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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 419593, 12 pages
http://dx.doi.org/10.1155/2013/419593
Research Article

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

Department of Applied Mathematics, University of Science and Technology Beijing, Beijing 100083, China

Received 28 January 2013; Accepted 31 March 2013

Academic Editor: Junjie Wei

Copyright © 2013 Shengyu Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [17]. 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 [1215], and others contain immune delay [1619]. 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 [1619]. 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.

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:

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

Theorem 1. For system (2) and (3), the uninfected equilibrium always exists;(1)if , a CTL-absent infection equilibrium exists, where (2)if , there exists a CTL-present infection equilibrium , where

3. 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 . Before analyzing the stability of system (2), we now show that the solutions of system (2) are bounded.

Theorem 2. Let , and be the solution of system (2) satisfying initial conditions (7), then there exists such that , , , and hold after sufficiently large time .

Proof. Let where . It follows from (2) that
Therefore, for all large , where is an arbitrarily small positive constant. Thus, , and for some positive constant .

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

Proof. Construct a Lyapunov function where . The derivative of along positive solutions of system (2) is given as follows:
On substituting and into (11), we derive that
If , then for all and . So the uninfected equilibrium is stable. Clearly, it follows from (12) that if and only if , and . Therefore, the largest invariant set in the set is the singleton . By LaSalle invariance principle, it follows that the equilibrium is globally asymptotically stable.

Theorem 4. For system (2), if and , CTL-absent infection equilibrium is globally asymptotically stable.

Proof. Define a Lyapunov function
Calculating the derivative of along positive solutions of system (2), it follows that
At CTL-absent infection equilibrium , on substituting , and into (14), we obtain that
Noting that
therefore,
Since and the equality holds if and only if and . If , then . So, if and , then for all and . Clearly, it follows from (17) that if and only if , and , thus the largest invariant set in the set is the singleton . Therefore, the global asymptotic stability of follows from the LaSalle’s invariance principle.

Theorem 5. For system (2), if and , CTL-present infection equilibrium is globally asymptotically stable.

Proof. Define a Lyapunov function
Calculating the derivative of along positive solutions of system (2), we obtain that
At CTL- present infection equilibrium , on substituting , , , and into (19), it follows that
Noting that it follows from (20) and (21) that
Since and the equality holds if and only if , and . If , then . Therefore, if and , it follows from (22) that for all , and . Clearly, it follows from (22) that if and only if , , , and . So the largest invariant set in the set is the singleton . By LaSalle invariance principle, we conclude that the equilibrium 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 be the Banach space of continuous functions mapping from the interval to with the topology of uniform convergence, where

The initial conditions for system (3) are given as follows: where . It is clear to see that all solutions of system (3) satisfying the initial conditions (24) are positive for all . By the similar method to Theorem 2, we can get the following theorem.

Theorem 6. Let , and be the solution of system (3) satisfying the initial conditions (24), then there exists such that , and hold after sufficiently large time .

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

Proof. If , construct a Lyapunov functional where . Calculating the derivative of along positive solutions of system (3), it follows that
On substituting and into (26), we derive that
If , we have for all , , , and . It follows that the uninfected equilibrium is stable. Clearly, it follows from (27) that if and only if , , , and . Therefore the largest invariant set in the set is the singleton . By LaSalle invariance principle [21], we can conclude that the equilibrium 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 for the DDE model.

Theorem 8. For system (3), if and , then CTL-absent infection equilibrium is globally asymptotically stable.

Proof. Define a lyapunov functional
Calculating the derivative of along positive solutions of system (3), it follows that
At CTL-absent infection equilibrium , on substituting , and into (29), we obtain that
It follows from (16) and (30) that
Since and the equality holds if and only if , and . If , then . Therefore, if and , it follows from (31) that for all , and . It is readily seen from (31) that if and only if , and . Thus the largest invariant set in the set is the singleton . Then the global asymptotic stability of 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 for the DDE model. Next, we analyze stability and Hopf bifurcation at the CTL-present equilibrium .

Firstly, the linearized equations of system (3) at are given as follows:

The characteristic equation of system (32) at takes the form where

Theorem 9. Suppose , if , then the CTL-present equilibrium of system (3) is locally asymptotically stable.

Proof. If , (33) becomes
Since , and , by the Routh-Hurwitz criteria, it follows that where
Let
If for all , then . Since and , it follows that for all , thus , that is, .
Noting that
it follows that . Therefore, all the roots of (35) have negative real parts. This completes the proof of Theorem 9.

For , the all roots of have negative real roots in Theorem 9. By the continuous dependence of roots of on the parameters, it follows that there exists such that for , all roots of (33) satisfy

and when , . To determine and the associated purely imaginary roots .

Suppose that is a solution of (33), it follows that

Separating the real and imaginary parts, it follows that

From (44), we obtain that where

It follows from (44) that where

Letting , (47) becomes

Denote . Then we have

Suppose that (49) has positive real roots. Without loss of generality, we assume that it has positive real roots, defined by , respectively. Then (47) has positive real roots

From (45), we have where , , then are a pair of purely imaginary roots of (33) with .

Differentiating (33) implicitly with respect to , we obtain that

Thus,

On substituting (45) into (54), we obtain where . If we suppose that , then

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 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 , where

Remark 11. If is the last simple positive root of (49), then we have . From (56), we obtain .

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 , , , , , , , , , and , then and the CTL-present equilibrium . From (49), we obtain that
Equation (58) has no positive roots, and all roots have negative real parts. Therefore, the equilibrium is locally asymptotically stable for all (e.g., , see Figure 1).

419593.fig.001
Figure 1: The CTL-present infection equilibrium of system (3) is locally asymptotically stable when and .

Example 14. If we select parameters , , , , , , , , , and , then and the CTL-present equilibrium . It follows from (49) that
Equation (59) has only one positive real root and any other roots have negative real parts. Thus, . In addition, it is easy to show that . Therefore, Theorem 10 is satisfied.
If , the CTL-present infection equilibrium of system (3) is locally asymptotically stable (see Figure 2). If , then the CTL-present infection equilibrium of system (3) becomes unstable, and the Hopf bifurcation occurs (see Figure 3).

419593.fig.002
Figure 2: The CTL-present infection equilibrium of system (3) is locally asymptotically stable when and .
419593.fig.003
Figure 3: The bifurcating periodic solution from the equilibrium of system (3) occurs when .

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 for system (2) and (3) has been given by the LaSalle’s invariance principle when the basic reproductive ratio ; 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 when . We also obtained the global asymptotic stability of a CTL-absent infection equilibrium for system (2) and (3) when and . 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 and CTL-absent infection equilibrium for the DDE model.

When , we show that the CTL-present infection equilibrium is locally asymptotically stable when the delay is small, and with the increase of the delay the stability of 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 [1719]. 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).

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