^{1}

Two virus infection models with antibody immune response and chronically infected cells are proposed and analyzed. Bilinear incidence rate is considered in the first model, while the incidence rate is given by a saturated functional response in the second one. One main feature of these models is that it includes both short-lived infected cells and chronically infected cells. The chronically infected cells produce much smaller amounts of virus than the short-lived infected cells and die at a much slower rate. Our mathematical analysis establishes that the global dynamics of the two models are determined by two threshold parameters

In recent years, many mathematical models have been proposed to study the dynamics of viral infections such as the human immunodeficiency virus (HIV), the hepatitis C virus (HCV), and the hepatitis B virus (HBV) (see, e.g., [

It is observed that the basic and global properties of model (

In this paper, we propose two virus infection models with antibody immune response and chronically infected cells. In the first model, bilinear incidence rate which is based on the law of mass-action is considered. The second model generalizes the first one where the incidence rate is given by a saturation functional response. The global stability of all equilibria of the models is established using the method of Lyapunov function. We prove that the global dynamics of the models are determined by two threshold parameters

In this section we propose a viral dynamics model with antibody immune response, taking into consideration the chronically infected cells. Based on the mass-action principle, we assume that the incidence rate of infection is bilinear; that is, the infection rate per virus and per uninfected cell is constant:

We note that model (

There exist positive numbers

To show the boundedness of the solutions we let

System (

From (

if

if

if

In this section, we study the global stability of all the equilibria of system (

For system (

Define a Lyapunov function

For system (

Define the following Lyapunov function:

For system (

We consider a Lyapunov function

In model (

All the variables and parameters have the same meanings as given in model (

Similar to the previous section, we can define two threshold parameters

if

if

if

In this section, we study the global stability of all the equilibria of system (

For system (

Define a Lyapunov function

For system (

Construct a Lyapunov function as follows:

For system (

We consider a Lyapunov function as follows:

We now use simple numerical simulations to illustrate our theoretical results for the two models. In both models we will fix the following data: ^{−3} day^{−1},^{−1}, ^{3} day^{−1}, ^{−1}, ^{−1}, ^{3} day^{−1}, and ^{−1}. The other parameters will be chosen below. All computations were carried out by MATLAB.

In this section, we perform simulation results for model (

^{3} day^{−1}. Using these data we compute ^{−3}, while the concentrations of short-lived infected cells, chronically infected cells, free viruses, and antibody immune cells are decaying and tend to zero.

^{3} day^{−1}. In this case,

^{3} day^{−1}. Then we compute

The evolution of uninfected cells for model (

The evolution of short-lived infected cells for model (

The evolution of chronically infected cells for model (

The evolution of free viruses for model (

The evolution of antibody immune cells for model (

We note that the values of the parameters

Figures

In this section, we perform simulation results to check Theorems ^{3}. We have the following cases.

^{3} day^{−1}. Using these data, we compute

^{3} day^{−1}. This will give

^{3} day^{−1}. This yields

The evolution of uninfected cells for model (

The evolution of short-lived infected cells for model (

The evolution of chronically infected cells for model (

The evolution of free viruses for model (

The evolution of antibody immune cells for model (

From the definition of the parameter

Figures

In this paper, we have proposed two virus infection models with antibody immune response taking into account the chronically infected cells. In the first model we have assumed that the incidence rate of infection is bilinear while in the second model the incidence rate is given by saturation functional response. We have shown that the dynamics of the models are fully determined by two threshold parameters

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

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 130-078-D1434. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors are also grateful to Professor Malay Banerjee and to the anonymous reviewers for constructive suggestions and valuable comments, which improve the quality of the paper.

^{+}T cells

^{+}T cells and macrophages