Dynamics Analysis of an HIV Infection Model including Infected Cells in an Eclipse Stage

Abstract

In this paper, an HIV infection model including an eclipse stage of infected cells is considered. Some quicker cells in this stage become productively infected cells, a portion of these cells are reverted to the uninfected class, and others will be latent down in the body. We consider CTL-response delay in this model and analyze the effect of time delay on stability of equilibrium. It is shown that the uninfected equilibrium and CTL-absent infection equilibrium are globally asymptotically stable for both ODE and DDE model. And we get the global stability of the CTL-present equilibrium for ODE model. For DDE model, we have proved that the CTL-present equilibrium is locally asymptotically stable in a range of delays and also have studied the existence of Hopf bifurcations at the CTL-present equilibrium. Numerical simulations are carried out to support our main results.

1. Introduction

In recent years, mathematical models have been done on the viral dynamics of HIV. In the basic mathematical modeling of viral dynamics, the description of the virus infection process has three populations: uninfected target cells, productively infected cells, and free viral particles [1–7]. In this model, infected cells are assumed to produce new virions immediately after target cells are infected by a free virus.

However, there are many biological steps between viral infection of target cells and the production of HIV-1 virions. In 2007, Rong and coworkers [8] studied an extension of the basic model of HIV-1 infection. The main feature of their model is that an eclipse stage for the infected cells is included and a portion of these cells are reverted to the uninfected class. Perelson et al. [9] presented this kind of cell early in 1993. Buonomo and Vergas-De-León [10] have performed the global stability analysis of this model. Perelson et al. [1] put forward another model in 1997. He divided infected cells into two kinds: long-lived productively infected cells and latently infected cells. Latently infected cells are also activated into productively infected cells [11]. Motivated by their work and now we concern the progression of infected cells from this eclipse phase to the productive, and a portion of these cells are reverted to the uninfected class or are latent down in the body.

In most virus infections, cytotoxic T lymphocytes (CTLs) play a critical role in antiviral defense by attacking virus-infected cells. Therefore, the dynamics of HIV infection with CTL response has received much attention in the past decades, some include the immune response without immune delay [12–15], and others contain immune delay [16–19]. Some HIV infection models with CTL-response describe only the interaction among uninfected target cells, productively infected cells, CTLs [12, 14, 20]. The most basic model can be written as 𝑑𝑥𝑑𝑡=𝑠−𝑑𝑥−𝛽𝑥𝑦,𝑑𝑦𝑑𝑡=𝛽𝑥𝑦−𝛼𝑦−𝑝𝑦𝑧,𝑑𝑧𝑑𝑡=𝑓𝑥,𝑦,𝑧−𝑟𝑧, (1) where 𝑥,𝑦, and 𝑧 represent the concentration of uninfected target cells, productively infected cells, CTLs at time 𝑡, respectively. Parameters 𝑠 and 𝑑 are the birth rate and death rate of uninfected cells, respectively. The uninfected cells become infected at rate of 𝛽𝑥𝑦. Productively infected cells are produced at rate 𝛽𝑥𝑦, 𝛼 is the death rate of productively infected cells, 𝑝 is the strength of the lytic component, and 𝑟 is the death rate of CTLs. Function 𝑓(𝑥,𝑦,𝑧) describes the rate of immune response activated by the infected cells. Wang et al. [14] assumed that the production of CTLs depends only on the population of infected cells and gave 𝑓(𝑥,𝑦,𝑧)=𝑐𝑦. Ji et al. [12] assumed that the production of CTLs also depends on the population of CTL cells and chose the former 𝑓(𝑥,𝑦,𝑧)=𝑐𝑦𝑧.

In this paper, we also consider the dynamics of HIV infection with CTL response and give 𝑓(𝑥,𝑦,𝑧)=𝑐𝑦𝑧. Meanwhile, our model also concludes an eclipse stage of infected cells. After the eclipse stage, some quicker infected cells which become productively infected cells are obviously attacked by CTLs. Other infected cells which will be reverted to the uninfected class or be latent down in the body do not have the ability to express HIV and will not cause CTL immune response. Therefore, we only take the immune response to productively infected cells into account and ignore the attack to latently infected cells by CTLs. So we get the following ODE: 𝑑𝑥𝑑𝑡=𝑠−𝛽𝑥𝑦−𝑑𝑥+𝛿𝑤,𝑑𝑤𝑑𝑡=𝛽𝑥𝑦−𝛿+𝜂+𝑞𝑤,𝑑𝑦𝑑𝑡=𝑞𝑤−𝑝𝑦𝑧−𝛼𝑦,𝑑𝑧𝑑𝑡=𝑐𝑦𝑧−𝑟𝑧, (2) where 𝑤 represents the concentration of infected cells in the eclipse stage at time 𝑡. Infected cells in the eclipse phase revert to the uninfected class at a constant rate 𝛿. In addition, they may alternatively progress to the productively infected class at the rate 𝑞 or die at the rate 𝜂. But some authors believe that time delay cannot be ignored in models for immune response [16–19]. In this paper, 𝜏 represents CTL-response delay, that is, the time between antigenic stimulation and generating CTLs. We investigated the effect of a time delay on system (2) to obtain the following DDE model: 𝑑𝑥𝑑𝑡=𝑠−𝛽𝑥𝑦−𝑑𝑥+𝛿𝑤,𝑑𝑤𝑑𝑡=𝛽𝑥𝑦−𝛿+𝜂+𝑞𝑤,𝑑𝑦𝑑𝑡=𝑞𝑤−𝑝𝑦𝑧−𝛼𝑦,𝑑𝑧()𝑧()𝑑𝑡=𝑐𝑦𝑡−𝜏𝑡−𝜏−𝑟𝑧. (3)

Our paper is organized as follows: the three equilibriums on system (2) and (3) are given in the next section. In Section 3, the global stability of the ODE model is discussed. The analysis of the stability for this DDE model is carried out in Section 4. Finally, some numerical simulations are carried out to support our analytical results, and some conclusions are presented.

2. The Existence of the Equilibrium of System

In system (2) and (3), the basic reproduction numbers for viral infection and for CTL response are given as follows: 𝑅₀=𝑞𝑠𝛽,𝑅𝑑𝛼𝛿+𝜂+𝑞₁=𝑐𝑞𝑠𝛽.𝑐𝑑𝛼𝛿+𝜂+𝑞+𝛼𝛽𝑟𝜂+𝑞 (4)

It is clear that 𝑅₀>𝑅₁ always holds. For system (2) and (3), there exists three equilibriums.

Theorem 1. For system (2) and (3), the uninfected equilibrium 𝐸₀(𝑠/𝑑,0,0,0) always exists;(1)if 𝑅₀>1, a CTL-absent infection equilibrium 𝐸₁(𝑥₁,𝑤₁,𝑦₁,𝑧₁) exists, where 𝑥₁=𝛼𝛿+𝜂+𝑞,𝑤𝛽𝑞₁=𝑠1𝑛+𝑞1−𝑅₀,𝑦₁=𝑞𝛼𝑤₁,𝑧₁=0; (5)(2)if 𝑅₁>1, there exists a CTL-present infection equilibrium 𝐸₂(𝑥₂,𝑤₂,𝑦₂,𝑧₂), where 𝑥₂=𝑐𝛿+𝜂+𝑞𝑤𝛽𝑟²,𝑤₂=𝑠𝛽𝑟,𝑦𝑐𝑑𝛿+𝜂+𝑞+𝛽𝑟𝜂+𝑞₂=𝑟𝑐,𝑧₂=𝛼𝑝𝑅₁−1. (6)

3. The Global Stability of the ODE Model

The initial conditions for system (2) are given as follows: 𝑥(0)>0,𝑤(0)>0,𝑦(0)>0,𝑧(0)>0. (7)

It is clear that all solutions of system (2) are positive for 𝑡>0. Before analyzing the stability of system (2), we now show that the solutions of system (2) are bounded.

Theorem 2. Let (𝑥(𝑡),𝑤(𝑡),𝑦(𝑡), and 𝑧(𝑡)) be the solution of system (2) satisfying initial conditions (7), then there exists 𝑀>0 such that 𝑥(𝑡)<𝑀, 𝑤(𝑡)<𝑀, 𝑦(𝑡)<𝑀, and 𝑧(𝑡)<𝑀 hold after sufficiently large time 𝑡.

Proof. Let 𝐿(𝑡)(𝑡)(𝑡)(𝑡)+𝑝=𝑥+𝑤+𝑦𝑐𝑧(𝑡), (8) where 𝑘=min{𝑑,𝜂,𝛼,𝑟}. It follows from (2) that 𝑑𝐿(𝑡)(𝑡)(𝑡)(𝑡)−𝑑𝑡=𝑠−𝑑𝑥−𝜂𝑤−𝛼𝑦𝑝𝑟𝑐𝑧(𝑡)𝑥(𝑡)(𝑡)(𝑡)+𝑝<𝑠−𝑘+𝑤+𝑦𝑐𝑧(𝑡)=𝑠−𝑘𝐿(𝑡). (9)
Therefore, 𝐿(𝑡)<𝑠/𝑘+𝜀 for all large 𝑡, where 𝜀 is an arbitrarily small positive constant. Thus, 𝑥(𝑡)<𝑀,𝑤(𝑡)<𝑀, 𝑦(𝑡)<𝑀 and 𝑧(𝑡)<𝑀 for some positive constant 𝑀.

Theorem 3. If 𝑅₀<1, the uninfected equilibrium 𝐸₀ of system (2) is globally asymptotically stable.

Proof. Construct a Lyapunov function 𝑉₁𝑥,𝑤,𝑦,𝑧=𝑥−𝑥₀−𝑥₀𝑥ln𝑥₀+𝛿2𝑑+𝜂+𝑞𝑥₀𝑥−𝑥₀+𝑤²+𝑤+𝛿+𝜂+𝑞𝑞𝑝𝑦+𝛿+𝜂+𝑞𝑐𝑞𝑧, (10) where 𝑥₀=𝑠/𝑑. The derivative of 𝑉₁ along positive solutions of system (2) is given as follows: 𝑑𝑉₁=𝑑𝑡𝑥−𝑥₀𝑥+𝛿𝑠−𝑑𝑥−𝛽𝑥𝑦+𝛿𝑤𝑑+𝜂+𝑞𝑥₀𝑥−𝑥₀×𝑤++𝑤𝑠−𝑑𝑥−𝜂+𝑞𝛿+𝜂+𝑞𝑞𝑤+𝑝𝑞𝑤−𝛼𝑦−𝑝𝑦𝑧+𝛽𝑥𝑦−𝛿+𝜂+𝑞𝛿+𝜂+𝑞𝑞𝑦𝑧−𝑝𝑟𝛿+𝜂+𝑞𝑐𝑞𝑧. (11)
On substituting 𝑠=𝑑𝑥₀ and 𝛿𝑤(𝑥−𝑥₀)/𝑥=−𝛿𝑤((𝑥−𝑥₀)²/𝑥𝑥₀)+𝛿𝑤((𝑥−𝑥₀)/𝑥₀) into (11), we derive that 𝑑𝑉₁𝑑𝑡=−𝑑𝑥₀+𝛿𝑤+𝑑𝛿𝑥𝑑+𝜂+𝑞𝑥−𝑥₀²𝑥𝑥₀−𝛿𝜂+𝑞𝑑+𝜂+𝑞𝑥₀𝑤²+𝛼𝛿+𝜂+𝑞𝑞𝑅₀𝑦−−1𝑝𝑟𝛿+𝜂+𝑞𝑐𝑞𝑧. (12)
If 𝑅₀<1, then 𝑑𝑉₁/𝑑𝑡≤0 for all 𝑥>0,𝑤>0,𝑦>0 and 𝑧>0. So the uninfected equilibrium 𝐸₀ is stable. Clearly, it follows from (12) that 𝑑𝑉₁/𝑑𝑡=0 if and only if 𝑥=𝑥₀=𝑠/𝑑,𝑤=0,𝑦=0, and 𝑧=0. Therefore, the largest invariant set in the set {(𝑥,𝑤,𝑦,𝑧)∈𝑅⁴₊∣𝑑𝑉₁/𝑑𝑡=0} is the singleton {𝐸₀}. By LaSalle invariance principle, it follows that the equilibrium 𝐸₀ is globally asymptotically stable.

Theorem 4. For system (2), if 𝑅₁<1 and 1<𝑅₀≤1+(𝜂+𝑞)/𝛿, CTL-absent infection equilibrium 𝐸₁ is globally asymptotically stable.

Proof. Define a Lyapunov function 𝑉₂𝑥,𝑤,𝑦,𝑧=𝑥−𝑥₁−𝑥₁𝑥ln𝑥₁+𝑤−𝑤₁−𝑤₁𝑤ln𝑤₁+𝛽𝑝𝑥₁𝑦₁𝑐𝑞𝑤₁𝑧+𝛿2𝑑+𝜂+𝑞𝑥₁𝑥−𝑥₁+𝑤−𝑤₁²+𝛽𝑥₁𝑦₁𝑞𝑤₁𝑦−𝑦₁−𝑦₁𝑦ln𝑦₁. (13)
Calculating the derivative of 𝑉₂ along positive solutions of system (2), it follows that 𝑑𝑉₂=𝑥𝑑𝑡1−¹𝑥𝑑𝑥+𝑑𝑡𝛽𝑥₁𝑦₁𝑞𝑤₁𝑦1−¹𝑦𝑑𝑦+𝛿𝑑𝑡𝑑+𝜂+𝑞𝑥₁𝑥−𝑥₁+𝑤−𝑤₁×𝑑𝑥+𝑑𝑡𝑑𝑤+𝑤𝑑𝑡1−¹𝑤𝑑𝑤+𝑑𝑡𝛽𝑝𝑥₁𝑦₁𝑞𝑐𝑤₁𝑑𝑧=𝑥𝑑𝑡1−¹𝑥+𝑠−𝑑𝑥−𝛽𝑥𝑦+𝛿𝑤𝛽𝑥₁𝑦₁𝑞𝑤₁𝑦1−¹𝑦+𝛿𝑞𝑤−𝛼𝑦−𝑝𝑦𝑧𝑑+𝜂+𝑞𝑥₁𝑥−𝑥₁+𝑤−𝑤₁×𝑤+𝑤𝑠−𝑑𝑥−𝜂+𝑞1−¹𝑤𝑤+𝛽𝑥𝑦−𝛿+𝜂+𝑞𝛽𝑝𝑥₁𝑦₁𝑞𝑤₁𝑦𝑧−𝛽𝑝𝑟𝑥₁𝑦₁𝑐𝑞𝑤₁𝑧. (14)
At CTL-absent infection equilibrium 𝐸₁, on substituting 𝑠=𝛽𝑥₁𝑦₁+𝑑𝑥₁−𝛿𝑤₁,𝛼=𝑞(𝑤₁/𝑦₁), and 𝛿+𝜂+𝑞=𝛽𝑥₁𝑦₁/𝑤₁ into (14), we obtain that 𝑑𝑉₂=𝑥𝑑𝑡1−¹𝑥−𝑑𝑥−𝑥₁−𝛽𝑥𝑦+𝛽𝑥₁𝑦₁+𝛿𝑤−𝑤₁+𝛽𝑥₁𝑦₁𝑞𝑤₁𝑦1−¹𝑦𝑞𝑤−𝑞𝑤₁𝑦𝑦₁+𝛿−𝑝𝑦𝑧𝑑+𝜂+𝑞𝑥−𝑥₁+𝑤−𝑤₁×−𝑑𝑥−𝑥₁−𝜂+𝑞𝑤−𝑤₁+𝑤1−¹𝑤𝛽𝑥𝑦−𝛽𝑥₁𝑦₁𝑤𝑤₁+𝛽𝑝𝑥₁𝑦₁𝑞𝑤₁𝑦𝑧−𝛽𝑟𝑝𝑥₁𝑦₁𝑐𝑞𝑤₁𝑧. (15)
Noting that 𝛿𝑥1−𝑥₁𝑤−𝑤₁=−𝛿𝑤−𝑤₁𝑥−𝑥₁²𝑥𝑥₁+𝛿𝑥₁𝑥−𝑥₁𝑤−𝑤₁, (16)
therefore, 𝑑𝑉₂𝑑𝑡=−𝑑𝑥₁−𝛿𝑤₁+𝛿𝑤+𝑑𝛿𝑥𝑑+𝜂+𝑞𝑥−𝑥₁²𝑥𝑥₁−𝛿𝜂+𝑞𝑑+𝜂+𝑞𝑤−𝑤₁²−𝛽𝑥₁𝑦₁𝑥₁𝑥+𝑦₁𝑤𝑦𝑤₁+𝑥𝑦𝑤₁𝑥₁𝑦₁𝑤+−3𝛽𝑝𝑥₁𝑦₁𝑐𝑞𝑤₁𝑑𝛿+𝜂+𝑞+𝑟𝛽𝜂+𝑞𝛽𝑅𝜂+𝑞¹−1𝑧. (17)
Since 𝑥₁/𝑥+𝑦₁𝑤/𝑦𝑤₁+𝑥𝑦𝑤₁/𝑥₁𝑦₁𝑤−3≥0 and the equality holds if and only if 𝑥=𝑥₁,𝑤=𝑤₁ and 𝑦=𝑦₁. If 𝑅₀≤1+(𝜂+𝑞)/𝛿, then 𝑑𝑥₁−𝛿𝑤₁≥0. So, if 𝑅₁<1 and 1<𝑅₀≤1+(𝜂+𝑞)/𝛿, then 𝑑𝑉₂/𝑑𝑡≤0 for all 𝑥>0,𝑤>0,𝑦>0 and 𝑧>0. Clearly, it follows from (17) that 𝑑𝑉₂/𝑑𝑡=0 if and only if 𝑥=𝑥₁,𝑤=𝑤₁,𝑦=𝑦₁, and 𝑧=0, thus the largest invariant set in the set {(𝑥,𝑤,𝑦,𝑧)∈𝑅⁴₊∣𝑑𝑉₂/𝑑𝑡=0} is the singleton {𝐸₁}. Therefore, the global asymptotic stability of 𝐸₁ follows from the LaSalle’s invariance principle.

Theorem 5. For system (2), if 𝑅₁>1 and 𝑐𝑑(𝛿+𝜂+𝑞)−𝑟𝛽𝛿≥0, CTL-present infection equilibrium 𝐸₂ is globally asymptotically stable.

Proof. Define a Lyapunov function 𝑉₃𝑥,𝑤,𝑦,𝑧=𝑥−𝑥₂−𝑥₂𝑥ln𝑥₂+𝑤−𝑤₂−𝑤₂𝑤ln𝑤₂+𝛿2𝑑+𝜂+𝑞𝑥₂𝑥−𝑥₂+𝑤−𝑤₂²+𝛽𝑥₂𝑦₂𝑞𝑤₂𝑦−𝑦₂−𝑦₂𝑦ln𝑦₂+𝛽𝑝𝑥₂𝑦₂𝑐𝑞𝑤₂𝑧−𝑧₂−𝑧₂𝑧ln𝑧₂. (18)
Calculating the derivative of 𝑉₃ along positive solutions of system (2), we obtain that 𝑑𝑉₃=𝑥𝑑𝑡1−²𝑥𝑑𝑥+𝑑𝑡𝛽𝑥₂𝑦₂𝑞𝑤₂𝑦1−²𝑦𝑑𝑦+𝛿𝑑𝑡𝑑+𝜂+𝑞𝑥₂𝑥−𝑥₂+𝑤−𝑤₂𝑑𝑥+𝑑𝑡𝑑𝑤+𝑤𝑑𝑡1−²𝑤𝑑𝑤+𝑑𝑡𝛽𝑝𝑥₂𝑦₂𝑞𝑐𝑤₂𝑧1−²𝑧𝑑𝑧=𝑥𝑑𝑡1−²𝑥+𝑠−𝑑𝑥−𝛽𝑥𝑦+𝛿𝑤𝛽𝑥₂𝑦₂𝑞𝑤₂𝑦1−²𝑦+𝛿𝑞𝑤−𝛼𝑦−𝑝𝑦𝑧𝑢+𝜂+𝑞𝑥₂𝑥−𝑥₂+𝑤−𝑤₂×𝑤+𝑤𝑠−𝑑𝑥−𝜂+𝑞1−²𝑤𝑤+𝛽𝑥𝑦−𝛿+𝜂+𝑞𝛽𝑝𝑥₂𝑦₂𝑞𝑐𝑤₂𝑧1−²𝑧𝑐𝑦𝑧−𝑟𝑧. (19)
At CTL- present infection equilibrium 𝐸₂, on substituting 𝑠=𝛽𝑥₂𝑦₂+𝑑𝑥₂−𝛿𝑤₂, 𝛼=𝑞(𝑤₂/𝑦₂)−𝑝𝑧₂, 𝑦₂=𝑟/𝑐, and 𝛿+𝜂+𝑞=𝛽𝑥₂𝑦₂/𝑤₂ into (19), it follows that 𝑑𝑉₃=𝑥𝑑𝑡1−²𝑥×−𝑑𝑥−𝑥₂−𝛽𝑥𝑦+𝛽𝑥₂𝑦₂+𝛿𝑤−𝑤₂+𝛽𝑥₂𝑦₂𝑞𝑤₁𝑦1−²𝑦𝑞𝑤−𝑞𝑤₂𝑦𝑦₂+𝑝𝑦𝑧₂+𝛿−𝑝𝑦𝑧𝑑+𝜂+𝑞𝑥−𝑥₂+𝑤−𝑤₂×−𝑑𝑥−𝑥₂−𝜂+𝑞𝑤−𝑤₂+𝑤1−²𝑤𝛽𝑥𝑦−𝛽𝑥₂𝑦₂𝑤𝑤₂+𝛽𝑝𝑥₂𝑦₂𝑞𝑐𝑤₂𝑧1−²𝑧𝑐𝑦𝑧−𝑟𝑧. (20)
Noting that 𝛿𝑥1−𝑥₂𝑤−𝑤₂=−𝛿𝑤−𝑤₂𝑥−𝑥₂²𝑥𝑥₂+𝛿𝑥₂𝑥−𝑥₂𝑤−𝑤₂, (21) it follows from (20) and (21) that 𝑑𝑉₃𝑑𝑡=−𝑑𝑥₂+𝛿𝑤−𝑤₂𝑥−𝑥₂²𝑥𝑥₂+𝛿𝑥₂𝑥−𝑥₂𝑤−𝑤₂+𝛽𝑥₂𝑦₂1−𝑥𝑦𝑥₂𝑦₂−𝑥₂𝑥+𝑦𝑦₂+𝛽𝑥₂𝑦₂𝑤𝑤₂−𝑦𝑦₂−𝑦₂𝑤𝑦𝑤₂−+1𝛽𝑝𝑥₂𝑦₂𝑞𝑤₂𝑦−𝑦₂𝑧−𝑧₂−𝛿𝑥₂𝑑𝑑+𝜂+𝑞𝑥−𝑥₂²+𝑥−𝑥₂×𝑤−𝑤₂+𝜂+𝑞𝑑+𝜂+𝑞𝑤−𝑤₂²+𝛽𝑥₂𝑦₂𝑥𝑦𝑥₂𝑦₂−𝑤𝑤₂−𝑥𝑤₂𝑦𝑥₂𝑤𝑦₂++1𝛽𝑝𝑥₂𝑦₂𝑞𝑤₂𝑧−𝑧₂𝑦−𝑦₂=−𝑑𝑥₂−𝛿𝑤₂+𝛿𝑤+𝑑𝛿𝑥𝑑+𝜂+𝑞𝑥−𝑥₂²𝑥𝑥₂−𝛿𝜂+𝑞𝑑+𝜂+𝑞𝑤−𝑤₂²−𝛽𝑥₂𝑦₂𝑥₂𝑥+𝑦₂𝑤𝑦𝑤₂+𝑥𝑤₂𝑦𝑥₂𝑦₂𝑤−3. (22)
Since 𝑥₂/𝑥+𝑦₂𝑤/𝑦𝑤₂+𝑥𝑤₂𝑦/𝑥₂𝑦₂𝑤−3≥0 and the equality holds if and only if 𝑥=𝑥₂,𝑤=𝑤₂, and 𝑦=𝑦₂. If 𝑐𝑑(𝛿+𝜂+𝑞)−𝑟𝛽𝛿≥0, then 𝑑𝑥₂−𝛿𝑤₂=((𝑐𝑑(𝛿+𝜂+𝑞)−𝑟𝛽𝛿)/𝑟𝛽)𝑤₂≥0. Therefore, if 𝑅₁>1 and 𝑐𝑑(𝛿+𝜂+𝑞)−𝑟𝛽𝛿≥0, it follows from (22) that 𝑑𝑉₃/𝑑𝑡≤0 for all 𝑥>0,𝑤>0,𝑦>0, and 𝑧>0. Clearly, it follows from (22) that 𝑑𝑉₃/𝑑𝑡=0 if and only if 𝑥=𝑥₂, 𝑤=𝑤₂, 𝑦=𝑦₂, and 𝑧=𝑧₂. So the largest invariant set in the set {(𝑥,𝑤,𝑦,𝑧)∈𝑅⁴₊∣𝑑𝑉₃/𝑑𝑡=0} is the singleton {𝐸₂}. By LaSalle invariance principle, we conclude that the equilibrium 𝐸₂ is globally asymptotically stable.

4. The Stability Analysis of the DDE Model

In this section, we consider the stability of the delay model (3).

Let 𝐶=𝐶([−𝜏,0],𝑅⁴₊₀) be the Banach space of continuous functions mapping from the interval [−𝜏,0] to 𝑅⁴₊₀ with the topology of uniform convergence, where 𝑅⁴₊₀=𝑥₁,𝑥₂,𝑥₃,𝑥₄∣𝑥_𝑖≥0,𝑖=1,2,3,4. (23)

The initial conditions for system (3) are given as follows: 𝑥=𝜑₁(𝜃),𝑤=𝜑₂(𝜃),𝑦=𝜑₃(𝜃),𝑧=𝜑₄(𝜃)[],𝑥(0)(0)(0)(0),𝜃∈−𝜏,0>0,𝑤>0,𝑦>0,𝑧>0, (24) where (𝜑₁(𝜃),𝜑₂(𝜃),𝜑₃(𝜃),𝜑₄(𝜃))∈𝐶([−𝜏,0],𝑅⁴₊₀). It is clear to see that all solutions of system (3) satisfying the initial conditions (24) are positive for all 𝑡≥0. By the similar method to Theorem 2, we can get the following theorem.

Theorem 6. Let 𝑥(𝑡),𝑤(𝑡),𝑦(𝑡), and 𝑧(𝑡) be the solution of system (3) satisfying the initial conditions (24), then there exists 𝑀>0 such that 𝑥(𝑡)<𝑀,𝑤(𝑡)<𝑀,𝑦(𝑡)<𝑀, and 𝑧(𝑡)<𝑀 hold after sufficiently large time 𝑡.

Theorem 7. If 𝑅₀<1, the uninfected equilibrium 𝐸₀ of system (3) is globally asymptotically stable.

Proof. If 𝑅₀<1, construct a Lyapunov functional 𝑊₁𝑥,𝑤,𝑦,𝑧=𝑥−𝑥₀−𝑥₀𝑥ln𝑥₀+𝛿2𝑑+𝜂+𝑞𝑥₀𝑥−𝑥₀+𝑤²++𝑤𝛿+𝜂+𝑞𝑞𝑝𝑦+𝛿+𝜂+𝑞𝑧+𝑝𝑐𝑞𝛿+𝜂+𝑞𝑞^𝑡_𝑡−𝜏𝑦(𝜃)𝑧(𝜃)𝑑𝜃, (25) where 𝑥₀=𝑠/𝑑. Calculating the derivative of 𝑊₁ along positive solutions of system (3), it follows that 𝑑𝑊₁=𝑑𝑡𝑥−𝑥₀𝑥+𝛿𝑠−𝑑𝑥−𝛽𝑥𝑦+𝛿𝑤𝑑+𝜂+𝑞𝑥₀𝑥−𝑥₀×𝑤++𝑤𝑠−𝑑𝑥−𝜂+𝑞𝛿+𝜂+𝑞𝑞−𝑝𝑞𝑤−𝛼𝑦−𝑝𝑦𝑧+𝛽𝑥𝑦𝛿+𝜂+𝑞𝑤+𝛿+𝜂+𝑞𝑞−𝑦𝑧𝑝𝑟𝛿+𝜂+𝑞𝑐𝑞𝑧. (26)
On substituting 𝑠=𝑑𝑥₀ and 𝛿𝑤(𝑥−𝑥₀)/𝑥=−𝛿𝑤((𝑥−𝑥₀)²/𝑥𝑥₀)+𝛿𝑤((𝑥−𝑥₀)/𝑥₀) into (26), we derive that 𝑑𝑊₁𝑑𝑡=−𝑑𝑥₀+𝛿𝑤+𝑑𝛿𝑥×𝑑+𝜂+𝑞𝑥−𝑥₀²𝑥𝑥₀−𝛿𝑤²𝜂+𝑞𝑑+𝜂+𝑞𝑥₀+𝛼𝛿+𝜂+𝑞𝑞𝑅₀−1𝑦−𝑝𝑟𝛿+𝜂+𝑞𝑐𝑞𝑧. (27)
If 𝑅₀<1, we have 𝑑𝑊₁/𝑑𝑡≤0 for all 𝑥>0, 𝑤>0, 𝑦>0, and 𝑧>0. It follows that the uninfected equilibrium 𝐸₀ is stable. Clearly, it follows from (27) that 𝑑𝑊₁/𝑑𝑡=0 if and only if 𝑥=𝑠/𝑑, 𝑤=0, 𝑦=0, and 𝑧=0. Therefore the largest invariant set in the set {(𝑥,𝑤,𝑦,𝑧)∈𝑅⁴₊∣𝑑𝑊₁/𝑑𝑡=0} is the singleton {𝐸₀}. By LaSalle invariance principle [21], we can conclude that the equilibrium 𝐸₀ is globally asymptotically stable.

From the above analysis, we can obtain that the time delay has no effect on the stability of the uninfected equilibrium 𝐸₀ for the DDE model.

Theorem 8. For system (3), if 𝑅₁<1 and 1<𝑅₀≤1+(𝜂+𝑞)/𝛿, then CTL-absent infection equilibrium 𝐸₁ is globally asymptotically stable.

Proof. Define a lyapunov functional 𝑊₂𝑥,𝑤,𝑦,𝑧=𝑥−𝑥₁−𝑥₁𝑥ln𝑥₁+𝛽𝑥₁𝑦₁𝑞𝑤₁𝑦−𝑦₁−𝑦₁𝑦ln𝑦₁+𝛿2𝑑+𝜂+𝑞𝑥₁𝑥−𝑥₁+𝑤−𝑤₁²+𝑤−𝑤₁−𝑤₁𝑤ln𝑤₁+𝛽𝑝𝑥₁𝑦₁𝑐𝑞𝑤₁𝑧+𝛽𝑝𝑥₁𝑦₁𝑞𝑤₁^𝑡_𝑡−𝜏𝑦(𝜃)𝑧(𝜃)𝑑𝜃. (28)
Calculating the derivative of 𝑊₂ along positive solutions of system (3), it follows that 𝑑𝑊₂=𝑥𝑑𝑡1−¹𝑥+𝑠−𝑑𝑥−𝛽𝑥𝑦+𝛿𝑤𝛽𝑥₁𝑦₁𝑞𝑤₁𝑦1−¹𝑦+𝛿𝑞𝑤−𝛼𝑦−𝑝𝑦𝑧𝑑+𝜂+𝑞𝑥₁𝑥−𝑥₁+𝑤−𝑤₁×𝑤+𝑤𝑠−𝑑𝑥−𝜂+𝑞1−¹𝑤𝑤−𝛽𝑥𝑦−𝛿+𝜂+𝑞𝛽𝑝𝑟𝑥₁𝑦₁𝑐𝑞𝑤₁𝑧+𝛽𝑝𝑥₁𝑦₁𝑞𝑤₁𝑦𝑧. (29)
At CTL-absent infection equilibrium 𝐸₁, on substituting 𝑠=𝛽𝑥₁𝑦₁+𝑑𝑥₁−𝛿𝑤₁,𝛼=𝑞(𝑤₁/𝑦₁), and 𝛿+𝜂+𝑞=𝛽𝑥₁𝑦₁/𝑤₁ into (29), we obtain that 𝑑𝑊₂=𝑥𝑑𝑡1−¹𝑥−𝑑𝑥−𝑥₁−𝛽𝑥𝑦+𝛽𝑥₁𝑦₁+𝛿𝑤−𝑤₁+𝛽𝑥₁𝑦₁𝑞𝑤₁𝑦1−¹𝑦𝑞𝑤−𝑞𝑤₁𝑦𝑦₁+𝛿−𝑝𝑦𝑧𝑑+𝜂+𝑞𝑥−𝑥₁+𝑤−𝑤₁×−𝑑𝑥−𝑥₁−𝜂+𝑞𝑤−𝑤₁+𝑤1−¹𝑤𝛽𝑥𝑦−𝛽𝑥₁𝑦₁𝑤𝑤₁−𝛽𝑟𝑝𝑥₁𝑦₁𝑐𝑞𝑤₁𝑧+𝛽𝑝𝑥₁𝑦₁𝑞𝑤₁𝑦𝑧. (30)
It follows from (16) and (30) that 𝑑𝑊₂𝑑𝑡=−𝑑𝑥₁+𝛿𝑤−𝑤₁𝑥−𝑥₁²𝑥𝑥₁+𝛿𝑥₁𝑥−𝑥₁𝑤−𝑤₁+𝛽𝑥₁𝑦₁1−𝑥𝑦𝑥₁𝑦₁−𝑥₁𝑥+𝑦𝑦₁+𝛽𝑥₁𝑦₁𝑤𝑤₁−𝑦𝑦₁−𝑦₁𝑤𝑦𝑤₁−+1𝛽𝑥₁𝑦₁𝑞𝑤₁𝑦1−¹𝑦𝑝𝑦𝑧+𝛽𝑥₁𝑦₁𝑥𝑦𝑥₁𝑦₁−𝑤𝑤₁−𝑥𝑤₁𝑦𝑥₁𝑤𝑦₁−𝛿+1𝑥₁𝑑𝑑+𝜂+𝑞𝑥−𝑥₁²+𝑥−𝑥₁×𝑤−𝑤₁+𝜂+𝑞𝑑+𝜂+𝑞𝑤−𝑤₁²−𝛽𝑟𝑝𝑥₁𝑦₁𝑐𝑞𝑤₁𝑧+𝛽𝑝𝑥₁𝑦₁𝑞𝑤₁𝑦𝑧=−𝑑𝑥₁−𝛿𝑤₁+𝛿𝑤+𝑑𝛿𝑥×𝑑+𝜂+𝑞𝑥−𝑥₁²𝑥𝑥₁−𝛿𝜂+𝑞𝑑+𝜂+𝑞𝑤−𝑤₁²−𝛽𝑥₁𝑦₁𝑥₁𝑥+𝑦₁𝑤𝑦𝑤₁+𝑥𝑦𝑤₁𝑥₁𝑦₁𝑤+−3𝛽𝑝𝑥₁𝑦₁𝑐𝑞𝑤₁𝑑𝛿+𝜂+𝑞+𝑟𝛽𝜂+𝑞𝛽𝑅𝜂+𝑞¹−1𝑧. (31)
Since 𝑥₁/𝑥+𝑦₁𝑤/𝑦𝑤₁+𝑥𝑦𝑤₁/𝑥₁𝑦₁𝑤−3≥0 and the equality holds if and only if 𝑥=𝑥₁,𝑤=𝑤₁, and 𝑦=𝑦₁. If 𝑅₀≤1+(𝜂+𝑞)/𝛿, then 𝑑𝑥₁−𝛿𝑤₁≥0. Therefore, if 𝑅₁<1 and 1<𝑅₀≤1+(𝜂+𝑞)/𝛿, it follows from (31) that 𝑑𝑊₂/𝑑𝑡≤0 for all 𝑥>0,𝑤>0,𝑦>0, and 𝑧>0. It is readily seen from (31) that 𝑑𝑊₂/𝑑𝑡=0 if and only if 𝑥=𝑥₁,𝑤=𝑤₁,𝑦=𝑦₁, and 𝑧=0. Thus the largest invariant set in the set {(𝑥,𝑤,𝑦,𝑧)∈𝑅⁴₊∣𝑑𝑊₂/𝑑𝑡=0} is the singleton {𝐸₁}. Then the global asymptotic stability of 𝐸₁ follows from the LaSalle’s invariance principle [21].

From the above analysis, we obtain that the time delay has no effect on the stability of the CTL-absent infection equilibrium 𝐸₁ for the DDE model. Next, we analyze stability and Hopf bifurcation at the CTL-present equilibrium 𝐸₂.

Firstly, the linearized equations of system (3) at 𝐸₂ are given as follows: 𝑑𝑥𝑑𝑡=−𝛽𝑦₂+𝑑𝑥(𝑡)−𝛽𝑥₂𝑦(𝑡)+𝛿𝑤(𝑡),𝑑𝑤𝑑𝑡=𝛽𝑦₂𝑥(𝑡)+𝛽𝑥₂𝑦(𝑡)−𝛿+𝜂+𝑞𝑤(𝑡),𝑑𝑦(𝑡)−𝑑𝑡=𝑞𝑤𝛼+𝑝𝑧₂𝑦(𝑡)−𝑝𝑦₂𝑧(𝑡),𝑑𝑧𝑑𝑡=𝑐𝑧₂𝑦(𝑡−𝜏)+𝑐𝑦₂𝑧(𝑡−𝜏)−𝑟𝑧(𝑡). (32)

The characteristic equation of system (32) at 𝑂(0,0,0,0) takes the form 𝐺(𝜆)=𝜆⁴+𝑀₁𝜆³+𝑀₂𝜆²+𝑀₃𝜆+𝑀₄−𝑁₁𝜆³+𝑁₂𝜆²+𝑁₃𝜆+𝑁₄e^−𝜆𝜏=0, (33) where 𝑀₁=𝛽𝑦₂+𝑑+𝛿+𝜂+𝑞+𝛼+𝑝𝑧₂𝑀+𝑟,₂=𝛽𝑦₂+𝜂+𝑞+𝑑𝛿+𝜂+𝑞𝛽𝑦₂+𝑑𝛼+𝑝𝑧₂+𝑟𝛽𝑦₂+𝑑+𝛿+𝜂+𝑞+𝛼+𝑝𝑧₂,𝑀₃=𝑟𝛽𝑦₂+𝜂+𝑞+𝑑𝛿+𝜂+𝑞𝛽𝑦₂+𝑑𝛼+𝑝𝑧₂+𝛽𝑦₂𝜂+𝑞𝛼+𝑝𝑧₂,𝑀₄=𝑟𝛽𝑦₂𝜂+𝑞𝛼+𝑝𝑧₂,𝑁₁𝑁=𝑟,₂=𝑟𝛽𝑦₂+𝑑+𝛿+𝜂+𝑞+𝛼+𝑝𝑧₂−𝑐𝑝𝑦₂𝑧₂,𝑁₃=𝑟𝛽𝑦₂+𝜂+𝑞+𝑑𝛿+𝜂+𝑞𝛽𝑦₂+𝑑𝛼+𝑝𝑧₂−𝑐𝑝𝑦₂𝑧₂𝛽𝑦₂,𝑁+𝑑+𝛿+𝜂+𝑞₄=𝑟𝛽𝑦₂𝜂+𝑞𝛼+𝑝𝑧₂−𝑐𝑝𝑦₂𝑧₂𝛽𝑦₂𝜂+𝑞+𝑑𝛿+𝜂+𝑞. (34)

Theorem 9. Suppose 𝜏=0, if 𝑅₁>1, then the CTL-present equilibrium 𝐸₂ of system (3) is locally asymptotically stable.

Proof. If 𝜏=0, (33) becomes 𝜆⁴+𝑀₁−𝑁₁𝜆³+𝑀₂−𝑁₂𝜆²+𝑀₃−𝑁₃𝜆+𝑀₄−𝑁₄=0. (35)
Since 𝑅₁>1,𝑥₂>0,𝑤₂>0,𝑦₂>0, and 𝑧₂>0, by the Routh-Hurwitz criteria, it follows that 𝐻₁=𝑀₁−𝑁₁=𝛽𝑦₂+𝑑+𝛿+𝜂+𝑞+𝛼+𝑝𝑧₂𝐻>0,₂=𝑀₁−𝑁₁𝑀₂−𝑁₂−𝑀₃−𝑁₃=𝛽𝑦₂+𝜂+𝑞+𝑑𝛿+𝜂+𝑞𝛽𝑦₂+𝑑𝛼+𝑝𝑧₂+𝑐𝑝𝑦₂𝑧₂×𝛽𝑦₂𝑑+𝛿+𝜂+𝑞+𝛼+𝑝𝑧₂−𝛽𝑦₂𝜂+𝑞𝛼+𝑝𝑧₂−𝑐𝑝𝑦₂𝑧₂𝛽𝑦₂=𝑑++𝑑+𝛿+𝜂+𝑞𝛿+𝜂+𝑞𝛽𝑦₂+𝑑𝛼+𝑝𝑧₂×𝛽𝑦₂+𝑑+𝛿+𝜂+𝑞+𝛼+𝑝𝑧₂+𝛽𝑦₂𝜂+𝑞𝛽𝑦₂+𝑑+𝛿+𝜂+𝑞+𝑐𝑝𝑦₂𝑧₂𝛼+𝑝𝑧₂𝐻>0,₃=𝑀₁−𝑁₁𝑀₂−𝑁₂𝑀₃−𝑁₃−𝑀₁−𝑁₁𝑀₄−𝑁₄−𝑀₃−𝑁₃²=𝛽𝑦₂+𝜂+𝑞+𝑑𝛿+𝜂+𝑞𝛽𝑦₂+𝑑𝛼+𝑝𝑧₂+𝑐𝑝𝑦₂𝑧₂×𝛽𝑦₂+𝑑+𝛿+𝜂+𝑞+𝛼+𝑝𝑧₂×𝛽𝑦₂𝜂+𝑞𝛼+𝑝𝑧₂+𝑐𝑝𝑦₂𝑧₂𝛽𝑦₂−+𝑑+𝛿+𝜂+𝑞𝛽𝑦₂+𝑑+𝛿+𝜂+𝑞+𝛼+𝑝𝑧₂²×𝑐𝑝𝑦₂𝑧₂𝛽𝑦₂−𝜂+𝑞+𝑑𝛿+𝜂+𝑞𝛽𝑦₂𝜂+𝑞𝛼+𝑝𝑧₂+𝑐𝑝𝑦₂𝑧₂𝛽𝑦₂+𝑑+𝛿+𝜂+𝑞²=𝐴𝑐𝑝𝑦₂𝑧₂²+𝐵𝑐𝑝𝑦₂𝑧₂+𝐶, (36) where 𝐴=𝛼+𝑝𝑧₂𝛽𝑦₂+𝑑+𝛿+𝜂+𝑞>0,𝐵=𝛽𝑦₂+𝑑²+𝛽𝑦₂𝛿𝛼+𝑝𝑧₂×𝛽𝑦₂+𝑑+𝛼+𝑝𝑧₂+𝛽𝑦₂+𝑑²+𝛽𝑦₂𝛿×𝛿𝛼+𝑝𝑧₂+𝛽𝑑𝑦₂+𝑑²𝜂+𝑞𝛼+𝑝𝑧₂−𝛽𝑦₂𝜂+𝑞²𝛼+𝑝𝑧₂,𝐶=𝛽𝑦₂𝜂+𝑞𝛼+𝑝𝑧₂𝛽𝑦₂×+𝑑+𝛿+𝜂+𝑞𝛽𝑦₂𝜂+𝑞+𝑑𝛿+𝜂+𝑞𝛽𝑦₂+𝑑𝛼+𝑝𝑧₂+𝛽𝑦₂𝜂+𝑞𝛼+𝑝𝑧₂²×𝑑𝛿+𝜂+𝑞+𝛽𝑦₂+𝑑𝛼+𝑝𝑧₂>0. (37)
Let 𝐷=𝛽𝑦₂𝜂+𝑞²𝛼+𝑝𝑧₂,ℎ=𝑐𝑝𝑦₂𝑧₂𝑓(ℎ)>0,=𝐴ℎ²𝑔(ℎ)+𝐵ℎ+𝐶,=𝐴ℎ²−𝐷ℎ+𝐶. (38)
If 𝑔(ℎ)>0 for all ℎ>0, then 𝑓(ℎ)>0. Since Δ₁=𝐷²−4𝐴𝐶=−4𝛼+𝑝𝑧₂³𝛽𝑦₂+𝑑+𝛿+𝜂+𝑞𝛽𝑦₂×𝑑+𝜂+𝑞𝛿+𝜂+𝑞𝛽𝑦₂+𝑑𝛼+𝑝𝑧₂−4𝛼+𝑝𝑧₂²𝛽𝑦₂+𝑑+𝛿𝛽𝑦₂×𝜂+𝑞𝛽𝑦₂×+𝑑+𝛿+𝜂+𝑞𝛽𝑦₂𝜂+𝑞+𝑑𝛿+𝜂+𝑞𝛽𝑦₂+𝑑𝛼+𝑝𝑧₂−4𝛼+𝑝𝑧₂³𝜂+𝑞²𝛽𝑦₂𝛽𝑦₂+𝑑+𝛿+𝜂+𝑞×𝑑𝛿+𝜂+𝑞𝛽𝑦₂+𝑑−4𝛼+𝑝𝑧₂²𝜂+𝑞³𝛽𝑦₂²𝛽𝑦₂+𝑑+𝛿−3𝛽𝑦₂𝜂+𝑞²𝛼+𝑝𝑧₂²<0, (39) and 𝑔(0)=𝐶>0, it follows that 𝑔(ℎ)>0 for all ℎ>0, thus 𝑓(ℎ)>0, that is, 𝐻₃>0. 𝐻₄=𝑀₁−𝑁₁𝑀₃−𝑁₃001𝑀₂−𝑁₂𝑀₄−𝑁₄00𝑀₁−𝑁₁𝑀₃−𝑁₃001𝑀₂−𝑁₂𝑀₄−𝑁₄=𝑀₄−𝑁₄𝐻₃. (40)
Noting that 𝑀₄−𝑁₄=𝑐𝑝𝑦₂𝑧₂𝛽𝑦₂𝜂+𝑞+𝑑𝛿+𝜂+𝑞>0, (41)
it follows that 𝐻₄>0. Therefore, all the roots of (35) have negative real parts. This completes the proof of Theorem 9.

For 𝜏=0, the all roots of 𝐺(𝜆)=0 have negative real roots in Theorem 9. By the continuous dependence of roots of 𝐺(𝜆)=0 on the parameters, it follows that there exists 𝜏>0 such that for 𝜏∈[0,𝜏), all roots of (33) satisfy 𝐺(𝜆)=0,Re(𝜆)<0,for𝜏∈[0,𝜏), (42)

and when 𝜏=𝜏, Re(𝜆)=0. To determine 𝜏 and the associated purely imaginary roots 𝜔𝑖(𝜔>0).

Suppose that 𝜆=𝜔𝑖(𝜔>0) is a solution of (33), it follows that 𝜔⁴−𝑀₁𝜔³𝑖−𝑀₂𝜔²𝑖+𝑀₃−𝜔𝑖−𝑁₁𝜔³𝑖−𝑁₂𝜔²+𝐵₃𝜔𝑖+𝐵₄(cos𝜔𝜏−𝑖sin𝜔𝜏)=0. (43)

Separating the real and imaginary parts, it follows that 𝑁₄−𝑁₂𝜔²cos𝜔𝜏+𝑁₃𝜔−𝑁₁𝜔³sin𝜔𝜏=𝜔⁴−𝑀₂𝜔²+𝑀₄,𝑁₁𝜔³−𝑁₃𝜔cos𝜔𝜏+𝑁₄−𝑁₂𝜔²sin𝜔𝜏=𝑀₁𝜔³−𝑀₃𝜔. (44)

From (44), we obtain that 1cos𝜔𝜏=Δ𝐴₁𝜔⁶+𝐴₂𝜔⁴+𝐴₃𝜔²+𝐴₄,𝜔sin𝜔𝜏=−Δ𝐵₁𝜔⁶+𝐵₂𝜔⁴+𝐵₃𝜔²+𝐵₄, (45) where 𝐴₁=𝑀₁𝑁₁−𝑁₂,𝐴₂=𝑁₄+𝑀₂𝑁₂−𝑀₁𝑁₃−𝑀₃𝑁₁,𝐴₃=𝑀₃𝑁₃−𝑀₂𝑁₄−𝑀₄𝑁₂,𝐴₄=𝑀₄𝑁₄,𝐵₁=𝑁₁,𝐵₂=𝑀₁𝑁₂−𝑁₃−𝑀₂𝑁₁,𝐵₃=𝑀₂𝑁₃+𝑀₄𝑁₁−𝑀₃𝑁₂−𝑀₁𝑁₄,𝐵₄=𝑀₃𝑁₄−𝑀₄𝑁₃,Δ=𝑁₄−𝑁₂𝜔²²+𝑁₃𝜔−𝑁₁𝜔³²>0. (46)

It follows from (44) that 𝜔⁸+𝑞₁𝜔⁶+𝑞₂𝜔⁴+𝑞₃𝜔²+𝑞₄=0, (47) where 𝑞₁=𝑀²₁−2𝑀₂−𝑁²₁,𝑞₂=𝑀²₂+2𝑀₄−2𝑀₁𝑀₃+2𝑁₁𝑁₃−𝑀²₂,𝑞₃=𝑀²₃−2𝑀₂𝑀₄+2𝑁₂𝑁₄−𝑀²₃,𝑞₄=𝑀²₄−𝑁²₄. (48)

Letting 𝑢=𝜔², (47) becomes 𝑢⁴+𝑞₁𝑢³+𝑞₂𝑢²+𝑞₃𝑢+𝑞₄=0. (49)

Denote 𝐺(𝑢)=𝑢⁴+𝑞₁𝑢³+𝑞₂𝑢²+𝑞₃𝑢+𝑞₄. Then we have 𝐺^′(𝑢)=4𝑢³+3𝑞₁𝑢²+2𝑞₂𝑢+𝑞₃. (50)

Suppose that (49) has positive real roots. Without loss of generality, we assume that it has 𝑛(1≤𝑛≤4) positive real roots, defined by 𝑢₁<𝑢₂<⋯<𝑢_𝑛, respectively. Then (47) has 𝑛 positive real roots 𝜔₁=√𝑢₁,𝜔₂=√𝑢₂,…,𝜔_𝑛=√𝑢_𝑛. (51)

From (45), we have 𝜏^𝑗_𝑙=1𝜔_𝑙𝐴arccos¹𝜔⁶_𝑙+𝐴₂𝜔⁴_𝑙+𝐴₃𝜔²_𝑙+𝐴₄𝑁₄−𝑁₂𝜔²_𝑙²+𝑁₃𝜔_𝑙−𝑁₁𝜔³_𝑙²+2𝑗𝜋, (52) where 𝑙=1,2,…,𝑛, 𝑗=0,1,2,3,…, then ±𝜔_𝑙i are a pair of purely imaginary roots of (33) with 𝜏^𝑗_𝑙.

Differentiating (33) implicitly with respect to 𝜏, we obtain that 𝑑𝜆𝑑𝜏⁻¹=−4𝜆³+3𝑀₁𝜆²+2𝑀₂𝜆+𝑀₃e^𝜆𝜏𝜆𝑁₁𝜆³+𝑁₂𝜆²+𝑁₃𝜆+𝑁₄+3𝑁₁𝜆²+2𝑁₂𝜆+𝑁₃𝜆𝑁₁𝜆³+𝑁₂𝜆²+𝑁₃𝜆+𝑁₄−𝜏𝜆. (53)

Thus, Re𝑑𝜆𝑑𝜏⁻¹_𝜏=𝜏^𝑗_𝑙=1𝜔_𝑙Δ3𝑀₁𝜔²_𝑙−𝑀₃×𝑁₁𝜔³_𝑙−𝑁₃𝜔_𝑙cos𝜔_𝑙𝜏+𝑁₄−𝑁₄𝜔²_𝑙sin𝜔_𝑙𝜏+4𝜔³_𝑙−2𝑀₂𝜔_𝑙×𝑁₄−𝑁₂𝜔²_𝑙cos𝜔_𝑙𝜏−𝑁₁𝜔³_𝑙−𝑁₃𝜔_𝑙sin𝜔_𝑙𝜏+𝑁₃−3𝑁₁𝜔²_𝑙𝑁₁𝜔³_𝑙−𝑁₃𝜔_𝑙+2𝑁₂𝜔_𝑙𝑁₄−𝑁₂𝜔²_𝑙. (54)

On substituting (45) into (54), we obtain Re𝑑𝜆𝑑𝜏⁻¹_𝜏=𝜏^𝑗_𝑙=1Δ4𝜔⁶_𝑙+3𝑀²₁−2𝑀₂−𝑁²₁𝜔⁴_𝑙+2𝑀²₂−𝑁²₂+2𝑀₄+2𝑁₁𝑁₃−2𝑀₁𝑀₃𝜔²_𝑙+𝑀²₃−𝑁²₃+2𝑁₂𝑁₄−2𝑀₂𝑀₄=𝐺^′𝑢_𝑙Δ, (55) where Δ=(𝑁₄−𝑁₂𝜔²)²+(𝑁₃𝜔−𝑁₁𝜔³)²>0. If we suppose that 𝐺^′(𝑢)≠0, then signRe𝑑𝜆𝑑𝜏_𝜏=𝜏^𝑗_𝑙=signRe𝑑𝜆𝑑𝜏⁻¹_𝜏=𝜏^𝑗_𝑙𝐺=sign^′𝑢_𝑙Δ=sign𝐺^′𝑢_𝑙. (56)

Applying Theorem 9 and the Hopf bifurcation theorem for functional differential equation [22] from (56), we derive the existence of a Hopf bifurcation as follows.

Theorem 10. Suppose that (49) has at least one simple positive root and 𝑢 is the last such root, then there is a Hopf bifurcation for the system (3) as 𝜏 passes upwards through 𝜏 leading to a periodic solution that bifurcates from 𝐸₂, where 1𝜏=𝜔𝐴𝑎𝑟𝑐𝑐𝑜𝑠¹𝜔⁶_𝑙+𝐴₂𝜔⁴_𝑙+𝐴₃𝜔²_𝑙+𝐴₄𝑁₄−𝑁₂𝜔²_𝑙²+𝑁₃𝜔_𝑙−𝑁₁𝜔³_𝑙²+2𝑗𝜋. (57)

Remark 11. If𝑢 is the last simple positive root of (49), then we have 𝐺^′(𝑢)>0. From (56), we obtain Re[𝑑𝜆/𝑑𝜏]_𝜏=𝜏>0.

Remark 12. In this paper, we construct a few Lyapunov functions (functionals) to prove the global stability of steady states of ODE model (DDE model). This function (functional) can also prove the global stability of steady states of other viral infections models with cure rate [7, 10]. Moreover, the method studying the existence of Hopf bifurcations applies to other viral infections models with immune delay [2, 18, 19].

5. Numerical Simulations

In this section, we perform numerical calculation to support our theoretical analysis of this paper.

Example 13. If we choose parameters 𝑠=1.5, 𝛽=0.3, 𝛼=0.1, 𝑝=0.015, 𝑐=0.12, 𝑟=0.8, 𝑑=0.1, 𝛿=0.6, 𝜂=0.4, and 𝑞=0.7, then 𝑅₁=1.33>1 and the CTL-present equilibrium 𝐸₂(1.1751,1.3825,6.6667,1.6283). From (49), we obtain that 𝑢⁴+10.5385𝑢³+3.9615𝑢²+0.0729𝑢+0.0151=0. (58)
Equation (58) has no positive roots, and all roots have negative real parts. Therefore, the equilibrium is locally asymptotically stable for all 𝜏≥0 (e.g., 𝜏=1.5, see Figure 1).

Figure 1 

The CTL-present infection equilibrium 𝐸2 of system (3) is locally asymptotically stable when 𝑅1=1.33>1 and 𝜏=1.5.

Example 14. If we select parameters 𝑠=10, 𝛽=0.1, 𝛼=0.4, 𝑝=0.3, 𝑐=0.5, 𝑟=0.9, 𝑑=1, 𝛿=1, 𝜂=1, and 𝑞=4, then 𝑅₁=1.45>1 and the CTL-present equilibrium 𝐸₂(8.8107,0.2642,1.7997,0.6244). It follows from (49) that 𝑢⁴+40.7439𝑢³+41.9038𝑢²−1.9589𝑢−2.0726=0. (59)
Equation (59) has only one positive real root 𝑢=0.2217 and any other roots have negative real parts. Thus, 𝜔=√𝑢=0.4709. In addition, it is easy to show that 𝜏=0.948. Therefore, Theorem 10 is satisfied.
If 𝜏=0.2<𝜏, the CTL-present infection equilibrium 𝐸₂ of system (3) is locally asymptotically stable (see Figure 2). If 𝜏=5>𝜏, then the CTL-present infection equilibrium 𝐸₂ of system (3) becomes unstable, and the Hopf bifurcation occurs (see Figure 3).

Figure 2 

The CTL-present infection equilibrium 𝐸2 of system (3) is locally asymptotically stable when 𝑅1=1.45>1 and 𝜏=0.2<𝜏.

Figure 3 

The bifurcating periodic solution from the equilibrium 𝐸2 of system (3) occurs when 𝜏=5.

6. Conclusion

In this paper, we have studied an HIV infection model including infected cells in an eclipse stage and CTL immune response. The global stability of the uninfected equilibrium 𝐸₀ for system (2) and (3) has been given by the LaSalle’s invariance principle when the basic reproductive ratio 𝑅₀<1; it shows that the disease will be controlled. Compared with the earlier modeling studies on the immune response of HIV infection [14, 18, 20], our analysis reveals the existence of a CTL-absent infection equilibrium 𝐸₁(𝑥₁,𝑤₁,𝑦₁,0) when 𝑅₀>1. We also obtained the global asymptotic stability of a CTL-absent infection equilibrium 𝐸₁ for system (2) and (3) when 𝑅₁<1 and 1<𝑅₀<1+(𝜂+𝑞)/𝛿. This indicates that there is a persistent HIV infection with no humeral and cellular immune responses. Furthermore, we can see that the time delay has no effect on the stability of the uninfected equilibrium 𝐸₀ and CTL-absent infection equilibrium 𝐸₁ for the DDE model.

When 𝑅₁>1, we show that the CTL-present infection equilibrium 𝐸₂ is locally asymptotically stable when the delay 𝜏 is small, and with the increase of the delay 𝜏 the stability of 𝐸₂ may destabilize and lead to Hopf bifurcation. This suggests that, with the HIV infection developing, the proviral load and CTL frequency can either stabilize at a constant level or show oscillations. Similar phenomenon was also observed in [17–19]. The HIV dynamics model without immune delay is globally stable [14, 15]. In this paper, we show that the HIV infection model including infected cells in an eclipse stage and CTL immune response without immune delay is globally stable; and for the model with immune delay, Hopf bifurcation appears under some conditions.

Acknowledgments

This work was supported by National Natural Science Foundation of China (61174209, 11071013), the Basic Scientific Research Foundation of Central University (FRF-BR-11-048B, FRF-BR-12-004), and the Basic Theory Research Foundation for Engineering Research Institute of USTB (YJ2012-001).