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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 781407, 9 pages
http://dx.doi.org/10.1155/2013/781407
Research Article

Global Dynamics of Virus Infection Model with Antibody Immune Response and Distributed Delays

1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt
3Department of Mathematics, Faculty of Arts and Science, Qassim University, Buraidah 71511, Saudi Arabia

Received 8 August 2013; Accepted 7 October 2013

Academic Editor: Victor S. Kozyakin

Copyright © 2013 A. M. Elaiw 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

We present qualitative behavior of virus infection model with antibody immune response. The incidence rate of infection is given by saturation functional response. Two types of distributed delays are incorporated into the model to account for the time delay between the time when uninfected cells are contacted by the virus particle and the time when emission of infectious (matures) virus particles. Using the method of Lyapunov functional, we have established that the global stability of the steady states of the model is determined by two threshold numbers, the basic reproduction number and antibody immune response reproduction number . We have proven that if , then the uninfected steady state is globally asymptotically stable (GAS), if , then the infected steady state without antibody immune response is GAS, and if , then the infected steady state with antibody immune response is GAS.

1. Introduction

In the past ten years there has been a growing interest in modeling viral infections for the study and characterization of host infection dynamics. The mathematical models, based on biological interactions, present a framework which can be used to obtain new insights into the viral dynamics and to interpret experimental data. Many authors have formulated mathematical models to describe the population dynamics of several viruses such as, human immunodeficiency virus (HIV) [114], hepatitis B virus (HBV) [15, 16], and hepatitis C virus (HCV) [17, 18]. During viral infections, the host immune system reacts with antigen-specific immune response. In particular, B cells play a critical role in antiviral defense by attacking free virus particles by making antibodies to clear antigens circulating in blood and lymph. The antibody immune response is more effective than the cell-mediated immune in some diseases like in malaria infection [19].

Mathematical models for virus dynamics with the antibody immune response have been developed in [2026]. The basic virus dynamics model with antibody immune response was introduced by Murase et al. [21] as where , , , and represent the populations of uninfected cells, infected cells, virus, and B cells at time , respectively; and are the recruited rate and death rate constants of uninfected cells, respectively; is the infection rate constant; is the number of free virus produced during the average infected cell life span; is the death rate constant of infected cells; is the clearance rate constant of the virus; and are the recruited rate and death rate constants of B cells; and is the B cells neutralizion rate. Model (1) is based on the assumption that the infection could occur and that the viruses are produced from infected cells instantaneously, once the uninfected cells are contacted by the virus particles. Other accurate models incorporate the delay between the time viral entry into the uninfected cell and the time the production of new virus particles, modeled with discrete time delay or distributed time delay using functional differential equations (see e.g., [714, 27]). In [714, 27], the viral infection models are presented without taking into consideration the antibody immune response. In [25, 28], global stability of viral infection models with antibody immune response and discrete delays has been studied.

In model (1) the infection rate is assumed to be bilinear in and however, this bilinear incidence rate associated with the mass action principle is insufficient to describe the infection process in detail [29, 30]. For example, a less than linear response in could occur due to saturation at high virus concentration, where the infectious fraction is high so that exposure is very likely. Thus, it is reasonable to assume that the infection rate is given by saturation functional response [31]. In [26], a virus infection model with antibody immune response and with saturation incidence rate has been considered. However, the time delay was not considered.

In this paper, we assume that the infection rate is given by saturation functional response. We incorporate two types of distributed delays into the model to account for the time delay between the time when uninfected cells are contacted by the virus particle and the time of emission of infectious (matures) virus particles. The global stability of this model is established using Lyapunov functionals, which are similar in nature to those used in [32]. We prove that the global dynamics of this model is determined by the basic reproduction number and antibody immune response reproduction number . If , then the uninfected steady state is globally asymptotically stable (GAS), if , then the infected steady state without antibody immune response is GAS, and if , then the infected steady state with antibody immune response is GAS.

2. The Model

In this section we propose a delay mathematical model of viral infection with saturation functional response which describes the interaction of the virus with uninfected and infected cells, taking into account the effect of antibody immune response. Consider where is a positive constant, and all the variables and parameters of the model have the same meanings as given in model (1). To account for the time lag between viral contacting a target cell and the production of new virus particles, two distributed intracellular delays are introduced. It is assumed that the uninfected cells that are contacted by the virus particles at time become infected cells at time , where is a random variable with a probability distribution over the interval and is limit superior of this delay. The factor accounts for the probability of surviving the time period of delay, where is the death rate of infected cells but not yet virus producer cells. On the other hand, it is assumed that a cell infected at time starts to yield new infectious virus at time where is distributed according to a probability distribution over the interval and is limit superior of this delay. The factor accounts for the probability of surviving during the time period of delay, where is constant. All the parameters are supposed to be positive.

The probability distribution functions and are assumed to satisfy and , and where is a positive number. Then Let and The initial conditions for system (2)–(5) take the form where ,, where is the Banach space of continuous functions mapping the interval into . By the fundamental theory of functional differential equations [33], system (2)–(5) has a unique solution satisfying the initial conditions (8).

2.1. Nonnegativity and Boundedness of Solutions

In the following, we establish the nonnegativity and boundedness of solutions of (2)–(5) with initial conditions (8).

Proposition 1. Let be any solution of (2)–(5) satisfying the initial conditions (8); then , and are all nonnegative for and ultimately bounded.

Proof. First, we prove that for all . Assume that loses its non-negativity on some local existence interval for some constant , and let be such that . From (2) we have . Hence, for some , where is sufficiently small. This leads to contradiction, and hence for all . Now from (3), (4), and (5) we have confirming that , and for all . By a recursive argument, we obtain , and for all .
Next we show the boundedness of the solutions. From (2) we have . This implies . Let ; then where . Hence, , where . Since , then . On the other hand, let ; then where . Hence, , where . Since , and , then and .
Therefore, , and are ultimately bounded.

2.2. Steady States

We define the basic reproduction number for system (2)–(5) as and the antibody immune response reproduction number Clearly . It can be seen that system (2)–(5) has an uninfected steady state , where . In addition to , the system has an infected steady state without immune response and infected steady state with immune response , where From the above we have the following:(i)if , then there exists a positive steady state ;(ii)if , then there exists a positive steady state .

3. Global Stability

In this section, we prove the global stability of the steady states of system (2)–(5) employing the method of Lyapunov functional which is used in [32] for SIR epidemic model with distributed delay. Next we will use the following notation: for any . We also define a function as It is clear that for any and has the global minimum .

Theorem 2. If , then is GAS.

Proof. Define a Lyapunov functional as follows: The time derivative of along the trajectories of (2)–(5) satisfies If , then for all . By Theorem 5.3.1 in [33], the solutions of system (2)–(5) limit to , the largest invariant subset of . Clearly, it follows from (17) that if and only if , , and . Noting that is invariant for each element of , we have , and , and then . From (4) we derive that This yields . Hence, if and only if , , , and . From LaSalle’s Invariance Principle, is GAS.

Theorem 3. If , then is GAS.

Proof. We construct the following Lyapunov functional: The time derivative of along the trajectories of (2)–(5) is given by Using the steady state conditions for : we have Using the following equalities: then collecting terms of (22), we obtain Thus, if , then , and , and hence, if , then for all . By Theorem 5.3.1 in [33], the solutions of system (2)–(5) limit to , the largest invariant subset of . It can be seen that if and only if ,, and ; that is, From (25), if , then , and hence equals zero at . LaSalle’s Invariance Principle implies global stability of .

Theorem 4. If , then is GAS.

Proof. We construct the following Lyapunov functional: The time derivative of along the trajectories of (2)–(5) is given by Using the steady state conditions for : we obtain Using the following equalities: and collecting terms of (29), we obtain Thus, if , then , and and . By Theorem 5.3.1 in [33], the solutions of system (2)–(5) limit to , the largest invariant subset of . It can be seen that if and only if ,, and ; that is, Then from (32) , and hence at the steady state . LaSalle’s Invariance Principle implies global stability of .

4. Conclusion

In this paper, we have proposed a virus infection model which describes the interaction of the virus with the uninfected and infected cells taking into account the antibody immune response. The infection rate is given by saturation functional response. Two types of distributed time delays have been incorporated into the model to describe the time needed for infection of uninfected cell and virus replication. Using the method of Lyapunov functional, we have established that the global dynamics of the model is determined by two threshold parameters and . The basic reproduction number viral infection determines whether a chronic infection can be established, and the antibody immune response reproduction number determines whether a persistent antibody immune response can be established. We have proven that if , then the uninfected steady state is GAS, and the viruses are cleared. If , then the infected steady state without antibody immune response is GAS, and the infection becomes chronic but without persistent antibody immune response. If , then the infected steady state with antibody immune response is GAS, and the infection is chronic with persistent antibody immune response. We note that the effect of the time delay appears in the parameters and . Since and , then the intracellular delay can reduce the parameters and . As a consequence, ignoring the delay will produce overestimation of .

Conflict of Interests

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

Acknowledgments

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors are also grateful to Professor Victor Kozyakin and to the anonymous reviewers for constructive suggestions and valuable comments, which improved the quality of the article.

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