Global Stability of Delayed Viral Infection Models with Nonlinear Antibody and CTL Immune Responses and General Incidence Rate

The dynamical behaviors for a five-dimensional viral infection model with three delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses, and nonlinear incidence rate are investigated. The threshold values for viral infection, antibody response, CTL immune response, CTL immune competition, and antibody competition, respectively, are established. Under certain assumptions, the threshold value conditions on the global stability of the infection-free, immune-free, antibody response, CTL immune response, and interior equilibria are proved by using the Lyapunov functionals method, respectively. Immune delay as a bifurcation parameter is further investigated. The numerical simulations are performed in order to illustrate the dynamical behavior of the model.


Introduction
In recent years, many authors have formulated and studied mathematical models which describe the dynamics of virus population in vivo. These provide insights in our understanding of HIV (human immunodeficiency virus) and other viruses, such as HBV (hepatitis B virus) and HCV (hepatitis C virus) . In particular, the global stability of steady states for these models will give us a detailed information and enhance our understanding about the viral dynamics.
During viral infections, the immune system reacts against virus. The antibody and CTL play the crucial roles in preventing and modulating infections. The antibody response is implemented by the functioning of immunocompetent B lymphocytes. The CTL immune response has the ability to suppress the virus replication in vivo. Hence, in order to prevent virus infection, an effective vaccine needs both strong neutralizing antibody and CTL immune responses [1,2,14,[18][19][20][21][22][23][25][26][27][28][29][30][31][32]. Based on these, it is of interest for us to investigate whether sustained oscillations are the result of delayed viral infection model. This provides us with the motivation to conduct our work. In [ where , , V, , and denote the concentrations of susceptible host cells, infected cells, free virus, antibody responses, and CTL immune responses, respectively. The local and global stability of the infection-free equilibrium and infected equilibrium and the existence of Hopf bifurcation are 2 Computational and Mathematical Methods in Medicine obtained. Furthermore, by using the Nyquist criterion, the estimation of the length of the delay to preserve stability of the infected equilibrium is obtained.
Motivated by the work in [1,2,20,21], in the present paper we propose a general viral infection model with three time delays which describes the interactions of antibody, CTL immune responses, and nonlinear incidence rate where ( ) denotes the intrinsic growth rate of uninfected target cells accounting for both production and natural mortality. In the literature of virus dynamics, the typical forms of the growth rate are ( ) = − and ( ) = − + (1− / ), where , , , are positive real numbers [4-13, 15, 16, 18, 20-23, 26-32, 34].
It is also assumed that the death rates of the infected target cells, viruses, antibody, and CTLs depend on their concentrations. These rates are given by 1 ( ), 2 (V), ℎ 3 ( ), and 4 ( ), respectively. The neutralization rate of viruses and the activation rate of B cells are proportional to the product of the removal rates of the viruses and B cells. Let 2 (V) 3 ( ) and 2 (V) 3 ( ) be the neutralization rate of viruses and activation rate of B cells, respectively. The typical forms can be seen as V and V [1,2,20,21,31,32]. Accordingly, let 1 ( ) 4 ( ) and 1 ( ) 4 ( ) be the killing rate of infected cells and the birth rate of the CTL cells, respectively. The typical forms are and that appear in several papers [1,2,14,20,22,27,30,34].
For model (2), based on the epidemiological background, we assume that virus production occurs after the virus entry by the time delay 1 . The probability of surviving the time period from − 1 to is − 1 1 . Let 2 be the maturation time of the newly produced viruses. The constant − 2 2 denotes the surviving rate of virus during the delay period. Antigenic stimulation generating CTL cell may need a period of time 3 .
In this paper, our purpose is to investigate the dynamical properties of model (2), including the local and global stability of equilibria. The reproduction numbers for viral infection, antibody response, CTL immune response, CTL immune competition, and antibody competition, respectively, are calculated. By using Lyapunov functionals and LaSalle's invariance principle, the threshold conditions for the global asymptotic stability of infection-free equilibrium 0 , immune-free equilibrium 1 , infection equilibrium 2 only with antibody response, and infection equilibrium 3 only with CTL immune response and infection equilibrium 4 with both antibody and CTL immune responses when the delay 3 = 0, respectively, are established. By using the linearization method, the instability of equilibria 0 , 1 , 2 , and 3 , respectively, is also established. Furthermore, by using the numerical simulation method, we will discuss the existence of the Hopf bifurcation and stability switches at equilibria 3 and 4 when 3 > 0.
The organization of this paper is as follows. In the next section, the basic properties of model (2) for the positivity and boundedness of solutions, the threshold values, and the existence of equilibria are discussed. In Section 3, the threshold conditions on the global stability and instability of equilibria 0 , 1 , and 2 are proved. When 3 = 0, the threshold conditions on the global stability and instability for equilibria 3 and 4 are stated and proved. In Section 4, the numerical simulations are given to further discuss the stability of equilibria 3 and 4 when 3 > 0. It is shown that the Hopf bifurcation and stability switches at these equilibria occur as 3 increases. In the last section, we offer a brief conclusion.
From ( 1 ) we easily obtain that ( ) > 0 for all 0 < < and ( ) < 0 for all > . Assumption ( 1 ) shows that the number of healthy cells has a maximum capacity in the absence of infection. When < , ( ) has a positive growth; if > it has a negative growth. Assumption ( 2 ) implies that there are no new infected cells (i.e., ( , V) = 0) without healthy cells ( = 0) or virus (V = 0). The higher the number of healthy cells is, the higher the number of healthy cells which are infected in the unit time will be. Similarly, the higher the amount of virus V is, the higher the number of healthy cells which are infected in the unit time will be. Assumption ( 3 ) assumes that the death rates of the infected target cells , virus V, antibodies , and CTLs depend on their concentrations. If these numbers , V, , increase, the corresponding rates 1 ( ), 2 (V), ℎ 3 ( ), and 4 ( ) will increase, and the ratio ( )/ is no less than a positive constant for = 1, 2, 3, 4. Finally, assumption ( 4 ) indicates that both the rate of new infections of target cells and the virus clearance rate increase according to the level of virus. However, the corresponding ratio is nonincreasing.
We consider the existence of infection equilibrium 3 = ( 3 , 3 , V 3 , 3 , 0) with only CTL immune response. From the third and fourth equations of (4), we obtain unique 3 = −1 Define the constant which is called the CTL immune response reproductive number of model (2). Solving the second equation for yields Therefore, 3 exists and is unique if 2 > 1.
Define the constants which are called the CTL immune response competitive reproductive number and the antibody response competitive reproductive number of model (2), respectively. Solving the second equation for yields a unique Solving the third equation for , we further obtain a unique Therefore, 4 exists and is unique if 3 > 1 and 4 > 1. ( 2 ) and ( 4 ), we obtain 1 < 0 and 2 < 0 . In fact,

Stability of Equilibrium 0
Proof. Consider conclusion (a). Define a Lyapunov functional 1 ( ) as follows: Computational and Mathematical Methods in Medicine 5 Calculating the time derivative of 1 ( ) along solutions of model (2), we obtain It follows that Note that [36], 0 is globally asymptotically stable.
Remark 4. Theorem 3 shows that if only equilibrium 0 exists, then it is globally asymptotically stable, and delays 1 , 2 , and 3 do not impact the stability of 0 . 1 . Firstly, we introduce two lemmas which will be used in the proof of Theorem 7.
By ( 2 ) and ( 4 ), it follows that sign This completes the proof.
Next, consider conclusion (b). By computing, the characteristic equation of the linearization system of model (2) at 1 is Computational and Mathematical Methods in Medicine Hence, there is also a positive root * such that 1 ( * ) = 0. Therefore, when 1 > 1 or 2 > 1, 1 is unstable. This completes the proof.
Remark 8. Theorem 7 shows that if only equilibria 0 and 1 exist, then 1 is globally asymptotically stable, and delays 1 , 2 , and 3 do not impact the stability of 1 .

Stability of Equilibrium 2 .
We firstly have the following Lemma.
Remark 11. Theorem 10 shows that if only equilibria 0 , 1 , and 2 exist, then when 3 ≤ 1 and 1 > 1, 2 is globally asymptotically stable, and delays 1 , 2 , and 3 do not impact the stability of 2 . 3 . On the stability analysis of equilibrium 3 , we only discuss the following case: 1 ≥ 0, 2 ≥ 0, and 3 = 0. Other cases, 1 ≥ 0, 2 ≥ 0, and 3 ≥ 0, are numerically verified for bifurcation phenomena and stability switches of 3 but the analytic analysis is left as an open problem. Before the proof of theorem, we have the following Lemma.

Numerical Simulations
In the above section, we obtain the global asymptotic stability of equilibria 3 and 4 when the delay 3 = 0. In this section, by using the numerical simulation, it is shown that the Hopf bifurcation and stability switches occur at equilibria 3 and 4 in the case 3 > 0.
Example 17. Corresponding to model (2), we consider the following model:  Figures 1-4, we see that as 3 increases the complex dynamical behaviors of equilibrium 3 occur.
In Figures 1-8, we denote by (a) the time-series of ( ), by (b) the time-series of ( ), by (c) the time-series of V( ), by (d) the time-series of ( ), and by (e) the time-series of ( ).   Example 18. Corresponding to model (2), we consider the following model:

Discussion
In this paper we have considered an in-host model with intracellular delay 1 , virus replication delay 2 , and immune response delay 3 , given by (2) together with assumptions ( 1 )-( 4 ), which describes the dynamics among uninfected cells, infected cells, virus, CTL responses, and antibody responses. The model allows for general target-cell dynamics ( ), including a nonlinear incidence ( , V), discrete delays, and state-dependent removal functions ( = 1, 2, 3, 4). This general model includes many existing models in the literature as special cases. Dynamical analysis of model (2) shows that 1 , 2 , and 3 play different roles in the stability of the equilibria. Particularly, we see that 3 may impact the stability of equilibria 3 and 4 .
By the analysis, we have shown that when 0 ≤ 1, 0 is globally asymptotically stable, which means that the virus is cleared up. When 0 > 1, 1 ≤ 1, and 2 ≤ 1, 1 is globally asymptotically stable, which means that the infection is successful, but the establishments of both antibody and CTLs immune responses are unsuccessful. When 1 > 1 and 4 , we have obtained that for special case, 3 = 0, 1 ≥ 0, and 2 ≥ 0, when 3 > 1 and 4 > 1, 4 is globally asymptotically stable, that is, susceptible cells, infected cells, free virus, CTLs, and antibodies coexist in vivo.
Based on Theorems 13 and 15, we obtain that the intracellular delay 1 and virus replication delay 2 for model (2) do not cause Hopf bifurcation. Moreover, 0 plays a crucial role in virus infection dynamics. Actually, in model (2), 0 is a decreasing function on time delay 1 . When all other parameters are fixed and delay 1 is sufficiently large, 0 becomes less than one, only infection-free equilibrium 0 exists, and the virus is cleared in the host. By biological meanings, intracellular delay plays a positive role in virus infection process in order to eliminate virus. Sufficiently large intracellular delay makes the virus development slower and the virus has been controlled and disappeared. This gives us some suggestions on new drugs to prolong the time of infected cells producing virus. However, by the recent research of Li and Shu [37], in the case of the coexistence of mitosis rate of the target cells and an intracellular delay in the viral infection model, the intracellular delay produces Hopf bifurcation only when the mitosis rate is sufficiently large.
When 3 > 0, by numerical simulations, it is shown that the Hopf bifurcation and stability switches occur at equilibria 3 and 4 as 3 increases. Figures 1-4 indicate that 3 remains stable as 3 > 0 is small, and along with the increase of 3 , equilibrium 3 becomes unstable and periodic oscillations appear. It shows that stability switches occur as delay 3 increases. Similarly, from Figures 5-8, we see that along with the increases of 3 > 0 the dynamical behaviors of model (53) at equilibrium 4 appear as very large diversification. Particularly, when 3 is small enough, 4 is asymptotically stable and when 3 is increasing, the stability switches occur at equilibrium 4 , and when 4 is unstable, a Hopf bifurcation occurs. Finally, when 3 is enough large, equilibrium 4 always is unstable. Summarizing these discussions, we have the conclusion that 3 affects markedly the stability of equilibria 3 and 4 . From the numerical simulations, we observe that immune response delay 3 can cause Hopf bifurcation. Upon primary infection, the sustained oscillations from the Hopf bifurcation imply that the pathogen may not always be cleared entirely with the CTL responses which usually occur in a few days after serum conversion. As the increase of immune delay 3 , we know that the drug prevents virus from continuing through their cell cycle, thus trapping them at some point during interphase, where the cells die from natural causes. Then susceptible cells, infected cells, free virus, CTLs, and antibodies reach a stable level in the host. When immune delay 3 continuously increases, the activation of the immune cell is to fight against the malignant virus cells. Thus susceptible cells, infected cells, free virus, CTLs, and antibodies exhibit sustained periodic oscillations in the chronic phase of infection. This explains the fact that the immune response delay plays negative roles in controlling disease progression.
Observing all obtained results in this paper, we can directly put forward the following open questions which need to be further studied in the future.
For one, in addition to 1 , 2 , and 3 , antibody response delay 4 is also considered, whether the results obtained in this paper can be extended to a virus infection model with nonlinear incidence rate and four time delays. For another, we obtain the Hopf bifurcation and stability switches at equilibria 3 and 4 for model (2) only by using the numerical simulation method for special examples (52) and (53). Up to now, the theoretical analysis and results in this aspect are few and rough. Therefore, a systemic and complete theoretical analysis and results will be a very estimable and significative subject.