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

Miao, Hui; Teng, Zhidong; Li, Zhiming

doi:https://doi.org/10.1155/2016/3903726

Computational and Mathematical Methods in Medicine

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Abstract Introduction Preliminaries Discussion Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2016 | Article ID 3903726 | https://doi.org/10.1155/2016/3903726

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

Hui Miao,¹Zhidong Teng,¹and Zhiming Li¹

Academic Editor: Fabien Crauste

Received12 Aug 2016

Accepted22 Nov 2016

Published15 Dec 2016

Abstract

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.

1. 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) [1–34]. 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, 18–23, 25–32]. 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 [2], Balasubramaniam et al. developed the viral infection model by incorporating immune delays and Beddington-DeAngelis incidence rate where , , , , 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 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 , where are positive real numbers [4–13, 15, 16, 18, 20–23, 26–32, 34].

We assume that the incidence of new infections of target cells occurs at a rate . This form of incident rate is general to encompass several forms such as bilinear incidence [4, 13], saturated incidence [16], Holling type II functional response [15], and Crowley-Martin incidence [12, 35], where , , and are positive constants.

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 , , , and , 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 and be the neutralization rate of viruses and activation rate of B cells, respectively. The typical forms can be seen as and [1, 2, 20, 21, 31, 32]. Accordingly, let and 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 . The probability of surviving the time period from to is . Let be the maturation time of the newly produced viruses. The constant denotes the surviving rate of virus during the delay period. Antigenic stimulation generating CTL cell may need a period of time .

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 , immune-free equilibrium , infection equilibrium only with antibody response, and infection equilibrium only with CTL immune response and infection equilibrium with both antibody and CTL immune responses when the delay , respectively, are established. By using the linearization method, the instability of equilibria , , , and , 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 and when .

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 , , and are proved. When , the threshold conditions on the global stability and instability for equilibria and are stated and proved. In Section 4, the numerical simulations are given to further discuss the stability of equilibria and when . It is shown that the Hopf bifurcation and stability switches at these equilibria occur as increases. In the last section, we offer a brief conclusion.

2. Preliminaries

Let and . denotes the space of continuous functions mapping interval into with norm for any .

The initial conditions for any solutions of model (2) are given as follows: where By the fundamental theory of functional differential equation [36], model (2) admits a unique solution satisfying initial conditions (3).

In this paper, we firstly introduce the following assumptions:() is continuously differentiable. There exists such that and .() is continuously differentiable; for , ; if and only if or ; and for all and ; for all () is strictly increasing on ; ; and there exists such that for any ; and .() is nonincreasing with respect to for .

From we easily obtain that for all and for all . Assumption 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 implies that there are no new infected cells (i.e., ) without healthy cells () or virus (). 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 is, the higher the number of healthy cells which are infected in the unit time will be. Assumption assumes that the death rates of the infected target cells , virus , antibodies , and CTLs depend on their concentrations. If these numbers increase, the corresponding rates , , , and will increase, and the ratio is no less than a positive constant for . Finally, assumption 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.

Using an argument similar to [14] we have the following result.

Theorem 1. Assume that hold. Let be the solution of model (2) with initial conditions (3); then is positive and ultimately bounded.

Next, we discuss the existence and uniqueness of equilibria of model (2). We know that any equilibrium of model (2) satisfies

It is clear from (4) that model (2) has a unique infection-free equilibrium . When , from (4) we have , , , and . Solving these equations, we have , , , and . When , from (4) we have , , , , and Solving these equations, we have , , , and . Therefore, besides equilibrium , model (2) only has the following four possible equilibria: , , , and .

The existence of immune-free equilibrium is equivalent to the existence of positive solution of the following equations: By , the inverse function exists. Solving , we have with and , where is the unique positive root of equation Define Then and

Define the basic reproduction number for viral infection Note that Thus, if , then This implies that there exists such that The value of is given by . ensures that has a unique positive solution Therefore, exists if .

Next we show that is a unique immune-free equilibrium. Otherwise, there exists another Without of loss of generality, we assume that , and then Meanwhile, and . By and , we have and Since , we obtain and For another, we have This is a contradiction. Thus is a unique equilibrium.

We consider the existence of infection equilibrium with only antibody response. It is clear that Define By and , we obtain Since and , there exists a unique such that Then, we have

Define the constant which is called the antibody response reproductive number of model (2). Solving from (4), we obtain that Therefore, exists and is unique if

We consider the existence of infection equilibrium with only CTL immune response. From the third and fourth equations of (4), we obtain unique and Define By and , we obtain Since and , there exists a unique such that

Define the constant which is called the CTL immune response reproductive number of model (2). Solving the second equation for yields Therefore, exists and is unique if

Lastly, we consider the existence of infection equilibrium with both antibody and CTL immune responses. From the fourth and fifth equation of (4), we obtain unique and Define By and , we obtain Since and , there exists a unique such that

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, exists and is unique if and

Remark 2. From and , we obtain and In fact,

3. Stability Analysis

3.1. Stability of Equilibrium

Theorem 3. (a) If , then infection-free equilibrium is globally asymptotically stable.
(b) If , then is unstable.

Proof. Consider conclusion (a). Define a Lyapunov functional as follows: Calculating the time derivative of along solutions of model (2), we obtain Note that , and It follows that Note that if and only if , , , , and . So, the maximal compact invariant set in is singleton . By LaSalle’s invariance principle [36], is globally asymptotically stable.
Next, we consider conclusion (b). By computing, the characteristic equation of the linearization system of model (2) at is where When , we have and . Hence, there is such that . Therefore, when , is unstable. This completes the proof.

Remark 4. Theorem 3 shows that if only equilibrium exists, then it is globally asymptotically stable, and delays , , and do not impact the stability of

3.2. Stability of Equilibrium

Firstly, we introduce two lemmas which will be used in the proof of Theorem 7.

Lemma 5. Suppose that hold and Let and satisfy and Then for equilibrium ,

Proof. Since , we have By and , we get Using , we have By and , it follows that This completes the proof.

Lemma 6. Suppose that hold and Let , , and satisfy , , and Then, for equilibrium ,

Proof. Since and , we have Since , one has By and , we get , and By and , we further have This completes the proof.

Theorem 7. Let . (a) If and , then immune-free equilibrium is globally asymptotically stable. (b) If or , then is unstable.

Proof. Consider conclusion (a). Denote with . Define a Lyapunov functional as follows: Calculating the derivative of along solutions of model (2), we obtain where Therefore, Note that , and Lemmas 5 and 6 imply that and if and It then follows from the monotonicity of and that . We have , and if and only if , , , , and From LaSalle’s invariance principle [36], we finally have that equilibrium of model (2) is globally asymptotically stable when , , and .
Next, consider conclusion (b). By computing, the characteristic equation of the linearization system of model (2) at is where and When , we have . Hence, there is a positive root . When , we have and . Hence, there is also a positive root such that . Therefore, when or , is unstable. This completes the proof.

Remark 8. Theorem 7 shows that if only equilibria and exist, then is globally asymptotically stable, and delays , , and do not impact the stability of

3.3. Stability of Equilibrium

We firstly have the following Lemma.

Lemma 9. Suppose and . Let be the solution of equation (4) with and Then for equilibrium .

Proof. Since satisfies (4), we have , , and Compared with , we obtain and When , we get Since it follows that if and This completes the proof.

Theorem 10. Let . (a) If , then antibody response equilibrium is globally asymptotically stable.
(b) If , then is unstable.

Proof. Consider conclusion (a). Define a Lyapunov functional as follows: Calculating the derivative of along solutions of model (2), we obtain where Therefore, Note that , and Since , we have , and if and only if , , , and From LaSalle’s invariance principle [36], we finally have that is globally asymptotically stable when and .
Next, consider conclusion (b). By computing, the characteristic equation of linearization system of model (2) at is where and where When , we have and . Hence, there is also a positive root such that . Therefore, when , is unstable. This completes the proof.

Remark 11. Theorem 10 shows that if only equilibria , , and exist, then when and , is globally asymptotically stable, and delays , , and do not impact the stability of .

3.4. Stability of Equilibrium

On the stability analysis of equilibrium , we only discuss the following case: , , and . Other cases, , , and , are numerically verified for bifurcation phenomena and stability switches of but the analytic analysis is left as an open problem. Before the proof of theorem, we have the following Lemma.

Lemma 12. Suppose and . Let be the solution of (4) with and Then for equilibrium , .

Proof. Since satisfies (4), we have , , and Compared with , we get and When , we obtain Since it follows that if and . This completes the proof.

Theorem 13. Let . (a) If and , then infection equilibrium with only CTL response is globally asymptotically stable.
(b) If , then is unstable.

Proof. We first consider conclusion (a). Define a Lyapunov functional as follows: Calculating the derivative of along solutions of model (2), we obtain that Note that , and Since , we have , and if and only if , , , and From LaSalle’s invariance principle [36], we finally have that is globally asymptotically stable when , , , and .
Next, we consider conclusion (b). By computing, the characteristic equation of the linearization system of model (2) at is where where When , we have . Hence, there is a positive root . Therefore, when , is unstable for any , , and . This completes the proof.

Remark 14. Theorem 13 shows that if only equilibria , , , and exist, then when , , and is globally asymptotically stable, and delays and do not impact the stability of

3.5. Stability of Equilibrium

On the stability analysis of equilibrium , we here only discuss the following case: , , and . However, for the cases , , and , the theoretical analysis is very complicated. We will give numerical analysis for this case in the next section.

Theorem 15. If , , and , then infection equilibrium with both antibody and CTL immune responses is globally asymptotically stable.

Proof. Define a Lyapunov functional as follows: Using the above similar method, we obtain Note that , and Obviously, we have , and if and only if , , and From LaSalle’s invariance principle [36], we finally have that is globally asymptotically stable when , , and . This completes the proof.

Remark 16. Theorem 15 shows that if equilibria , , , , and exist, then when , , and , is globally asymptotically stable, and delays and do not impact the stability of

4. Numerical Simulations

In the above section, we obtain the global asymptotic stability of equilibria and when the delay . In this section, by using the numerical simulation, it is shown that the Hopf bifurcation and stability switches occur at equilibria and in the case

Example 17. Corresponding to model (2), we consider the following model: where are constants. We have , , and It can easily verify that hold. Taking , , , , , , , , , , , , , , , , , , and , choose as free parameter. By computing, , , and . From Figures 1–4, we see that as increases the complex dynamical behaviors of equilibrium occur.

(a)

(b)

(c)

(d)

(e)

(a)

(b)

(c)

(d)

(e)

(a)

(b)

(c)

(d)

(e)

(a)

(b)

(c)

(d)

(e)

In Figures 1–8, we denote by (a) the time-series of , by (b) the time-series of , by (c) the time-series of , by (d) the time-series of , and by (e) the time-series of .

(a)

(b)

(c)

(d)

(e)

(a)

(b)

(c)

(d)

(e)

(a)

(b)

(c)

(d)

(e)

(a)

(b)

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Example 18. Corresponding to model (2), we consider the following model: where are constants. We have , , and It is easily verified that hold.

Taking , , , , , , , , , , , , , , , , and , choose as free parameter. By computing, , , and . From Figures 5–8, we see that as increases the complex dynamical behaviors of equilibrium occur.

5. Discussion

In this paper we have considered an in-host model with intracellular delay , virus replication delay , and immune response delay , given by (2) together with assumptions , 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 , discrete delays, and state-dependent removal functions . This general model includes many existing models in the literature as special cases. Dynamical analysis of model (2) shows that , , and play different roles in the stability of the equilibria. Particularly, we see that may impact the stability of equilibria and .

By the analysis, we have shown that when , is globally asymptotically stable, which means that the virus is cleared up. When , , and is globally asymptotically stable, which means that the infection is successful, but the establishments of both antibody and CTLs immune responses are unsuccessful. When and , is globally asymptotically stable, which implies that the antibody response is established, but the infected cells are too weak to stimulate CTL immune response. With respect to the analysis of , we consider special cases , , and ; when and , is globally asymptotically stable, which means that the CTL immune response is determined, but the viral loads are so small that it cannot activate the antibody response. About the stability of , we have obtained that for special case, , , and , when and , 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 and virus replication delay for model (2) do not cause Hopf bifurcation. Moreover, plays a crucial role in virus infection dynamics. Actually, in model (2), is a decreasing function on time delay . When all other parameters are fixed and delay is sufficiently large, becomes less than one, only infection-free equilibrium 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 , by numerical simulations, it is shown that the Hopf bifurcation and stability switches occur at equilibria and as increases. Figures 1–4 indicate that remains stable as is small, and along with the increase of , equilibrium becomes unstable and periodic oscillations appear. It shows that stability switches occur as delay increases. Similarly, from Figures 5–8, we see that along with the increases of the dynamical behaviors of model (53) at equilibrium appear as very large diversification. Particularly, when is small enough, is asymptotically stable and when is increasing, the stability switches occur at equilibrium , and when is unstable, a Hopf bifurcation occurs. Finally, when is enough large, equilibrium always is unstable. Summarizing these discussions, we have the conclusion that affects markedly the stability of equilibria and From the numerical simulations, we observe that immune response delay 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 , 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 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 , , and , antibody response delay 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 and 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.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant nos. 11271312, 11261056, and 11661076), the Doctorial Subjects Foundation of The Ministry of Education of China (Grant no. 2013651110001), and the Natural Science Foundation of Xinjiang University (starting research fund for the Xinjiang University doctoral graduates, Grant no. BS150202).

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Copyright © 2016 Hui Miao 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.

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