DYNAMICS OF A CLASS OF VIRAL INFECTION MODELS WITH DIFFUSION

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The aim of this work is to study the dynamics of a class of viral infection models with diffusion and loss of viral particles due to the absorption into uninfected cells. We prove the global stability of equilibria by constructing suitable Lyapunov functionals for two cases: continuous and discrete. Also, some examples are given to illustrate the theoretical results.


INTRODUCTION
Reaction-diffusion equations modeling several phenomena in different fields such as physics, biology, economics, etc. These equations describe the variations in concentration or density distributed in space under the influence of two processes: the local interactions between species and the diffusion that causes the propagation of species in space. In population dynamics, the terms of diffusion correspond to a random movement of individuals and the terms of reaction describe their reproduction [1].
Recently, reaction-diffusion equations are used to describe the dynamics of viral infections and to obtain information on the mechanisms of these viral infections in vivo. In [2], the authors introduced a mathematical model formulated by partial differential equations (PDEs) to describe the hepatitis B virus (HBV) infection that represents a major global health problem.
They assumed that infection rate is bilinear and they ignored the absorption of the virus by the uninfected cells. The importance of our work is to consider both nonlinear incidence rate and absorption of the virus by the uninfected cells. Therefore, we propose a generalized viral infection model governed by the following nonlinear system of PDEs: where T (x,t), I(x,t) and V (x,t) are the densities of susceptible cells, infected cells and free virus at position x and time t, respectively. Susceptible cells are produced at rate λ , die at rate dT and become infected at rate f (T, I,V )V . Infected cells die at rate aI. Free viruses are produced from infected cells at rate kI and are removed at rate µV . d v is the diffusion coefficient, ∆ is the Laplacien operator, and i ∈ {0, 1} denotes the absorption effect.
As in [3,4], we suppose that the function f (T, I,V ) is continuously differentiable in I R 3 + and satisfies the following hypotheses: It is very important to note that our model represented by system (1), extends and improves many cases exiting in the literature. For instance, if f (T, I,V ) = β T and i = 0, we get the basic PDE model proposed in [2]. Further, the more recent model presented by Yang and Zhou in [5] is a special case of system (1).
In this work, we are interested in system (1) according to two purposes. The first is to investigate the dynamics of system (1) with initial values and Neumann boundary conditions where Ω is a bounded domain in I R n with smooth boundary ∂ Ω, ∂ ∂ n is an outward normal vector of ∂ Ω. The second purpose is to propose some applications and an numerical method that preserves the qualitative properties of the continuous model (1).
The rest of this paper is organized as follows. In section 2, we analysis the continuous version by showing well-posedness, equilibria and global stability. The discrete version is treated in section 3. In section 4, we give some applications of our analytical results. Finally, we conclude our work in section 5.

ANALYSIS OF CONTINUOUS VERSION
We first show the well-posedness of the model by proving the global existence, uniqueness, non-negativity and boundedness of solution of model (1) under (2). After, we determine the basic reproduction number, study steady states of the model (1) and discuss the global stability of the infection-free equilibrium and the chronic infection equilibrium. Proof. For any ϕ = (ϕ 1 , ϕ 2 , ϕ 3 ) T ∈ C and x ∈ Ω, we define F = (F 1 , F 2 , F 3 ) : C −→ I R 3 by Then system (1) can be rewritten as the following abstract functional differential equation So, we can get where δ = min{a, d}. Hence, This implies that T and I are bounded. From the boundedness of I and system (1), we deduce that V satisfies the following system be a solution to the ordinary differential .
By the comparison principale [7], we get V (x,t) ≤ V (t). Hence, From the above, we have proved that T (x,t), I(x,t) and V (x,t) are bounded on Ω × [0,t max ).
Therefore, it follows from the standard theory for semi-linear parabolic systems [8] that t max = +∞. This completes the proof.
Next, we study the existence of steady states of model (1). Obviously, model (1) has an Then the basic reproduction number of (1) is given by To find the other equilibrium of system (1), we resolve Adding the first two equations of (5), we get Using the third equation of (5), we have Replacing T,V into the second equation of (5), we obtain We define a function h on 0, λ a as follows It is easy to see that h( λ a ) = −µa < 0 and Clearly, we have a positive equilibrium E * (T * , I * ,V * ) when R 0 > 1.
For any I ∈ 0, λ a , we have Using hypotheses (H 2 ) and (H 3 ), we prove the uniqueness of the chronic infection equilibrium E * (T * , I * ,V * ). By this computation, we get the following result. Proof. Construct a Lyapunov functional for system (1) at E 0 as follows The time derivative of L 0 along the solution of system (1) satisfies Using the hypothesis (H 3 ), we get Since f (T, I,V ) is strictly monotonically increasing with respect to T , we have From LaSalle's invariance principle [9], we deduce that E 0 is globally asymptotically stable if R 0 ≤ 1. Similarly to [3], we can easily prove the instability of E 0 when R 0 > 1.
For the global stability of the chronic infection equilibrium E * , we assume that R 0 > 1 and f satisfies the following hypothesis Theorem 2.4. If R 0 > 1 and the hypthesis (H 4 ) holds, then the chronic infection equilibrium E * is globally asymptotically stable when i = 0.
Proof. Let us define the following Lyapunov functional: with g(z) = z − 1 − ln z. The function g(z) has its minimum 0 at z = 1. So, g(z) ≥ 0 for all z > 0.
The time derivative of L 1 along the solution of system (1) satisfies Since we have Since f (T, I,V ) is strictly monotonically increasing with respect to T , we have Based on the hypothesis (H 4 ), we have If follows from LaSalle's invariance principle that E * is globally asymptotically stable when R 0 > 1.

ANALYSIS OF DISCRETE VERSION
In this section, we discretize system (1) by using 'mixed' Euler method that is mixture of both forward and backward Euler method [10]. The choice of this numerical method is motivated by the work of Hattaf et al. [11].
Let Ω = [p, q] with p, q ∈ I R. Denote t m = m t and x n = p + n x, where t and x = q − p N are time and space step sizes, respectively. Let So, our discrete model is as follows where n ∈ {0, 1, ..., N} and m ∈ I N. The discrete initial and boundary conditions are It is clear that discrete system (6) and continuous (1) has the same equilibrium points. First, we establish that the solution of system (6) is nonnegative and bounded. Proof. System (6) can be writen as Thus, for any j ∈ {0, 1, ..., N}, we have Note that B is a M-matrix. Thus, from the third equation of (7), we have Therefore, by the method of induction, the solution remains nonnegative for all m ∈ I N.
Next, we prove the boundedness of the solution. Define a sequence G m = T m n + I m n . Then By mathematical induction, we have This implies that {G m n } is bounded. Therefore, T m n and I m n are also bounded.
By the third equation of (7), we get This completes the proof.
Next, we will establish the global stability of the infection-free equilibrium and chronic infection equilibrium for system (6). First, we need the following lemma.  Proof. Consider a Lyapunov functional Then, we get Consider Then we have Using Lemma 3.2, we obtain We have k f (T 0 , 0, 0) Hence, Thus, E 0 is globally asymptotically stable if R 0 ≤ 1. Proof. Consider the following Lyapunov functional: According to the Lemma 3.2, we consider Hence,

APPLICATIONS
Here, we give some examples for which we apply our theoretical results.
Example 1: Consider the following system where n ∈ {0, 1, ..., N} and m ∈ I N. The function F(x) is a twice differentiable and satisfies The discrete initial and boundary conditions are This model was studied by Yang and Zhou [5], which is a special case of our model (6), it suffices to take It is easy to show that the function f verified the four assumptions (H 1 )-(H 4 ). Also, the basic reproduction number of system (8) is given by By applying Theorems 3.3 and 3.4, we obtain the following result.
(i) If R 0 ≤ 1, then the infection-free equilibrium E 0 of system (8) is globally asymptotically stable.
(ii) If R 0 > 1 and i = 0, then the chronic infection equilibrium E * of system (8) is globally asymptotically stable.
Example 2: Consider the following system System (11) is a particular case of our model (6), it suffices to take f (T, I,V ) = β T α 0 + α 1 I + α 2 V + α 3 IV , where β > 0 is the infection rate and α 0 , α 1 , α 2 , α 3 are non-negative constants. This functional response was introduced Hattaf and Yousfi [4] and it covers various types of incidence rate existing in the literature.
Obviously, the function f satisfied the four assumptions (H 1 )-(H 4 ). Therefore, we get the following result.
(i) If R 0 ≤ 1, then the infection-free equilibrium E 0 of system (11) is globally asymptotically stable.
(ii) If R 0 > 1 and i = 0, then the chronic infection equilibrium E * of system (11) is globally asymptotically stable.

CONCLUSION
In this paper, we have proposed a class of virus infection model with diffusion and general incidence function. The continuous and discrete versions are rigorously analyzed by established the well-posedness of solutions and the global stability of equilibria. Furthermore, the discrete model and the corresponding results presented in the recent work [5] are improved and generalized.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.