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

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 R 0 and antibody immune response reproduction number


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) [1][2][3][4][5][6][7][8][9][10][11][12][13][14], 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 cellmediated immune in some diseases like in malaria infection [19].
Mathematical models for virus dynamics with the antibody immune response have been developed in [20][21][22][23][24][25][26].The basic virus dynamics model with antibody immune response was introduced by Murase et al. [21] as ẋ () =  −  () −  () V () , ẏ () =  () V () −  () , V () =  () − V () − V ()  () , ż () = V ()  () −  () , (1) where (), (), V(), 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., [7][8][9][10][11][12][13][14]27]).In [7][8][9][10][11][12][13][14]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 V 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 V 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  0 and antibody immune response reproduction number  1 .If  0 ≤ 1, then the uninfected steady state is globally asymptotically stable (GAS), if  1 ≤ 1 <  0 , then the infected steady state without antibody immune response is GAS, and if  1 > 1, then the infected steady state with antibody immune response is GAS.

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 [0, ℎ] and ℎ 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 [0, ] 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.

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

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  ∈ {, , V, }.We also define a function  : (0, ∞) → [0, ∞) as It is clear that () ≥ 0 for any  > 0 and  has the global minimum (1) = 0.

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  0 and  1 .The basic reproduction number viral infection  0 determines whether a chronic infection can be established, and the antibody immune response reproduction number  1 determines whether a persistent antibody immune response can be established.We have proven that if  0 ≤ 1, then the uninfected steady state is GAS, and the viruses are cleared.If  1 ≤ 1 <  0 , then the infected steady state without antibody immune response is GAS, and the infection becomes chronic but without persistent antibody immune response.If  1 > 1, 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 0 <  ≤ 1 and 0 <  ≤ 1, then the intracellular delay can reduce the parameters  0 and  1 .As a consequence, ignoring the delay will produce overestimation of  0 .