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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

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) [

Mathematical models for virus dynamics with the antibody immune response have been developed in [

In model (

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 [

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

The probability distribution functions

In the following, we establish the nonnegativity and boundedness of solutions of (

Let

First, we prove that

Next we show the boundedness of the solutions. From (

Therefore,

We define the basic reproduction number for system (

if

if

In this section, we prove the global stability of the steady states of system (

If

Define a Lyapunov functional

If

We construct the following Lyapunov functional:

If

We construct the following Lyapunov functional:

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

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

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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^{+}T Cells and Macrophages