡₀,𝜓for𝑡₀+𝑛|𝑟|≤𝑡<𝑡₀+𝜎.(2.6)(ii)If 𝜙∈𝐶^+𝑟, 𝜙≠0, 𝑡₀∈ℝ and 𝑥(𝑡,𝑡₀,𝜙) is defined on [𝑡₀,𝑡₀+𝜎) with 𝜎>𝑛|𝑟|, then 0<𝑥𝑡,𝑡₀,𝜙for𝑡₀+𝑛|𝑟|≤𝑡<𝑡₀+𝜎.(2.7)

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. Let 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 be an 𝑛×𝑛 matrix, and let 𝑃₁,…,𝑃_𝑛 be distinct points of the complex plane. For each nonzero element 𝑎_𝑖𝑗 of 𝐴, connect 𝑃_𝑖 to 𝑃_𝑗 with a directed line 𝑃_𝑖𝑃_𝑗. 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 𝑃_𝑖,𝑃_𝑗 with 𝑖≠𝑗, there is a directed path 𝑃_𝑖𝑃_𝑘1,𝑃_𝑘1𝑃_𝑘2,…,𝑃_𝑘𝑟−1𝑃_𝑗,(2.8) connecting 𝑃_𝑖 and 𝑃_𝑗. 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: 𝑢̇₁(𝑡)=𝑎¹𝑢₂(𝑡−𝜏)1+𝛼𝑢₂(𝑡−𝜏)−𝑎𝑢¹𝑢̇₂(𝑡),(𝑡)=𝑘𝑢₁(𝑡)−𝑎²𝑢₂(𝑡)1+𝛼𝑢₂(𝑡)−𝑢𝑢²(𝑡)(2.9) with initial conditions 𝑢_𝑖(𝑠)=𝜙_𝑖(𝑠)≥0,𝑠∈[−𝜏,0),𝜙_𝑖(0)>0,𝜙_𝑖∈𝐶[−𝜏,0),ℝ₊(𝑖=1,2).(2.10)

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^∗(𝑢^∗1,𝑢^∗2), where 𝑢^∗1=𝑎¹𝑘𝑎₁−𝑎𝑎₂−𝑎𝑢𝛼𝑎𝑘𝑎₁−𝑎𝑎₂,𝑢^∗2=𝑘𝑎¹−𝑎𝑎₂−𝑎𝑢𝛼𝑎𝑢.(2.11)

The characteristic equation of system (2.9) at the positive equilibrium 𝐴⁰ takes the form 𝜆²+𝑝₁𝜆+𝑝₀+𝑞₀𝑒^−𝜆𝜏=0,(2.12) where 𝑝₀=𝑎𝑎₂+𝑢,𝑝₁=𝑎+𝑎₂+𝑢,𝑞₀=−𝑘𝑎₁.(2.13) Noting that 𝑝₁>0,𝑝₀+𝑞₀=𝑎𝑢−𝑘𝑎₁−𝑎𝑎₂,(2.14) 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 that 𝑝²¹−2𝑝₀=𝑎²+(𝑎₂+𝑢)²>0. If 𝑘𝑎₁−𝑎𝑎₂<𝑎𝑢, then 𝑝²⁰−𝑞²⁰>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 𝜆²+𝑔₁𝜆+𝑔₀+ℎ₀𝑒^−𝜆𝜏=0,(2.15) where 𝑔₀=𝑎𝑢+𝑎²1+𝛼𝑢^∗2²,𝑔¹=𝑎+𝑢+𝑎²1+𝛼𝑢^∗2²,ℎ₀=−𝑘𝑎¹1+𝛼𝑢^∗2².(2.16) Note that 𝑔₁>0,𝑔₀+ℎ₀=𝑎𝑢𝑎𝑢−𝑘𝑎¹−𝑎𝑎₂𝑘𝑎₁−𝑎𝑎₂.(2.17) 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 𝑔²¹−2𝑔₀=𝑎²+𝑢+𝑎²1+𝛼𝑢^∗2²²>0.(2.18) If 𝑘𝑎₁−𝑎𝑎₂>𝑎𝑢, then 𝑔²⁰−ℎ²⁰>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)If 𝑘𝑎₁−𝑎𝑎₂>𝑎𝑢, then the positive equilibrium 𝐴^∗(𝑢^∗1,𝑢^∗2) is globally stable. (ii)If 𝑘𝑎₁−𝑎𝑎₂<𝑎𝑢, then the equilibrium 𝐴⁰(0,0) is globally stable.

Proof. We represent the right-hand side of (2.9) by 𝑓(𝑡,𝑥_𝑡)=(𝑓₁(𝑡,𝑥_𝑡),𝑓₂(𝑡,𝑥_𝑡)), and set 𝐿(𝑡,⋅)=𝐷_𝜙𝑓(𝑡,𝜙).(2.19) By direct calculation, we have 𝐿₁(𝑡,ℎ)=𝑎¹1+𝛼𝜙₂(−𝜏)²𝐿ℎ₂(−𝜏)−𝑎ℎ₁(0),₂(𝑡,ℎ)=𝑘ℎ₁(0)−𝑎²1+𝛼𝜙₂(0)²ℎ₂(0)−𝑢ℎ₂(0).(2.20) 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 matrix 𝐴(𝑡) takes the form⎛⎜⎜⎜⎝−𝑎𝑎¹1+𝛼𝜙₂(−𝜏)²𝑘−𝑎²1+𝛼𝜙₂(0)²−𝑢⎞⎟⎟⎟⎠.(2.21) Clearly, the matrix 𝐴(𝑡) is irreducible for each 𝑡∈ℝ.
From the definition of 𝐴(𝑡) and 𝜂_𝑖𝑗, 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, 𝜂_𝑖𝑗(𝜃,𝑡)=0, 𝑖≠𝑗 for 𝜃≤−𝜏, and 𝜂_𝑖𝑗(⋅,𝑡)∈BV[−𝜏,0], where 𝜂_𝑖𝑗 is a positive Borel measure on [−𝜏,0]. Therefore, 𝜂_𝑖𝑗(⋅,𝑡)>0. Thus, for each 𝑗, there is 𝑖≠𝑗 such that 𝜂_𝑖𝑗(−𝑟_𝑗+𝜀,𝑡)=𝜂_𝑖𝑗(−𝜏+𝜀,𝑡)>0 for all 𝑡∈ℝ and for 𝜀>0 sufficiently small, 𝑖=1,2. Hence, (h3) holds.
Thus, the conditions of Lemma 2.1 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=𝐶^+𝜏 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^∗(𝑢^∗1,𝑢^∗2) 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 ℛ₀=𝜆𝛽(𝑘−𝑎)𝑎𝑢𝑑.(3.1)  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 𝑥^∗=𝑎𝑢1+𝛼𝑣^∗𝛽(𝑘−𝑎),𝑦^∗=𝑢𝑘−𝑎𝑣^∗,𝑣^∗=𝜆𝛽(𝑘−𝑎)−𝑎𝑢𝑑𝑎𝑢(𝛼𝑑+𝛽).(3.2)

The characteristic equation of system (1.4) at the infection free equilibrium 𝐸⁰ is of the form (𝑠+𝑑)𝑠²+𝑝₁𝑠+𝑝₀+𝑞₀𝑒^−𝑠𝜏=0,(3.3) where 𝑝₀=𝑎𝑢+𝜆𝛽𝑑,𝑝¹=𝑎+𝑢+𝜆𝛽𝑑,𝑞⁰=𝑘𝜆𝛽𝑑.(3.4) Obviously, (3.3) always has a negative real root 𝑠=−𝑑. All other roots of (3.3) are determined by 𝑠²+𝑝₁𝑠+𝑝₀+𝑞₀𝑒^−𝑠𝜏=0.(3.5) 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.

If i𝜔(𝜔>0) is a solution of (3.3), by calculating, we have the following: 𝜔⁴+𝑝²¹−2𝑝₀𝜔²+𝑝²⁰−𝑞²⁰=0.(3.6) Note that 𝑝²¹−2𝑝₀=𝑎²+𝑢+𝜆𝛽𝑝𝑑>0,₂₀−𝑞²⁰=1𝑑²[𝑎𝑢𝑑+𝜆𝛽(𝑘+𝑎)][𝑎𝑢𝑑+𝜆𝛽(𝑎−𝑘)].(3.7) If ℛ₀<1, then 𝑝²⁰−𝑞²⁰>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 𝑠³+𝑔₂𝑠²+𝑔₁𝑠+𝑔₀+ℎ₁𝑠+ℎ₀𝑒^−𝑠𝜏=0,(3.8) where 𝑔₀=𝑎𝑑𝑢+𝑑𝛽𝑥^∗(1+𝛼𝑣^∗)²+𝑢𝛽𝑣^∗𝑔₁1+𝛼𝑣^∗,=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.(3.9) When 𝜏=0, (3.8) becomes 𝑠³+𝑔₂𝑠²+𝑔₁+ℎ₁𝑠+𝑔₀+ℎ₀=0.(3.10) By direct calculation, we have 𝑔₀+ℎ₀=𝑎𝑢(𝛼𝑑+𝛽)𝑣^∗1+𝛼𝑣^∗𝑔>0,₂𝑔₁+ℎ₁−𝑔₀+ℎ₀=𝑎²𝑢𝛼𝑣^∗1+𝛼𝑣^∗+𝑎²𝜆𝑥^∗+𝑘𝑑𝛽𝑥^∗(1+𝛼𝑣^∗)²+𝑢+𝜆𝑥^∗+𝛽𝑥^∗×𝑎𝑢𝛼𝑣_∗(1+𝛼𝑣^∗)²1+𝛼𝑣^∗+(𝑢+𝑎)𝜆𝑥^∗+𝑑𝛽𝑥^∗(1+𝛼𝑣^∗)²>0.(3.11) Clearly, all roots of (3.10) have only negative real parts.

If i𝜔(𝜔>0) is a solution of (3.8), separating real and imaginary parts, it follows that 𝑔𝜔³−𝑔₁𝜔=ℎ₁𝜔cos𝜔𝜏−𝑔₀sin𝜔𝜏,₂𝜔²−𝑔₀=ℎ₁𝜔sin𝜔𝜏+ℎ₀cos𝜔𝜏.(3.12) Squaring and adding the two equations of (3.12), we derive that 𝜔⁶+𝐶₁𝜔⁴+𝐶₂𝜔²+𝐶₃=0,(3.13) where 𝐶₁=𝑔²²−2𝑔₁,𝐶₂=𝑔²¹−2𝑔₀𝑔₂−ℎ²¹,𝐶₃=𝑔²⁰−ℎ²⁰.(3.14) Clearly, 𝐶₃=𝑔²⁰−ℎ²⁰>0. It is easy to show that 𝐶₁=𝑎²+𝑢²+𝑑²+𝛽𝑥^∗𝐶1+𝛼𝑣^∗2𝑑+1+𝑢𝛼𝑣^∗+1+1+1𝛼𝑣^∗²>0,₂=𝑎²+𝑢²𝑑+𝛽𝑣^∗1+𝛼𝑣^∗²+𝑎𝑢𝛼𝑣^∗+2𝑎₂1+𝛼𝑣^∗𝑎𝑢+((𝑘1++𝑎𝛼)𝑣𝛽_∗𝑥)^∗2+𝑑𝑢𝛽²𝑥^∗𝑣^∗(1+𝛼𝑣^∗)³+𝑑²𝛽𝑥^∗(1+𝛼𝑣^∗)²2𝑢+(1+𝛽𝑥𝛼^∗𝑣^∗)²>0.(3.15) 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)If ℛ₀<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)If ℛ₀>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. Let ℛ₀>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 𝑈𝑈₁=limsup_𝑡→+∞𝑥(𝑡),𝑉₁=liminf_𝑡→+∞𝑥(𝑡),₂𝑈=limsup_𝑡→+∞𝑦(𝑡),𝑉₂=liminf_𝑡→+∞𝑦(𝑡),₃=limsup_𝑡→+∞𝑣(𝑡),𝑉₃=liminf_𝑡→+∞𝑣(𝑡).(4.1) Now, we claim that 𝑈₁=𝑉₁=𝑥^∗, 𝑈₂=𝑉₂=𝑦^∗, 𝑈₃=𝑉₃=𝑣^∗.
It follows from the first equation of system (1.4) that𝑥̇(𝑡)≤𝜆−𝑑𝑥(𝑡).(4.2) By comparison, we derive that 𝑈₁=limsup_𝑡→+∞𝑥(𝑡)≤𝜆𝑑∶=𝑀^𝑥1.(4.3) Hence, for 𝜀>0 sufficiently small, there exists a 𝑇₁>0 such that if 𝑡>𝑇₁, 𝑥(𝑡)≤𝑀^𝑥1+𝜀. We therefore, derive from the second and the third equations of system (1.4) that for 𝑡>𝑇₁+𝜏, 𝑦̇(𝑡)≤𝛽𝑀^𝑥1+𝜀𝑣(𝑡−𝜏)𝑣̇(𝑡1)≤+𝛼𝑘𝑣𝑦((𝑡𝑡)−−𝜏𝑢)𝑣(𝑡)−.𝑎𝑦(𝑡),(4.4) Consider the following auxiliary equations: 𝑢̇₁(𝑡)=𝛽𝑀^𝑥1+𝜀𝑢₂(𝑡−𝜏)1+𝛼𝑢₂(𝑡−𝜏)−𝑎𝑢¹𝑢̇₂(𝑡),(𝑡)=𝑘𝑢₁(𝑡)−𝑢𝑢₂(𝑡).(4.5) Since ℛ₀>1, by Lemma 2.5, it follows from (4.5) that lim_𝑡→+∞𝑢₁(𝑡)=𝑘𝛽𝑀^𝑥1+𝜀−𝑎𝑢lim_𝑡→+∞𝑘𝑎𝛼,𝑢₂(𝑡)=𝑘𝛽𝑀^𝑥1+𝜀−𝑎𝑢𝑎𝑢𝛼.(4.6) By comparison, we obtain that 𝑈₂=limsup_𝑡→+∞𝑦(𝑡)≤𝑘𝛽𝑀^𝑥1+𝜀−𝑎𝑢𝑈𝑘𝑎𝛼₃,=limsup_𝑡→+∞𝑣(𝑡)≤𝑘𝛽𝑀^𝑥1+𝜀−𝑎𝑢𝑎𝑢𝛼.(4.7) Since these inequalities are true for arbitrary 𝜀>0, it follows that 𝑈₂≤𝑀^𝑦1, 𝑈₃≤𝑀^𝑣1, where 𝑀^𝑦1=𝑘𝛽𝑀^𝑥1−𝑎𝑢𝑘𝑎𝛼,𝑀^𝑣1=𝑘𝛽𝑀^𝑥1−𝑎𝑢𝑎𝑢𝛼.(4.8) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₂≥𝑇₁+𝜏 such that if 𝑡>𝑇₂, 𝑦(𝑡)≤𝑀^𝑦1+𝜀,𝑣(𝑡)≤𝑀^𝑣1+𝜀.
For 𝜀>0 sufficiently small, we derive from the first equation of system (1.4) that for 𝑡>𝑇₂,𝑥̇(𝑡)≥𝜆−𝑑𝑥(𝑡)−𝛽𝑥(𝑡)𝑀^𝑣1+𝜀1+𝛼𝑀^𝑣1+𝜀.(4.9) A comparison argument shows that 𝑉₁=liminf_𝑡→+∞𝑥(𝑡)≥𝜆1+𝛼𝑀^𝑣1+𝜀𝑑+(𝛼𝑑+𝛽)𝑀^𝑣1+𝜀.(4.10) Since these inequalities are true for arbitrary 𝜀>0, it follows that 𝑉₁≥𝑁^𝑥1, where 𝑁^𝑥1=𝜆1+𝛼𝑀^𝑣1𝑑+(𝛼𝑑+𝛽)𝑀^𝑣1.(4.11) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₃≥𝑇₂ such that if 𝑡>𝑇₃, 𝑥(𝑡)≥𝑁^𝑥1−𝜀.
For 𝜀>0 sufficiently small, it follows from the second and the third equations of system (1.4) that for 𝑡>𝑇₃+𝜏,𝑦̇(𝑡)≥𝛽𝑁^𝑥1−𝜀𝑣(𝑡−𝜏)𝑣̇(𝑡)≥𝑘𝑦(𝑡)1−+𝛽𝛼𝑣𝑀(𝑡_𝑥1−𝜏)−𝑎𝑦(𝑡),+𝜀𝑣(𝑡)1+𝛼𝑣(𝑡)−𝑢𝑣(𝑡).(4.12) Consider the following auxiliary equations: 𝑢̇₁(𝑡)=𝛽𝑁^𝑥1−𝜀𝑢₂(𝑡−𝜏)1+𝛼𝑢₂(𝑡−𝜏)−𝑎𝑢¹𝑢̇₂(𝑡),(𝑡)=𝑘𝑢₁(𝑡)−𝛽𝑀^𝑥1+𝜀𝑢₂(𝑡)1+𝛼𝑢₂(𝑡)−𝑢𝑢²(𝑡).(4.13) Since (H1) holds, by Lemma 2.5, it follows from (4.13) that lim_𝑡→+∞𝑢₁(𝑡)=𝛽𝑁^𝑥1−𝜀𝑘𝛽𝑁^𝑥1−𝜀−𝑎𝛽𝑀^𝑥1+𝜀−𝑎𝑢lim_𝑡→+∞𝑎𝛼𝑘𝛽𝑁^𝑥1−𝜀−𝑎𝛽𝑀^𝑥1+𝜀,𝑢₂(𝑡)=𝑘𝛽𝑁^𝑥1−𝜀−𝑎𝛽𝑀^𝑥1+𝜀−𝑎𝑢𝑎𝑢𝛼.(4.14) By comparison, we derive that 𝑉₂=liminf_𝑡→+∞𝑦(𝑡)≥𝛽𝑁^𝑥1−𝜀𝑘𝛽𝑁^𝑥1−𝜀−𝑎𝛽𝑀^𝑥1+𝜀−𝑎𝑢𝑉𝑎𝛼𝑘𝛽𝑁^𝑥1−𝜀−𝑎𝛽𝑀^𝑥1+𝜀₃,=liminf_𝑡→+∞𝑣(𝑡)≥𝑘𝛽𝑁^𝑥1−𝜀−𝑎𝛽𝑀^𝑥1+𝜀−𝑎𝑢𝑎𝑢𝛼.(4.15) Since these two inequalities hold for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑉₂≥𝑁^𝑦1, 𝑉₃≥𝑁^𝑣1, where 𝑁^𝑦1=𝛽𝑁^𝑥1𝑘𝛽𝑁^𝑥1−𝑎𝛽𝑀^𝑥1−𝑎𝑢𝑎𝛼𝑘𝛽𝑁^𝑥1−𝑎𝛽𝑀^𝑥1,𝑁^𝑣1=𝑘𝛽𝑁^𝑥1−𝑎𝛽𝑀^𝑥1−𝑎𝑢𝑎𝑢𝛼.(4.16) Therefore, for 𝜀>0 sufficiently small, there exists a 𝑇₄≥𝑇₃+𝜏 such that if 𝑡>𝑇₄, 𝑦(𝑡)≥𝑁^𝑦1−𝜀,𝑣(𝑡)≥𝑁^𝑣1−𝜀.
For 𝜀>0 sufficiently small, it follows from the first equation of system (1.4) that for 𝑡>𝑇₄,𝑥̇(𝑡)≤𝜆−𝑑𝑥(𝑡)−𝛽𝑥(𝑡)𝑁^𝑣1−𝜀1+𝛼𝑁^𝑣1−𝜀.(4.17) By comparison, we derive that 𝑈₁=limsup_𝑡→+∞𝑥(𝑡)≤𝜆1+𝛼𝑁^𝑣1−𝜀𝑑+(𝛼𝑑+𝛽)𝑁^𝑣1−𝜀.(4.18) Since this is true for arbitrary 𝜀>0, it follows that 𝑈₁≤𝑀^𝑥2, where 𝑀^𝑥2=𝜆1+𝛼𝑁^𝑣1𝑑+(𝛼𝑑+𝛽)𝑁^𝑣1.(4.19) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₅≥𝑇₄ such that if 𝑡>𝑇₅, 𝑥(𝑡)≤𝑀^𝑥2+𝜀. It therefore, follows from the second and the third equations of system (1.4) that for 𝑡>𝑇₅+𝜏, 𝑦̇(𝑡)≤𝛽𝑀^𝑥2+𝜀𝑣(𝑡−𝜏)𝑣̇(𝑡)≤𝑘𝑦(𝑡)−1𝛽+𝑁𝛼𝑣_𝑥2(𝑡−𝜏)−𝑎𝑦(𝑡),−𝜀𝑣(𝑡−𝜏)1+𝛼𝑣(𝑡−𝜏)−𝑢𝑣(𝑡).(4.20) By Lemma 2.5 and a comparison argument, we derive from (4.20) that 𝑈₂=limsup_𝑡→+∞𝑦(𝑡)≤𝛽𝑀^𝑥2+𝜀𝑘𝛽𝑀^𝑥2+𝜀−𝑎𝛽𝑁^𝑥1−𝜀−𝑎𝑢𝑈𝑎𝛼𝑘𝛽𝑀^𝑥2+𝜀−𝑎𝛽𝑁^𝑥1−𝜀₃,=limsup_𝑡→+∞𝑣(𝑡)≤𝑘𝛽𝑀^𝑥2+𝜀−𝑎𝛽𝑁^𝑥1−𝜀−𝑎𝑢𝑎𝑢𝛼.(4.21) Since these inequalities are true for arbitrary 𝜀>0, it follows that 𝑈₂≤𝑀^𝑦2, 𝑈₃≤𝑀^𝑣2, where 𝑀^𝑦2=𝛽𝑀^𝑥2𝑘𝛽𝑀^𝑥2−𝑎𝛽𝑁^𝑥1−𝑎𝑢𝑎𝛼𝑘𝛽𝑀^𝑥2−𝑎𝛽𝑁^𝑥1,𝑀^𝑣2=𝑘𝛽𝑀^𝑥2−𝑎𝛽𝑁^𝑥1−𝑎𝑢𝑎𝑢𝛼.(4.22) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₆≥𝑇₅+𝜏 such that if 𝑡>𝑇₆, 𝑦(𝑡)≤𝑀^𝑦2+𝜀, 𝑣(𝑡)≤𝑀^𝑣2+𝜀.
Again, for 𝜀>0 sufficiently small, we derive from the first equation of system (1.4) that for 𝑡>𝑇₆,𝑥̇(𝑡)≥𝜆−𝑑𝑥(𝑡)−𝛽𝑥(𝑡)𝑀^𝑣2+𝜀1+𝛼𝑀^𝑣2+𝜀.(4.23) A comparison argument shows that 𝑉₁=liminf_𝑡→+∞𝑥(𝑡)≥𝜆1+𝛼𝑀^𝑣2+𝜀𝑑+(𝛼𝑑+𝛽)𝑀^𝑣2+𝜀.(4.24) Since this is true for arbitrary 𝜀>0, it follows that 𝑉₁≥𝑁^𝑥2, where 𝑁^𝑥2=𝜆1+𝛼𝑀^𝑣2𝑑+(𝛼𝑑+𝛽)𝑀^𝑣2.(4.25) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₇≥𝑇₆ such that if 𝑡>𝑇₇, 𝑥(𝑡)≥𝑁^𝑥2−𝜀.
For 𝜀>0 sufficiently small, it follows from the second and the third equations of system (1.4) that for 𝑡>𝑇₇+𝜏,𝑦̇(𝑡)≥𝛽𝑁^𝑥2−𝜀𝑣(𝑡−𝜏)𝑣̇(𝑡)≥𝑘𝑦(𝑡)1−+𝛽𝛼𝑣𝑀(𝑡_𝑥2−𝜏)−𝑎𝑦(𝑡),+𝜀𝑣(𝑡)1+𝛼𝑣(𝑡)−𝑢𝑣(𝑡).(4.26) Since (H1) holds, by Lemma 2.5 and a comparison argument, it follows from (4.26) that 𝑉₂=liminf_𝑡→+∞𝑦(𝑡)≥𝛽𝑁^𝑥2−𝜀𝑘𝛽𝑁^𝑥2−𝜀−𝑎𝛽𝑀^𝑥2+𝜀−𝑎𝑢𝑉𝑎𝛼𝑘𝛽𝑁^𝑥2−𝜀−𝑎𝛽𝑀^𝑥2+𝜀₃,=liminf_𝑡→+∞𝑣(𝑡)≥𝑘𝛽𝑁^𝑥2−𝜀−𝑎𝛽𝑀^𝑥2+𝜀−𝑎𝑢𝑎𝑢𝛼.(4.27) Since these two inequalities hold for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑉₂≥𝑁^𝑦2, 𝑉₃≥𝑁^𝑣2, where 𝑁^𝑦2=𝛽𝑁^𝑥2𝑘𝛽𝑁^𝑥2−𝑎𝛽𝑀^𝑥2−𝑎𝑢𝑎𝛼𝑘𝛽𝑁^𝑥2−𝑎𝛽𝑀^𝑥2,𝑁^𝑣2=𝑘𝛽𝑁^𝑥2−𝑎𝛽𝑀^𝑥2−𝑎𝑢𝑎𝑢𝛼.(4.28) Therefore, for 𝜀>0 sufficiently small, there exists a 𝑇₈≥𝑇₇+𝜏 such that if 𝑡>𝑇₈, 𝑦(𝑡)≥𝑁^𝑦2−𝜀, 𝑣(𝑡)≥𝑁^𝑣2−𝜀.
Continuing this process, we derive six sequences 𝑀^𝑥𝑛, 𝑀^𝑦𝑛, 𝑀^𝑣𝑛, 𝑁^𝑥𝑛, 𝑁^𝑦𝑛, 𝑁^𝑣𝑛  (𝑛=1,2,…) such that for 𝑛≥2,𝑀^𝑥𝑛=𝜆1+𝛼𝑁^𝑣𝑛−1𝑑+(𝛼𝑑+𝛽)𝑁^𝑣𝑛−1𝑀_𝑦𝑛,=𝛽𝑀^𝑥𝑛𝑘𝛽𝑀^𝑥𝑛−𝑎𝛽𝑁^𝑥𝑛−1−𝑎𝑢𝑀_𝑣𝑛𝑎𝛼𝑘𝛽𝑀^𝑥𝑛−𝑎𝛽𝑁^𝑥𝑛−1,=𝑘𝛽𝑀^𝑥𝑛−𝑎𝛽𝑁^𝑥𝑛−1−𝑎𝑢𝑁𝑎𝑢𝛼_𝑥𝑛,=𝜆1+𝛼𝑀^𝑣𝑛𝑑+(𝛼𝑑+𝛽)𝑀^𝑣𝑛𝑁_𝑦𝑛,=𝛽𝑁^𝑥𝑛𝑘𝛽𝑁^𝑥𝑛−𝑎𝛽𝑀^𝑥𝑛−𝑎𝑢𝑁_𝑣𝑛𝑎𝛼𝑘𝛽𝑁^𝑥𝑛−𝑎𝛽𝑀^𝑥𝑛,=𝑘𝛽𝑁^𝑥𝑛−𝑎𝛽𝑀^𝑥𝑛−𝑎𝑢𝑎𝑢𝛼.(4.29) Clearly, we have 𝑁^𝑥𝑛≤𝑉₁≤𝑈₁≤𝑀^𝑥𝑛,𝑁^𝑦𝑛≤𝑉₂≤𝑈₂≤𝑀^𝑦𝑛,𝑁^𝑣𝑛≤𝑉₃≤𝑈₃≤𝑀^𝑣𝑛.(4.30) It is easy to show that the sequences 𝑀^𝑥𝑛, 𝑀^𝑦𝑛, and 𝑀^𝑣𝑛 are nonincreasing, and the sequences 𝑁^𝑥𝑛, 𝑁^𝑦𝑛, 𝑁^𝑣𝑛 are nondecreasing. Hence, the limit of each sequence in 𝑀^𝑥𝑛, 𝑀^𝑦𝑛, 𝑀^𝑣𝑛, 𝑁^𝑥𝑛, 𝑁^𝑦𝑛, and 𝑁^𝑣𝑛 exists. Denote 𝑥=lim_𝑛→+∞𝑀^𝑥𝑛,𝑥=lim_𝑛→+∞𝑁^𝑥𝑛,𝑦=lim_𝑛→+∞𝑀^𝑦𝑛,𝑦=lim_𝑛→+∞𝑁^𝑦𝑛,𝑣=lim_𝑛→+∞𝑀^𝑣𝑛,𝑣=lim_𝑛→+∞𝑁^𝑣𝑛.(4.31) We, therefore, obtain from (4.29) and (4.31) that 𝑎𝑎((𝛼𝛼𝑑𝑑++𝛽𝛽))𝑥𝑥²₂−𝑘(𝛼𝑑+𝛽)𝑥𝑥=𝑎(𝜆𝛼−𝑢)𝑥−𝑘𝜆𝛼𝑥,(4.32)−𝑘(𝛼𝑑+𝛽)𝑥𝑥=𝑎(𝜆𝛼−𝑢)𝑥−𝑘𝜆𝛼𝑥.(4.33) (4.32) minus (4.33), 𝑎(𝛼𝑑+𝛽)𝑥²−𝑥²=(𝜆𝑎𝛼+𝜆𝑘𝛼−𝑎𝑢)𝑥−𝑥.(4.34) Assume that 𝑥≠𝑥. Then, we derive from (4.34) that 𝑥+𝑥=𝜆𝑎𝛼+𝜆𝑘𝛼−𝑎𝑢𝑎(𝛼𝑑+𝛽).(4.35) (4.32) plus (4.33), 𝑎(𝛼𝑑+𝛽)𝑥+𝑥²−2(𝑘+𝑎)(𝛼𝑑+𝛽)𝑥𝑥=(𝜆𝑎𝛼−𝜆𝑘𝛼−𝑎𝑢)𝑥+𝑥.(4.36) On substituting (4.35) into (4.36), it follows that 𝑥𝑥=𝜆𝑘𝛼𝑎(𝑘+𝑎)(𝛼𝑑+𝛽)²(𝜆𝑎𝛼+𝜆𝑘𝛼−𝑎𝑢).(4.37) Note that 𝑥>0, 𝑥>0. Let (H2) hold. It follows from (4.35) and (4.37) that 𝑥+𝑥²−4𝑥𝑥=(𝜆𝑎𝛼+𝜆𝑘𝛼−𝑎𝑢)𝑎²(𝛼𝑑+𝛽)²(𝜆𝑎𝛼+𝜆𝑘𝛼−𝑎𝑢)−4𝑘𝑘+𝜆𝑎𝑎𝛼.(4.38) Hence, we have (𝑥+𝑥)²−4𝑥𝑥<0. This is a contradiction. Accordingly, we have 𝑥=𝑥. We, therefore, derive from (4.31) that 𝑦=𝑦, 𝑣=𝑣. 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. If ℛ₀<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).
If ℛ₀<1, choose 𝜀>0 sufficiently small satisfying𝑘𝛽𝜆𝑑+𝜀<𝑎𝑢.(4.39) It follows from the first equation of system (1.4) that 𝑥̇(𝑡)≤𝜆−𝑑𝑥(𝑡).(4.40) By comparison, we derive that limsup_𝑡→+∞𝑥(𝑡)≤𝜆𝑑.(4.41) Hence, for 𝜀>0 sufficiently small satisfying (4.39), there exists a 𝑇₁>0 such that if 𝑡>𝑇₁, 𝑥(𝑡)≤(𝜆/𝑑)+𝜀. We, therefore, derive from the second and the third equations of system (1.4) that for 𝑡>𝑇₁+𝜏, 𝑦̇(𝑡)≤𝛽((𝜆/𝑑)+𝜀)𝑣(𝑡−𝜏)𝑣̇(𝑡1)≤+𝛼𝑘𝑣𝑦((𝑡𝑡)−−𝜏𝑢)𝑣(𝑡).−𝑎𝑦(𝑡),(4.42) Consider the following auxiliary equation: 𝑢̇₁(𝑡)=𝛽((𝜆/𝑑)+𝜀)𝑢²(𝑡−𝜏)1+𝛼𝑢₂(𝑡−𝜏)−𝑎𝑢¹𝑢̇₂(𝑡),(𝑡)=𝑘𝑢₁(𝑡)−𝑢𝑢₂(𝑡).(4.43) If ℛ₀<1, then by Lemma 2.5, it follows from (4.41) and (4.43) that lim_𝑡→+∞𝑢₁(𝑡)=0,lim_𝑡→+∞𝑢₁(𝑡)=0.(4.44) By comparison, we obtain that lim_𝑡→+∞𝑦(𝑡)=0,lim_𝑡→+∞𝑣(𝑡)=0.(4.45) Therefore, for 𝜀>0 sufficiently small, there is a 𝑇₂>𝑇₁+𝜏 such that if 𝑡>𝑇₂, 𝑦(𝑡)<𝜀, 𝑣(𝑡)<𝜀.
It follows from the first equation of system (1.4) that for 𝑡>𝑇₂,𝑥̇(𝑡)≥𝜆−𝑑𝑥(𝑡)−𝛽𝑥(𝑡)𝜀1+𝛼𝜀.(4.46) By comparison, we derive that liminf_𝑡→+∞𝑥(𝑡)≥𝜆(1+𝛼𝜀)𝑑+(𝛼𝑑+𝛽)(1+𝛼𝜀).(4.47) Letting 𝜀→0, it follows that liminf_𝑡→+∞𝑥(𝑡)≥𝜆𝑑.(4.48) Noting that (4.41) holds, we conclude that lim_𝑡→+∞𝑥(𝑡)=𝜆𝑑.(4.49) 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 ratio ℛ₀ 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.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11071254) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and the Science Research Foundation of JCB (no. JCB 1005).