Abstract

In this paper, we propose a fractional reaction-diffusion model in order to better understand the mechanisms and dynamics of hepatitis B virus (HBV) infection in human body. The infection transmission is modeled by Hattaf–Yousfi functional response, and the fractional derivative is in the sense of Caputo. The global stability of the model equilibria is analyzed by means of Lyapunov functionals. Finally, numerical simulations are presented to support our analytical results.

1. Introduction

In the recent years, fractional calculus has attracted the attention of many researchers. Hattaf [1] proposed a new fractional derivative with nonsingular kernel which generalizes many forms existing in the literature such as the Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. Furthermore, there are some new methods used to solve numerically fractional models considered to explain deeper investigations of real-world problems [2–6].

On the other hand, hepatitis B is a dangerous infectious disease caused by the hepatitis B virus (HBV). It affects the lives of 257 million people and responsible for the deaths of 56000 people every year according to World Health Organization (WHO) estimates [7]. Therefore, several mathematical models have been proposed and developed to describe the dynamics of HBV infection. For instance, Manna and Chakrabarty [8] proposed and analyzed the dynamics of HBV infection by taking into account the spacial mobility of both HBV DNA-containing capsids and viruses. Their work was an extension of that presented in [9]. Guo et al. [10] studied a nonlinear system of partial differential equations (PDEs) for HBV infection with three time delays, general incidence rate, and spatial diffusion only in the viruses. Hattaf and Yousfi [11] developed a mathematical HBV infection model with two modes of transmission, which allows the movement of HBV DNA-containing capsids and viruses, and three distributed delays. Since fractional-order models possess property of memory, Bachraou et al. [12] proposed a mathematical model governed by fractional differential equations (FDEs) to more explore the dynamic characteristics of the HBV infection. They have improved and generalized the ordinary differential equation (ODE) models [9, 13] and also the FDE models [14–16] by using Hattaf’s incidence rate [17] that includes the common types such as the bilinear incidence rate, the saturated incidence rate, and the Beddington–DeAngelis functional response [18, 19].

In this study, we present an extension of our model presented in [12] by considering the mobility of capsids and viruses. So, we propose the following mathematical model formulated by fractional partial differential equations (FPDEs) to better describe the dynamics of HBV infection under the effects of diffusion and memory:

The state variables , , , and are, respectively, the concentrations of uninfected liver cells, infected liver cells, and HBV DNA-containing capsids and virions at location and time . Uninfected liver cells are produced at constant rate , die at rate , and become infected by virus at rate . The parameters , , , and are, respectively, the cure rate of infected liver cells, the death rate of infected liver cells and capsids, the production rate of capsids from infected liver cells, the rate at which the capsids are converted to virions, and the viral clearance rate. The positive constants and are the diffusion coefficients of capsids and virus. is the Laplacian operator. The incidence function of (1) is described by Hattaf–Yousfi functional response [17] of the form , where the nonnegative constants , , measure the saturation, inhibitory, or psychological effects and the positive constant is the infection rate. This functional response covers several specific cases available in the literature such as the bilinear and saturation incidences, the Beddington–DeAngelis and Crowley–Martin functional responses, and the specific functional response introduced by Hattaf et al. [20]. Finally, is the Caputo fractional derivative of order . The choice of this type of fractional derivative is motivated by the fact that the fractional derivative of a constant is equal to zero, and was chosen in the interval to have the same initial conditions as those of the PDE systems. Furthermore, a recent study in [21] has shown that the fractional-order model gives better predictions than that of the integer model about the plasma viral load of the patients.

Throughout this paper, we consider system (1) with initial conditionsand zero-flux boundary conditions where is a bounded domain in with smooth boundary and denotes the outward normal derivative on . From the biological point of view, these boundary conditions indicate that the capsids and free viral particles do not move across the boundary .

The rest of the paper is organized as follows. The following section is devoted to the calculations of the basic reproduction number and steady states of model (1). The global dynamics of the FPDE model is analyzed in Section 3. To support the analytical results, we present some numerical simulations in Section 4. Finally, we end the paper with biological and mathematical conclusions in Section 5.

2. Equilibria of the FPDE Model

It is easy to verify that the only infection-free steady state of the FPDE model (1) is , where . Then, the basic reproduction number of (1) is given by

The other steady states verify the following system:

From (5)–(8), we obtain , , , and

implies that . So, we consider the function defined on interval by

We have , , and

If , we deduce that system (1) admits a unique infection equilibrium with , , , and .

We summarize the above discussions in the following result.

Theorem 1. (i)When , the FPDE model (1) has one infection-free steady state , where (ii)When , the FPDE model (1) has uniquely one chronic infection steady state , where , , , and

3. Global Dynamics

This section analyzes the global dynamics of the FPDE model (1).

Theorem 2. The infection-free steady state is globally asymptotically stable if .

Proof. Letwhere for . According to [22], we obtainBy , we have Then, when . In addition, is the largest invariant set in . By LaSalle’s invariance principle [23], is globally asymptotically stable if .

Theorem 3. The chronic infection steady state is globally asymptotically stable when and

Proof. Let Then, Since , , , and , we have Hence,Since , we have if and . The last condition is equivalent to Also, is the largest invariant set in . According to LaSalle’s invariance principle, we deduce that is globally asymptotically stable. This completes the proof.

4. Numerical Simulations

In this section, we present some numerical illustrations to support the obtained analytical results.

Let be the time step size, , and be the space step size, where is a positive integer. The grid points for the space are for and for time are for . From the Grünwald–Letnikov method [24], the Caputo fractional derivative is approximated as follows: where and are the fractional binomial coefficients with the recursion formula:

Let be the approximations of the solution of (1) at the discretized point . Then, by applying (21), we obtain

Hence,