Global Dynamics of a Delayed HIV-1 Infection Model with CTL Immune Response

Abstract

A delayed HIV-1 infection model with CTL immune response is investigated. By using suitable Lyapunov functionals, it is proved that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio for viral infection is less than or equal to unity; if the basic reproduction ratio for CTL immune response is less than or equal to unity and the basic reproduction ratio for viral infection is greater than unity, the CTL-inactivated infection equilibrium is globally asymptotically stable; if the basic reproduction ratio for CTL immune response is greater than unity, the CTL-activated infection equilibrium is globally asymptotically stable.

1. Introduction

Recently, many mathematical models have been developed to describe the infection with HIV-1 (human immunodeficiency virus 1). By investigating these models, researchers have gained much important knowledge about the HIV-1 pathogenesis and have enhanced progress in the understanding of HIV-1 infection (see, e.g., [1–4]). It is pointed out by the work of [5] that immune response is universal and necessary to eliminate or control the disease during viral infections. In particular, as a part of innate response, cytotoxic T lymphocytes (CTLs) play a particularly important role in antiviral defense by attacking infected cells. Thus, many authors have studied the mathematical modelling of viral dynamics with CTL immune response (see, e.g., [5–9]). In [7], Nowak and Bangham considered an HIV-1 infection model with CTL immune response which is described by the following differential equations: 𝑥̇(𝑡)(𝑡)(𝑡)𝑣(𝑡),𝑦̇(𝑡)(𝑡)𝑣(𝑡)(𝑡)(𝑡)𝑧(𝑡),𝑣̇(𝑡)(𝑡)(𝑡),𝑧̇(𝑡)(𝑡)𝑧(𝑡)(𝑡),=𝜆−𝑑𝑥−𝛽𝑥=𝛽𝑥−𝑎𝑦−𝑝𝑦=𝑘𝑦−𝑢𝑣=𝑐𝑦−𝑏𝑧 (1.1) where 𝑥(𝑡), 𝑦(𝑡), 𝑣(𝑡), and 𝑧(𝑡) represent the densities of uninfected target cells, infected cells, virions, and CTL cells at time 𝑡, respectively. Uninfected cells are produced at rate 𝜆, die at rate 𝑑, and become infected cells at rate 𝛽𝑥𝑣. Infected cells are produced from uninfected cells at rate 𝛽𝑥𝑣 and die at rate 𝑎. The parameter 𝑝 accounts for the strength of the lytic component. Free virions are produced from uninfected cells at rate 𝑘𝑦 and are removed at rate 𝑢. The parameter 𝑏 is the death rate for CTLs, and 𝑐𝑦𝑧 describes the rate of CTL immune response activated by the infected cells.

Moreover, infection rate plays an important role in the modelling of epidemic dynamics. Holling type-II functional response seems more reasonable than the bilinear incidence rate (see, [10]). In [11], by stability analysis, Song and Avidan obtained that the system with the bilinear incidence rate was an extreme case of the model with Holling type-II functional response term.

In [3, 4, 7], the researchers used ordinary differential equations to describe different aspects of the dynamics of the viral infections. However, in the real virus dynamics, infection processes are not instantaneous. Time delays are usually introduced for the purpose of accurate representations of this phenomena (see, e.g., [6, 12–17]). As pointed out in [12], there is a time delay between initial viral entry into a cell and subsequent viral production, and the effect of saturation infection of an HIV-1 model was studied. By using the Lyapunov-LaSalle type theorem, sufficient conditions were derived for the global stability of the infection-free equilibrium and the chronic-infection equilibrium. In addition, there is also a period between virions that have created within a cell, and the new virions are released from the cell (see, e.g., [6, 13, 17]). In [13], Zhu and Zou studied an HIV-1 model with discrete delays and found that large delays can help eliminate the virus. To the best of our knowledge, there are few works on the dynamics of HIV-1 system with CTL immune response, Holling type-II functional response, and two kinds of discrete delays. Therefore, we are concerned with the effect of the above factors on system (1.1).

Motivated by the works of Nowak and Bangham [7], Song and Avidan [11], in the present paper, we consider the following delay differential equations:𝑥̇(𝑡)(𝑡)−(𝑡)𝑣(𝑡)=𝜆−𝑑𝑥𝛽𝑥(𝑡),𝑦̇(𝑡)=1+𝛼𝑣𝛽𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁(𝑡)(𝑡)𝑧(𝑡),𝑣̇(𝑡)−𝑎𝑦−𝑝𝑦=𝑘𝑦𝑡−𝜏₂(𝑡),𝑧̇(𝑡)(𝑡)𝑧(𝑡)(𝑡),−𝑢𝑣=𝑐𝑦−𝑏𝑧 (1.2) where the parameters have the same meanings as in system (1.1), 𝜏₁ represents the time between viral entry into a target cell and the production of new virus particles and 𝜏₂ stands for a virus production period for new virions to be produced within and released from the infected cells.

The initial conditions for system (1.2) take the form𝑥(𝜃)=𝜙₁(𝜃),𝑦(𝜃)=𝜙₂(𝜃),𝑣(𝜃)=𝜙₃(𝜃),𝑧(𝜃)=𝜙₄(𝜃),𝜙_𝑖(𝜃)≥0,𝜃∈[−𝜏,0],𝜙_𝑖(0)>0(𝑖=1,2,3,4), (1.3) where (𝜙₁(𝜃),𝜙₂(𝜃),𝜙₃(𝜃),𝜙₄(𝜃))∈𝐶([−𝜏,0],ℝ⁴₊₀), the Banach space of continuous functions mapping the interval [−𝜏,0] into ℝ⁴₊₀, where 𝜏=max{𝜏₁,𝜏₂}, ℝ⁴₊₀={(𝑥₁,𝑥₂,𝑥₃,𝑥₄)∶𝑥_𝑖≥0,𝑖=1,2,3,4}.

It is well known by the fundamental theory of functional differential equations [18] that system (1.2) has a unique solution (𝑥(𝑡),𝑦(𝑡),𝑣(𝑡),𝑧(𝑡)) satisfying the initial conditions (1.3). It is easy to show that all solutions of system (1.2) with initial conditions (1.3) are defined on [0,+∞) and remain positive for all 𝑡>0.

This paper is organized as follows. In Section 2, by analyzing the basic reproduction ratio for viral infection and CTL immune response, the existence of three equilibria is established. Moreover, the ultimate boundedness of the solutions for system (1.2) is presented. In Section 3, by means of suitable Lyapunov functionals and LaSalle's invariant principle, we discuss the global stability of the infection-free equilibrium, the CTL-inactivated infection equilibrium, and the CTL-activated infection equilibrium, respectively. In Section 4, we carry out some numerical examples to illustrate the theoretical results. Finally, a discussion is given in Section 5 to end this work.

2. Preliminary Results

In this section, we discuss the existence of three equilibria and prove that all the solutions are positive and bounded.

Clearly, system (1.2) always has an infection-free equilibrium 𝐸₀=(𝑥₀,0,0,0)=(𝜆/𝑑,0,0,0).

Denoteℛ₀=𝑘𝜆𝛽𝑎𝑑𝑢,ℛ₁=𝑐𝑘𝜆𝛽𝑎.𝑏𝑘𝛽+𝑐𝑑𝑢+𝑏𝑑𝑘𝛼 (2.1) Here, ℛ₀ and ℛ₁ are called the basic reproduction ratios for viral infection and CTL immune response of system (1.2), respectively. It is easy to see that ℛ₀>ℛ₁ always holds. If ℛ₀>1, system (1.2) has a CTL-inactivated infection equilibrium 𝐸₁=(𝑥₁,𝑦₁,𝑣₁,𝑧₁) besides the equilibrium 𝐸₀, where𝑥₁=𝑎𝑢𝑘𝛽1+𝛼𝑣₁,𝑦₁=𝑢𝑘𝑣₁,𝑣₁=𝑑ℛ𝛽+𝑑𝛼⁰−1,𝑧₁=0. (2.2)

If ℛ₁>1, system (1.2) has a CTL-activated infection equilibrium 𝐸₂=(𝑥₂,𝑦₂,𝑣₂,𝑧₂) besides the equilibrium 𝐸₀ and 𝐸₁, where𝑥₂=𝜆1+𝛼𝑣₂𝛽𝑣₂+𝑑1+𝛼𝑣₂,𝑦₂=𝑏𝑐,𝑣₂=𝑏𝑘𝑐𝑢,𝑧₂=𝑎𝑝ℛ₁−1. (2.3)

Theorem 2.1. Supposing that (𝑥(𝑡),𝑦(𝑡),𝑣(𝑡),𝑧(𝑡)) is a solution of system (1.2) with initial conditions (1.3), then there exists 𝑀>0, such that all the solutions satisfy 𝑥(𝑡)<𝑀, 𝑦(𝑡)<𝑀, 𝑣(𝑡)<𝑀, 𝑧(𝑡)<𝑀 for sufficiently large time 𝑡.

Proof. Let 𝑁(𝑡)=𝑥𝑡−𝜏₁(𝑡)+𝑎+𝑦𝑣2𝑘𝑡+𝜏₂+𝑝𝑐𝑧(𝑡)𝑎,𝛿=min𝑑,2,𝑢,𝑏. (2.4) Since all solutions of system (1.2) are positive, simple calculation leads to 𝑑𝑁(𝑡)𝑑𝑡=𝜆−𝑑𝑥𝑡−𝜏₁−𝛽𝑥𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁+𝛽𝑥𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁(𝑡)(𝑡)𝑧(𝑡)+𝑎−𝑎𝑦−𝑝𝑦(𝑡)2𝑘𝑘𝑦−𝑢𝑣𝑡+𝜏₂+𝑝𝑐𝑐𝑦(𝑡)𝑧(𝑡)−𝑏𝑧(𝑡)=𝜆−𝑑𝑥𝑡−𝜏₁−𝑎2𝑦(𝑡)−𝑎𝑢𝑣2𝑘𝑡+𝜏₂−𝑏𝑝𝑐𝑧(𝑡)≤𝜆−𝛿𝑁(𝑡). (2.5) Therefore, we get 𝑁(𝑡)<(𝜆/𝛿)+𝜀≜𝑀 for sufficiently large time 𝑡, where 𝜀 is an arbitrarily small positive constant. Finally, all the solutions of system (1.2) are ultimately bounded by some positive constant. This completes the proof.

3. Global Stability

In this section, we study the global stability of each equilibrium of system (1.2) by using suitable Lyapunov functionals which are inspired by Xu [12] and McCluskey [19] and LaSalle's invariant principle.

Define the following function:𝑔(𝑥)=𝑥−1−ln𝑥. (3.1) Clearly, for 𝑥∈(0,+∞), 𝑔(𝑥) has the minimum at 𝑥=1 and 𝑔(1)=0.

Theorem 3.1. If ℛ₀≤1, the infection-free equilibrium 𝐸₀=(𝑥₀,0,0,0) of system (1.2) is globally asymptotically stable.

Proof. Let (𝑥(𝑡),𝑦(𝑡),𝑣(𝑡),𝑧(𝑡)) be any positive solution of system (1.2) with initial conditions (1.3). Define the following Lyapunov functional: 𝑉₀(𝑡)=𝑥(𝑡)−𝑥₀𝑥(𝑡)ln𝑥₀(𝑡)+𝑎+𝑦𝑘𝑣(𝑡)+𝑝𝑐𝑧(𝑡)+𝛽^𝑡_𝑡−𝜏1𝑥(𝜃)𝑣(𝜃)1+𝛼𝑣(𝜃)𝑑𝜃+𝑎^𝑡_𝑡−𝜏2𝑦(𝜃)𝑑𝜃. (3.2) Calculating the derivative of 𝑉₀(𝑡) along positive solutions of system (1.2), it follows that 𝑑𝑉𝑑𝑡⁰(𝑡)=𝑥1−⁰𝑥(𝑡)(t)−(𝑡)𝑣(𝑡)𝜆−𝑑𝑥𝛽𝑥(𝑡)+1+𝛼𝑣𝛽𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁(𝑡)(𝑡)𝑧(𝑡)+𝑎−𝑎𝑦−𝑝𝑦𝑘𝑘𝑦𝑡−𝜏₂(𝑡)+𝑝−𝑢𝑣𝑐(𝑡)𝑧(𝑡)(𝑡)+(𝑡)𝑣(𝑡)𝑐𝑦−𝑏𝑧𝛽𝑥(𝑡)−1+𝛼𝑣𝛽𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁+𝑎𝑦(𝑡)−𝑎𝑦𝑡−𝜏₂=𝑥1−⁰𝑥(𝑡)𝑑𝑥₀(𝑡)−(𝑡)𝑣(𝑡)−𝑑𝑥𝛽𝑥(𝑡)+(𝑡)𝑣(𝑡)1+𝛼𝑣𝛽𝑥(𝑡)−1+𝛼𝑣𝑎𝑢𝑘𝑣(𝑡)−𝑏𝑝𝑐𝑧(𝑡)𝑑=−𝑥(𝑡)𝑥(𝑡)−𝑥₀²−𝑏𝑝𝑐𝑧(𝑡)−𝑎𝑢𝑘−𝛽𝑥₀(𝑡)𝑣(𝑡).1+𝛼𝑣 (3.3)
Noting that ℛ₀≤1, we obtain (𝑎𝑢/𝑘)−(𝛽𝑥₀/(1+𝛼𝑣(𝑡)))≥(𝑎𝑢/𝑘)−(𝛽𝜆/𝑑)=(𝑎𝑢/𝑘)(1−ℛ₀)≥0. Hence, from (3.3), we have 𝑉^′₀(𝑡)≤0. By Theorem 5.3.1 in [18], solutions limit to ℳ₀, the largest invariant subset of {𝑉^′₀(𝑡)=0}. Let (𝑥(𝑡),𝑦(𝑡),𝑣(𝑡),𝑧(𝑡)) be the solution with initial function in ℳ₀. Then, from the invariance of ℳ₀, we obtain 𝑥(𝑡)=𝑥₀, 𝑣(𝑡)=0, and 𝑧(𝑡)=0 for any 𝑡. Further, from the third equation of system (1.2), we obtain 𝑦(𝑡)=0. Accordingly, it follows from LaSalle's invariance principal that the infection-free equilibrium 𝐸₀ is globally asymptotically stable for any positive time delays. This completes the proof.

Theorem 3.2. If ℛ₁≤1<ℛ₀, the CTL-inactivated infection equilibrium 𝐸₁ of system (1.2) is globally asymptotically stable.

Proof. Let (𝑥(𝑡),𝑦(𝑡),𝑣(𝑡),𝑧(𝑡)) be any positive solution of system (1.2) with initial conditions (1.3). Define the following Lyapunov functional: 𝑉₁(𝑡)=𝑥₁𝑔𝑥(𝑡)𝑥₁+𝑦₁𝑔𝑦(𝑡)𝑦₁+𝛽𝑥₁𝑣²₁𝑘𝑦₁1+𝛼𝑣₁𝑔𝑣(𝑡)𝑣₁+𝑝𝑐𝑧(𝑡)+𝑉₁₁(𝑡)+𝑉₁₂(𝑡), (3.4) where 𝑉₁₁(𝑡)=𝛽𝑥₁𝑣₁1+𝛼𝑣₁^𝑡_𝑡−𝜏2𝑔𝑦(𝜃)𝑦₁𝑉𝑑𝜃,₁₂(𝑡)=𝛽𝑥₁𝑣₁1+𝛼𝑣₁^𝑡_𝑡−𝜏1𝑔𝑥(𝜃)𝑣(𝜃)1+𝛼𝑣₁𝑥₁𝑣₁(1+𝛼𝑣(𝜃))𝑑𝜃. (3.5)
For clarity, we will calculate the derivatives of 𝑔[𝑥(𝑡)/𝑥₁], 𝑔[𝑦(𝑡)/𝑦₁], 𝑔[𝑣(𝑡)/𝑣₁], 𝑧(𝑡), 𝑉₁₁(𝑡), and 𝑉₁₂(𝑡) along positive solutions of system (1.2), respectively.
Since 𝜆=𝑑𝑥₁+𝛽𝑥₁𝑣₁/(1+𝛼𝑣₁) holds, it follows that 𝑑𝑔𝑥(𝑡)𝑑𝑡𝑥₁=1𝑥₁𝑥1−¹𝑥(𝑡)(𝑡)−(𝑡)𝑣(𝑡)𝜆−𝑑𝑥𝛽𝑥(𝑡)=𝑥1+𝛼𝑣1−¹𝑥(𝑡)𝑑𝑥₁−𝑑𝑥(𝑡)+𝛽𝑥₁𝑣₁1+𝛼𝑣₁−𝛽𝑥(𝑡)𝑣(𝑡)(𝑡)(𝑡)1+𝛼𝑣=−𝑑𝑥𝑥₁𝑥1−¹𝑥(𝑡)²+1𝑥₁𝑥1−¹𝑥(𝑡)𝛽𝑥₁𝑣₁1+𝛼𝑣₁−𝛽𝑥(𝑡)𝑣(𝑡)(𝑡)(𝑡)1+𝛼𝑣=−𝑑𝑥𝑥₁𝑥1−¹𝑥(𝑡)²+1𝑥₁𝛽𝑥₁𝑣₁1+𝛼𝑣₁𝑥1−¹𝑥(𝑡)𝑥(𝑡)𝑣(𝑡)1−1+𝛼𝑣(𝑡)1+𝛼𝑣₁𝑥₁𝑣₁(𝑡)=−𝑑𝑥𝑥₁𝑥1−¹𝑥(𝑡)²+1𝑥₁𝛽𝑥₁𝑣₁1+𝛼𝑣₁𝑥1−¹𝑥(𝑡)−𝑥(𝑡)𝑣(𝑡)1+𝛼𝑣(𝑡)1+𝛼𝑣₁𝑥₁𝑣₁+𝑣(𝑡)1+𝛼𝑣(𝑡)1+𝛼𝑣₁𝑣₁. (3.6) Noting that 𝑎=𝛽𝑥₁𝑣₁/𝑦₁(1+𝛼𝑣₁), we get that 𝑑𝑔𝑦(𝑡)𝑑𝑡𝑦₁=1𝑦₁𝑦1−¹𝑦(𝑡)𝛽𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁(𝑡)(𝑡)𝑧(𝑡=1−𝑎𝑦−𝑝𝑦)𝑦₁𝑦1−¹𝑦(𝑡)𝛽𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁−𝛽𝑥₁𝑣₁1+𝛼𝑣₁𝑦(𝑡)𝑦₁(𝑡)𝑧(𝑡=1−𝑝𝑦)𝑦₁𝑦1−¹𝑦(𝑡)𝛽𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁−𝛽𝑥₁𝑣₁1+𝛼𝑣₁𝑦(𝑡)𝑦₁−𝑝𝑦(𝑡)𝑧(𝑡)𝑦₁𝑦1−¹𝑦(𝑡)=1𝑦₁𝛽𝑥₁𝑣₁1+𝛼𝑣₁𝑥t−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁1+𝛼𝑣₁𝑥₁𝑣₁−𝑦(𝑡)𝑦₁−𝑦₁𝑦(𝑡)𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁1+𝛼𝑣₁𝑥₁𝑣₁−(𝑡)𝑧(𝑡)+1𝑝𝑦𝑦₁+𝑝𝑧(𝑡). (3.7) Since 𝑢=𝑘𝑦₁/𝑣₁ holds, it follows that 𝑑𝑔𝑣(𝑡)𝑑𝑡𝑣₁=1𝑣₁𝑣1−¹𝑣(𝑡)𝑘𝑦𝑡−𝜏₂(𝑡=1−𝑢𝑣)𝑣₁𝑣1−¹𝑣(𝑡)𝑘𝑦𝑡−𝜏₂−𝑘𝑦₁𝑣₁𝑣(𝑡=)𝑘𝑦₁𝑣₁𝑣1−¹𝑣(𝑡)𝑦𝑡−𝜏₂𝑦₁−𝑣(𝑡)𝑣₁=𝑘𝑦₁𝑣₁𝑦𝑡−𝜏₂𝑦₁−𝑣₁𝑣(𝑡)𝑦𝑡−𝜏₂𝑦₁−𝑣(𝑡)𝑣₁+1. (3.8) Calculating the derivatives of 𝑉₁₁(𝑡) and 𝑉₁₂(𝑡) shows that 𝑑𝑉𝑑𝑡¹¹(𝑡)=𝛽𝑥₁𝑣₁1+𝛼𝑣₁𝑦(𝑡)𝑦₁−𝑦𝑡−𝜏₂𝑦₁𝑦+ln𝑡−𝜏₂𝑦₁𝑦(𝑡)−ln𝑦₁,𝑑𝑉𝑑𝑡¹²(𝑡)𝑥(𝑡)𝑣(𝑡)=𝛽(𝑡)−𝑥1+𝛼𝑣𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁+𝑥₁𝑣₁1+𝛼𝑣₁𝑥ln𝑡−𝜏₁𝑣𝑡−𝜏₁(1+𝛼𝑣(𝑡))𝑥(𝑡)𝑣(𝑡)1+𝛼𝑣𝑡−𝜏₁. (3.9) We therefore derive from (3.6)–(3.9) that 𝑑𝑉𝑑𝑡¹(𝑡)(𝑡)𝑥=−𝑑𝑥1−¹𝑥(𝑡)²−𝛽𝑥₁𝑣₁1+𝛼𝑣₁𝑥₁𝑥(𝑡)𝑥−1−ln¹𝑥(𝑡)−𝛽𝑥₁𝑣₁1+𝛼𝑣₁1+𝛼𝑣(𝑡)1+𝛼𝑣₁(𝑡)−1−ln1+𝛼𝑣1+𝛼𝑣₁−𝛽𝑥₁𝑣₁1+𝛼𝑣₁𝑣₁𝑣(𝑡)𝑦𝑡−𝜏₂𝑦₁𝑣−1−ln¹𝑣(𝑡)𝑦𝑡−𝜏₂𝑦₁−𝛽𝑥₁𝑣₁1+𝛼𝑣₁𝑦₁𝑦(𝑡)𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁1+𝛼𝑣₁𝑥₁𝑣₁𝑦−1−ln¹𝑦(𝑡)𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁1+𝛼𝑣₁𝑥₁𝑣₁−𝛼𝛽𝑥₁𝑣(𝑡)−𝑣₁²1+𝛼𝑣₁²((𝑡𝑦1+𝛼𝑣))+𝑝₁−𝑏𝑐𝑧(𝑡)(𝑡)𝑥=−𝑑𝑥1−¹𝑥(𝑡)²−𝛼𝛽𝑥₁𝑣(𝑡)−𝑣₁²1+𝛼𝑣₁²((𝑡𝑦1+𝛼𝑣))+𝑝₁−𝑏𝑐𝑧(𝑡)−𝛽𝑥₁𝑣₁1+𝛼𝑣₁𝑔𝑣₁𝑦𝑡−𝜏₂𝑦₁𝑣(𝑡)−𝛽𝑥₁𝑣₁1+𝛼𝑣₁𝑔𝑥₁𝑥(𝑡)−𝛽𝑥₁𝑣₁1+𝛼𝑣₁𝑔1+𝛼𝑣(𝑡)1+𝛼𝑣₁−𝛽𝑥₁𝑣₁1+𝛼𝑣₁𝑔𝑦₁𝑦(𝑡)𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁1+𝛼𝑣₁𝑥₁𝑣₁. (3.10)
Noting that ℛ₁≤1, we derive that 𝑦₁−𝑏/𝑐=((𝑏𝑘𝛽+𝑐𝑑𝑢+𝑏𝑑𝑘𝛼)/𝑐𝑘(𝛽+𝑑𝛼))(𝑅₁−1)≤0. Hence, from (3.10), we have 𝑉^′₁(𝑡)≤0. Similar to Theorem 3.1, solutions limit to ℳ₁, the largest invariant subset of {𝑉^′₁(𝑡)=0}. Let (𝑥(𝑡),𝑦(𝑡),𝑣(𝑡),𝑧(𝑡)) be the solution with initial function in ℳ₁. Then, we obtain that 𝑥(𝑡)=𝑥₁,𝑣(𝑡)=𝑣₁(𝑡)𝑣,𝑧=0,¹𝑦𝑡−𝜏₂𝑦₁𝑣(𝑡)=𝑦₁𝑦(𝑡)𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁1+𝛼𝑣₁𝑥₁𝑣₁=1. (3.11) It is readily to show that 𝑥(𝑡)=𝑥(𝑡−𝜏₁)=𝑥₁, 𝑦(𝑡)=𝑦(𝑡−𝜏₂)=𝑦₁, 𝑣(𝑡)=𝑣(𝑡−𝜏₁)=𝑣₁, and 𝑧(𝑡)=0 for any 𝑡. Thus, it follows from LaSalle's invariance principal that the CTL-inactivated infection equilibrium 𝐸₁ is globally asymptotically stable for any positive time delays. This completes the proof.

Theorem 3.3. If ℛ₁>1, the CTL-activated infection equilibrium 𝐸₂ of system (1.2) is globally asymptotically stable.

Proof. Let (𝑥(𝑡),𝑦(𝑡),𝑣(𝑡),𝑧(𝑡)) be any positive solution of system (1.2) with initial conditions (1.3). We construct the following Lyapunov functional: 𝑉₂(𝑡)=𝑥₂𝑔𝑥(𝑡)𝑥₂+𝑦₂𝑔𝑦(𝑡)𝑦₂+𝛽𝑥₂𝑣²₂𝑘𝑦₂1+𝛼𝑣₂𝑔𝑣(𝑡)𝑣₂+𝑝𝑧₂𝑐𝑔𝑧(𝑡)𝑧₂+𝑉₂₁(𝑡)+𝑉₂₂(𝑡), (3.12) where 𝑉₂₁(𝑡)=𝛽𝑥₂𝑣₂1+𝛼𝑣₂^𝑡_𝑡−𝜏2𝑔𝑦(𝜃)𝑦₂𝑉𝑑𝜃,₂₂(𝑡)=𝛽𝑥₂𝑣₂1+𝛼𝑣₂^𝑡_𝑡−𝜏1𝑔𝑥(𝜃)𝑣(𝜃)1+𝛼𝑣₂𝑥₂𝑣₂(1+𝛼𝑣(𝜃))𝑑𝜃. (3.13)
Next, we will calculate the derivatives of 𝑔[𝑥(𝑡)/𝑥₂], 𝑔[𝑦(𝑡)/𝑦₂], 𝑔[𝑣(𝑡)/𝑣₂], 𝑔[𝑧(𝑡)/𝑧₂], 𝑉₂₁(𝑡), and 𝑉₂₂(𝑡) along positive solutions of system (1.2), respectively.
Similar to (3.6), we derive that 𝑑𝑔𝑥(𝑡)𝑑𝑡𝑥₂(𝑡)=−𝑑𝑥𝑥₂𝑥1−²𝑥(𝑡)²+1𝑥₂𝛽𝑥₂𝑣₂1+𝛼𝑣₂𝑥1−²𝑥(𝑡)−𝑥(𝑡)𝑣(𝑡)1+𝛼𝑣(𝑡)1+𝛼𝑣₂𝑥₂𝑣₂+𝑣(𝑡)1+𝛼𝑣(𝑡)1+𝛼𝑣₂𝑣₂. (3.14) Noting that 𝑎=𝛽𝑥₂𝑣₂/𝑦₂(1+𝛼𝑣₂)−𝑝𝑧₂, it follows that 𝑑𝑔𝑦(𝑡)𝑑𝑡𝑦₂=1𝑦₂𝑦1−²𝑦(𝑡)𝛽𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁(𝑡)(𝑡)𝑧(𝑡=1−𝑎𝑦−𝑝𝑦)𝑦₂𝑦1−²𝑦(𝑡)𝛽𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁−𝛽𝑥₂𝑣₂1+𝛼𝑣₂𝑦(𝑡)𝑦₂+𝑝𝑧₂𝑦(𝑡)(𝑡)𝑧(𝑡=1−𝑝𝑦)𝑦₂𝛽𝑥₂𝑣₂1+𝛼𝑣₂𝑦1−²𝑦(𝑡)𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁1+𝛼𝑣₂𝑥₂𝑣₂−𝑦(𝑡)𝑦₂+𝑝𝑦(𝑡)𝑦₂𝑦1−²𝑦(𝑡)𝑧₂(𝑡)=1−𝑧𝑦₂𝛽𝑥₂𝑣₂1+𝛼𝑣₂𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁1+𝛼𝑣₂𝑥₂𝑣₂−𝑦(𝑡)𝑦₂−𝑦₂𝑦(𝑡)𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁1+𝛼𝑣₂𝑥₂𝑣₂+𝑝+1𝑦₂𝑦(𝑡)−𝑦₂𝑧₂−𝑧(𝑡). (3.15) Similar to (3.8), we get that 𝑑𝑔𝑣(𝑡)𝑑𝑡𝑣₂=𝑘𝑦₂𝑣₂𝑦𝑡−𝜏₂𝑦₂−𝑣₂𝑣(𝑡)𝑦𝑡−𝜏₂𝑦₂−𝑣(𝑡)𝑣₂+1. (3.16) Calculating the derivative of 𝑔[𝑧(𝑡)/𝑧₂] shows that 𝑑𝑔𝑧(𝑡)𝑑𝑡𝑧₂=1𝑧₂𝑧1−²𝑧(𝑡)(𝑡)𝑧(𝑡)(𝑡)=1𝑐𝑦−𝑏𝑧𝑧₂𝑧1−²𝑧(𝑡)𝑐𝑦(𝑡)𝑧(𝑡)−𝑐𝑦₂𝑧(𝑡)=𝑐𝑧₂𝑧(𝑡)−𝑧₂𝑦(𝑡)−𝑦₂. (3.17) Similar to (3.9), it follows that 𝑑𝑉𝑑𝑡²¹(𝑡)=𝛽𝑥₂𝑣₂1+𝛼𝑣₂𝑦(𝑡)𝑦₂−𝑦𝑡−𝜏₂𝑦₂𝑦+ln𝑡−𝜏₂𝑦₂𝑦(𝑡)−ln𝑦₂,𝑑𝑉𝑑𝑡²²(𝑡)𝑥(𝑡)𝑣(𝑡)=𝛽(𝑡)−𝑥1+𝛼𝑣𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁+𝑥₂𝑣₂1+𝛼𝑣₂𝑥ln𝑡−𝜏₁𝑣𝑡−𝜏₁(1+𝛼𝑣(𝑡))𝑥(𝑡)𝑣(𝑡)1+𝛼𝑣𝑡−𝜏₁. (3.18) We therefore derive from (3.14)–(3.18) that 𝑑𝑉𝑑𝑡²(𝑡)(𝑡)𝑥=−𝑑𝑥1−²𝑥(𝑡)²−𝛼𝛽𝑥₂𝑣(𝑡)−𝑣₂²1+𝛼𝑣₂²((𝑡−1+𝛼𝑣))𝛽𝑥₂𝑣₂1+𝛼𝑣₂𝑔𝑦₂𝑦(𝑡)𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁1+𝛼𝑣₂𝑥₂𝑣₂−𝛽𝑥₂𝑣₂1+𝛼𝑣₂𝑔𝑥₂𝑥(𝑡)−𝛽𝑥₂𝑣₂1+𝛼𝑣₂𝑔1+𝛼𝑣(𝑡)1+𝛼𝑣₂−𝛽𝑥₂𝑣₂1+𝛼𝑣₂𝑔𝑣₂𝑦𝑡−𝜏₂𝑦₂𝑣(𝑡). (3.19)
Hence, from (3.19), we get 𝑉^′₂(𝑡)≤0. Similar to Theorem 3.1, solutions limit to ℳ₂, the largest invariant subset of {𝑉^′₂(𝑡)=0}. Let (𝑥(𝑡),𝑦(𝑡),𝑣(𝑡),𝑧(𝑡)) be the solution with initial function in ℳ₂. Then, it holds that 𝑥(𝑡)=𝑥₂,𝑣(𝑡)=𝑣₂,𝑦₂𝑦(𝑡)𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁1+𝛼𝑣𝑡−𝜏₁1+𝛼𝑣₂𝑥₂𝑣₂=𝑣₂𝑦𝑡−𝜏₂𝑦₂𝑣(𝑡)=1. (3.20) It is easy to show that 𝑥(𝑡)=𝑥(𝑡−𝜏₁)=𝑥₂, 𝑦(𝑡)=𝑦(𝑡−𝜏₂)=𝑦₂, and 𝑣(𝑡)=𝑣(𝑡−𝜏₁)=𝑣₂ for any time 𝑡. Moreover, from the second equation of system (1.2), we obtain 𝑧(𝑡)=𝑧₂. Therefore, it follows from LaSalle's invariance principal that the CTL-activated infection equilibrium 𝐸₂ is globally asymptotically stable for any positive time delays. This completes the proof.

4. Numerical Simulations

In the following, we give three examples to illustrate the main theoretical results above. All the parameters were obtained from [9, 16, 20].

Example 4.1. In system (1.2), we choose 𝜆=1, 𝑑=0.1, 𝛽=0.0005, 𝛼=0.000001, 𝑎=0.3, 𝑝=1, 𝑘=200, 𝑢=8, 𝑐=0.2, 𝑏=0.15, and 𝜏₁=10, 𝜏₂=10. It is easy to show that ℛ₀=0.833<1, and that system (1.2) has an infection-free equilibrium 𝐸₀=(10,0,0,0). By Theorem 3.1, we get that the infection-free equilibrium 𝐸₀ of system (1.2) is globally asymptotically stable. Numerical simulation illustrates this fact (see Figure 1).

(a)

(b)

(c)

(d)

Figure 1 

The infection-free equilibrium 𝐸0 of system (1.2) is globally asymptotically stable. Here 𝜏1=10, 𝜏2=10, and the initial value is (𝑥0=1,𝑦0=1,𝑣0=5,𝑧0=1).

Example 4.2. In system (1.2), we set 𝜆=2, 𝑑=0.09, 𝛽=0.0006, 𝛼=0.000001, 𝑎=0.3, 𝑝=1, 𝑘=240, 𝑢=8, 𝑐=0.1, 𝑏=0.3, and 𝜏₁=5, 𝜏₂=9. It is easy to show that 0.8333=ℛ₁≤1<ℛ₀=1.3333, and that system (1.2) has a CTL-inactivated infection equilibrium 𝐸₁=(16.6675,1.6664,49.9925,0). By Theorem 3.2, we obtain that the CTL-inactivated infection equilibrium 𝐸₁ of system (1.2) is globally asymptotically stable. Numerical simulation illustrates our result (see Figure 2).

(a)

(b)

(c)

(d)

Figure 2 

The CTL-inactivated infection equilibrium 𝐸1 of system (1.2) is globally asymptotically stable. Here 𝜏1=5, 𝜏2=9, and the initial value is (𝑥0=10,𝑦0=1,𝑣0=50,𝑧0=2).

Example 4.3. In system (1.2), let 𝜆=5, 𝑑=0.02, 𝛽=0.009, 𝛼=0.000001, 𝑎=0.25, 𝑝=1, 𝑘=240, 𝑢=5, 𝑐=0.1, 𝑏=0.15, and 𝜏₁=4, 𝜏₂=2. It is easy to show that ℛ₁=12.9341>1, and then system (1.2) has a CTL-activated infection equilibrium 𝐸₂=(7.4856,1.5000,72.0000,2.9835). By Theorem 3.3, we get that the CTL-activated infection equilibrium 𝐸₂ of system (1.2) is globally asymptotically stable. Numerical simulation illustrates this fact (see Figure 3).

(a)

(b)

(c)

(d)

Figure 3 

The CTL-activated infection equilibrium 𝐸2 of system (1.2) is globally asymptotically stable. Here 𝜏1=4, 𝜏2=2, and the initial value is (𝑥0=1,𝑦0=1,𝑣0=50,𝑧0=4).

5. Discussion

In this paper, we have studied the global dynamics of a delayed HIV-1 infection model with CTL immune response. By constructing suitable Lyapunov functionals, sufficient conditions have been derived for the global stability of three equilibria. It is easy to show that if the basic reproduction ratio for viral infection ℛ₀⩽1, infection-free equilibrium 𝐸₀ is globally asymptotically stable, and the virus is cleared up; if the basic reproduction ratio for CTL immune response ℛ₁ satisfies ℛ₁≤1<ℛ₀, the equilibrium 𝐸₁ is globally asymptotically stable, and the infection becomes chronic but without CTL immune response; if ℛ₁>1, system (1.2) has a CTL-activated infection equilibrium 𝐸₂ besides 𝐸₀ and 𝐸₁, which is globally asymptotically stable, and the infection turns to chronic with CTL immune response.

From Theorems 3.1, 3.2, and 3.3, we see that the delays 𝜏₁ and 𝜏₂ do not affect the global stability of the feasible equilibria and therefore do not induce periodic oscillations, and the possibility of Hopf bifurcations is therefore ruled out. On the other hand, if the basic reproduction ratio for CTL immune response ℛ₁>1, we can get𝑦₁−𝑦₂=𝑘𝑏𝛽+𝑐𝑑𝑢+𝑘𝑏𝑑𝛼ℛ𝑐𝑘𝛽+𝑑𝛼¹−1>0,𝑣₁−𝑣₂=𝑘𝑏𝛽+𝑐𝑑𝑢+𝑘𝑏𝑑𝛼ℛ𝑐𝑢𝛽+𝑑𝛼¹−1>0, (5.1) which shows that the number of infected cells and virions of the equilibrium 𝐸₁ is greater than the number of those of the equilibrium 𝐸₂. Hence, the CTL immune response plays an important role in the reduction of the infected cells and the free virions.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (no. 11071254).