Stability and Hopf Bifurcation in an HIV-1 Infection Model with Latently Infected Cells and Delayed Immune Response

AnHIV-1 infectionmodel with latently infected cells and delayed immune response is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria is established and the existence of Hopf bifurcations at the CTL-activated infection equilibrium is also studied. By means of suitable Lyapunov functionals and LaSalle’s invariance principle, it is proved that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio for viral infection


Introduction
Mathematical and computational models of the human immune response under HIV-1 infection have received great attention in recent years [1][2][3][4][5][6][7][8][9].It is a useful tool of better understanding disease dynamics and making prediction of disease outbreak and evaluations of prevention strategies and drug therapy strategies used against HIV-1 infection.
It is well known that when HIV-1 enters the body, it targets cells with CD4 receptors, including the CD4+ T-cells, the main driver of the immune response.Recent studies have shown that a significant proportion of CD4+ T-cells are infected by the virus, and that this specific population of Tcells might be preferentially infected [10].In human's immune system, cytotoxic T lymphocytes (CTLs) play an important role in antiviral defense by attacking infected cells.Therefore, it is important and yet has been a hot topic to formulate models to explain the exhaustion of the CD4+ T-cells and CTLs.Such models involve the concentrations of uninfected CD4+ T-cells, , infected CD4+ T-cells that are producing virus, , free virus, V, and CTLs, .A basic mathematical model describing HIV-1 infection dynamics that has been studied in [1,2] is of the form ẋ () =  −  () −  () V () , ẏ () =  () V () −  () −  ()  () , V () =  () − V () , ż () =  () −  () , (1) where uninfected, susceptible CD4+ T-cells are created from sources within the body at a rate, , uninfected CD4+ Tcells die at rate () and become infected at rate ()V(), where  is the rate constant describing the infection process; infected cells die at rate () and are lysed by CTLs at a rate ()(); free virus is produced from infected cells at rate () and is removed at rate ().The CTLs expand at a rate () and decay at a rate ().
Time delays cannot be ignored in models for immune response, especially the delay between viral appearance and the production of new immune particles [6][7][8][9].Wang et al. [6] studied the effects of the time delay for immune response on a three-dimensional system with ż =  ( − ) − .
By assuming that the production of CTLs also depends on the population of CTL cells, Canabarro et al. [7] investigated the effects of a time delay on a four-dimensional system with ż =  ( − )( − ) − .
However, in all the previous works mentioned above, they all neglected the fact that once in the cells not all viruses initiate active virus production.A large proportion of CD4+ T-cells are latently infected following the integration of proviral DNA into the host cell genome, some of which can remain quiescent for long periods of time before becoming activated [11].In [12], such cells are defined as latently infected cells.The capability of the HIV-1 to persist latent inside CD4+ T-cells is currently regarded as a barrier to recovery from infection.But till now, as far as we know, only a few works (see, [13,14]) concern the effects that latently infected cells are expected to have on HIV-1 infection process.In this paper, motivated by the works of [1,7,13], we modify the basic virus infection model and add a further state variable, , which represents the population of latently infected cells and propose a four-dimensional delayed HIV-1 infection model with latently infected cells.The model is given by where the parameters , , , , , , and  are the same as those defined in model (1).In our model, we assume that the free virus interacts with the uninfected cells to produce actively infected cells at rate ()()/[1 + ()] and latently infected cells at rate (1−)()()/[1+()] due to the saturation response of the infection rate, where 0 <  < 1 and  > 0. Latently infected cells containing proviral DNA die at rate () and become activated at rate ().The CTL response is activated at a rate proportional to the number of infected cells at a previous time, ( − )( − ), where  is the time delay of CTL response.The initial conditions for system (2) take the form where ( 1 (),  2 (),  3 (),  4 ()) ∈  ([−, 0],  4 +0 ) and the Banach space of continuous functions maps the interval Our primary goal is to carry out a complete mathematical analysis of system (2) and establish its global dynamics.The organization of this paper is as follows.In Section 2, by analyzing the corresponding characteristic equations, we study the local stability of an infection-free equilibrium and a CTL-inactivated infection equilibrium of model (2).In Section 3, we discuss the local stability and the existence of Hopf bifurcations at the CTL-activated infection equilibrium.In Section 4, by means of suitable Lyapunov functionals and LaSalle's invariance principle, we study the global stability of the infection-free equilibrium and the CTL-inactivated infection equilibrium with any  ≥ 0 and the CTL-activated infection equilibrium with  = 0, respectively.Numerical simulations are carried out in the last section to illustrate the main results.A brief remark is also given in this section to conclude our work.

Equilibria and Their Local Stability
In this section, we firstly establish the nonnegativity and boundedness of solutions of system (2) and then give sufficient conditions for the existence of each of feasible equilibria of system (2) and discuss the local stability of the infectionfree equilibrium and the CTL-inactivated infection equilibrium, respectively.Theorem 1.Let ((), (), (), V()) be any solution of system (2).Then, under the initial conditions (3), all solutions ((), (), (), V()) are nonnegative on [0, +∞) and ultimately bounded.
Next, we prove the ultimate boundedness of (2 This implies that () is ultimately bounded, and so are (), (), (), and V().Thus the solutions of (2) are ultimately bounded.This completes the proof.
Clearly, system (2) always has an infection-free equilibrium  0 ( 0 , 0, 0, 0), where  0 = /. Denote Here  0 and  1 are called the basic reproduction ratios for viral infection and CTL immune response of system (2), respectively.It is easy to show that  0 >  1 always holds.
As to the stability of  1 , we have the following result.
Then we discuss the location of the roots of the following equation Hence, () = 0 has one positive real root. 1 ( 1 ,  1 ,  1 , 0) is unstable.
If  1 < 1, then we prove that all roots of () = 0 have negative real parts.Assume that Re  ≥ 0, then from () = 0, we derive that which comes to a contradiction.Hence, Re  < 0. From the discussion above, we can see that, if  1 < 1, the CTLinactivated infection equilibrium  1 ( 1 ,  1 ,  1 , 0) is locally asymptotically stable.This completes the proof of Theorem 3.

Stability and Hopf Bifurcation at the CTL-Activated Infection Equilibrium
From the results above, when  1 > 1, the CTL-inactivated infection equilibrium  1 is unstable, and at the same time a CTL-activated infection equilibrium  * emerges.Now we regard  as a parameter to study the stability of  * and the existence of Hopf bifurcations.The characteristic equation of system (2) at the CTLactivated infection equilibrium  * ( * ,  * ,  * ,  * ) is of the form where 2 = ( +  + ) ( +  +  * ) +  ( +  * ) +  ( + ) ( When  = 0, (21) becomes It is easy to verify that Since by direct calculation, we can easily derive that Then for  = 0, according to Routh-Hurwitz criterion, all the roots of (21) have negative real parts if and only if the following conditions hold: (H1) From what has been discussed above, we have the following result.Theorem 4. Let  = 0. Then the CTL-activated infection equilibrium  * ( * ,  * ,  * ,  * ) of system (2) is locally asymptotically stable if  1 > 1 and the condition (H1) holds.
For  > 0, it is not easy to find rigorously local stability of  * .In the following, we will firstly investigate the existence of purely imaginary roots to (21) following the framework of that in [9].
Let () = () + () be the root of (21) near  =  Proof.Calculating the derivative with respective to , we obtain Then we derive from (37) that Therefore, we have where   =  2  .From (29) and (31), we get Then we get that Furthermore, it follows that sign [ (())  ] Then, this completes the proof of Theorem 5.
Applying the Hopf bifurcation theorem for functional differential equation [15], we derive the existence of a Hopf bifurcation at  * as stated in the following theorem.Theorem 6. Suppose that (31) has at least one simple positive root and  0 is the last such root.Then there is a Hopf bifurcation for system (2) as  passes through  0 leading to a periodic solution that bifurcates from  * , where Remark 7. As an example, we suppose that (31) has three simple positive roots, denoted by  1 <  2 <  3 , respectively.

Global Stability
In this part, we study the global stability of the infection-free equilibrium and the CTL-inactivated infection equilibrium and then discuss the global stability of the CTL-activated infection equilibrium when  = 0.The strategy of proofs is to use suitable Lyapunov functionals and LaSalle's invariance principle in [16].Define Clearly, for  ∈ (0, +∞),  () is nonnegative and has the global minimum at  = 1 and (1) = 0.
We are now in a position to establish the global stability of the CTL-inactivated infection equilibrium  1 of system (2).
Finally, we give the global stability result of  * when there is no time delay, that is,  = 0.

Numerical Simulations and Conclusions
In the following, we give examples to illustrate the main theoretical results above.In system ( 2 and the CTL-activated infection equilibrium  * exists.The values of these parameters satisfy the condition of Theorem 6.By calculation, we derive that  0 ≈ 0.2860 and  0 ≈ 0.3746. When  = 0.2 <  0 , the trajectory converges to the CTL-activated infection equilibrium  * (see Figure 1).While, when  = 1.2 >  0 , the CTL-activated infection equilibrium  * of (2) becomes unstable (see Figure 2), and the Hopf bifurcation occurs.
In this paper, we have investigated the global dynamics of an HIV-1 infection model with latently infected cells and delayed immune response.By giving the explicit expressions of two basic reproduction ratios  0 and  1 , we first gave sufficient conditions for the existence of each of feasible equilibria of system (2).Then, a detailed analysis on the local asymptotic stability of the equilibria was carried out.It was shown that, if  0 < 1 (hence,  1 and  * are not feasible), the infectionfree equilibrium  0 is locally asymptotically stable.If  1 < 1, the CTL-inactivated infection equilibrium  1 is locally asymptotically stable.If  1 > 1, the CTL-activated infection equilibrium  * exists.By giving Theorem 6, we studied the existence of Hopf bifurcations at  * .By constructing suitable Lyapunov functionals and using LaSalle's invariance principle, we proved that, for any time delay  ≥ 0, if  0 < 1, the infection-free equilibrium  0 is globally asymptotically stable.If  0 > 1 and  1 < 1, the CTL-inactivated infection equilibrium  1 is globally asymptotically stable.From this we can see that time delay has no effect on the stability of the infection-free equilibrium and the CTL-inactivated infection equilibrium.When  = 0, we also proved the global asymptotic stability of the CTL-activated infection equilibrium  * of system (2) whenever it exists.