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

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.


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][2][3][4][5][6][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 virusinfected 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][13][14][15], and others contain immune delay [16][17][18][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 where , , and  represent the concentration of uninfected target cells, productively infected cells, CTLs at time , respectively.Parameters  and  are the birth rate and death rate of uninfected cells, respectively.The uninfected cells become infected at rate of .Productively infected cells are produced at rate ,  is the death rate of productively infected cells,  is the strength of the lytic component, and  is the death rate of CTLs.Function (, , ) 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 (, , ) = .Ji et al. [12] assumed that the production of CTLs also depends on the population of CTL cells and chose the former (, , ) = .
In this paper, we also consider the dynamics of HIV infection with CTL response and give (, , ) = .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: where  represents the concentration of infected cells in the eclipse stage at time .Infected cells in the eclipse phase revert to the uninfected class at a constant rate .In addition, they may alternatively progress to the productively infected class at the rate  or die at the rate .But some authors believe that time delay cannot be ignored in models for immune response [16][17][18][19].In this paper,  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: 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.

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: It is clear that  0 >  1 always holds.For system (2) and (3), there exists three equilibriums.

The Global Stability of the ODE Model
The initial conditions for system (2) are given as follows: It is clear that all solutions of system (2) are positive for  > 0.
Before analyzing the stability of system (2), we now show that the solutions of system (2) are bounded.
Proof.Define a Lyapunov function Calculating the derivative of  3 along positive solutions of system (2), we obtain that Noting that it follows from ( 20) and ( 21) that

The Stability Analysis of the DDE Model
In this section, we consider the stability of the delay model (3).
From the above analysis, we can obtain that the time delay has no effect on the stability of the uninfected equilibrium  0 for the DDE model.Theorem 8.For system (3), if  1 < 1 and 1 <  0 ≤ 1 + ( + )/, then CTL-absent infection equilibrium  1 is globally asymptotically stable.
From the above analysis, we obtain that the time delay has no effect on the stability of the CTL-absent infection equilibrium  1 for the DDE model.Next, we analyze stability and Hopf bifurcation at the CTL-present equilibrium  2 .
Noting that it follows that  4 > 0. Therefore, all the roots of (35) have negative real parts.This completes the proof of Theorem 9.
Theorem 10.Suppose that (49) has at least one simple positive root and  is the last such root, then there is a Hopf bifurcation for the system (3) as  passes upwards through  leading to a periodic solution that bifurcates from  2 , where

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

Conclusion
In this paper, we have studied an HIV infection model including infected cells in an eclipse stage and CTL immune Journal of Applied Mathematics  response.The global stability of the uninfected equilibrium  0 for system (2) and (3) has been given by the LaSalle's invariance principle when the basic reproductive ratio  0 < 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  1 ( 1 ,  1 ,  1 , 0) when  0 > 1.We also obtained the global asymptotic stability of a CTL-absent infection equilibrium  1 for system (2) and (3) when  1 < 1 and 1 <  0 < 1 + ( + )/.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  0 and CTL-absent infection equilibrium  1 for the DDE model.
When  1 > 1, we show that the CTL-present infection equilibrium  2 is locally asymptotically stable when the delay  is small, and with the increase of the delay  the stability of  2 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][18][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.