Analysis of a Fractional Reaction-Diffusion HBV Model with Cure of Infected Cells

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.


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][3][4][5][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]. erefore, 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. eir 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. ey have improved and generalized the ordinary differential equation (ODE) models [9,13] and also the FDE models [14][15][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: e state variables U(x, t), I(x, t), C(x, t), and V(x, t) are, respectively, the concentrations of uninfected liver cells, infected liver cells, and HBV DNA-containing capsids and virions at location x and time t. Uninfected liver cells are produced at constant rate σ, die at rate δU, and become infected by virus at rate F(U, V)V. e parameters ε, ρ, κ, η, 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. e positive constants d C and d V are the diffusion coefficients of capsids and virus. Δ � n i�1 z 2 /zx 2 i is the Laplacian operator. e incidence function of (1) is described by Hattaf-Yousfi functional response [17] of the form F(U, V) � ((bU)/(α 0 + α 1 U + α 2 V + α 3 UV)), where the nonnegative constants α i , i � 0, 1, 2, 3, measure the saturation, inhibitory, or psychological effects and the positive constant b is the infection rate.
is 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, z α t is the Caputo fractional derivative of order α ∈ (0, 1]. e choice of this type of fractional derivative is motivated by the fact that the fractional derivative of a constant is equal to zero, and α was chosen in the interval (0, 1] 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. roughout this paper, we consider system (1) with initial conditions and zero-flux boundary conditions where Ω is a bounded domain in R n with smooth boundary zΩ and z/z n → denotes the outward normal derivative on zΩ.
From the biological point of view, these boundary conditions indicate that the capsids and free viral particles do not move across the boundary zΩ. e rest of the paper is organized as follows. e following section is devoted to the calculations of the basic reproduction number and steady states of model (1). e 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.

Equilibria of the FPDE Model
It is easy to verify that the only infection-free steady state of the FPDE model (1) en, the basic reproduction number of (1) is given by e other steady states verify the following system: ηC − ]V � 0.
We have If R 0 > 1, we deduce that system (1) admits a unique infection equilibrium P 1 (U 1 , We summarize the above discussions in the following result. (1) has uniquely one chronic infection steady state P 1 (U 1 ,

Global Dynamics
is section analyzes the global dynamics of the FPDE model (1).

Theorem 2.
e infection-free steady state P 0 is globally asymptotically stable if R 0 ≤ 1.

Numerical Simulations
In this section, we present some numerical illustrations to support the obtained analytical results.
Let Δt be the time step size, Ω � [x min , x max ], and Δx � (x max − x min )/N be the space step size, where N is a positive integer. e grid points for the space are x i � x min + iΔx for i ∈ 0, . . . , N { } and for time are t m � mΔt for m ∈ IN. From the Grünwald-Letnikov method [24], the Caputo fractional derivative is approximated as follows: where l m � ((l(x i , 0)t − α m )/(Γ(1 − α))) and β α j are the fractional binomial coefficients α j with the recursion formula: Let (U m i , I m i , C m i , V m i ) be the approximations of the solution (U, I, C, V) of (1) at the discretized point (x i , t m ).

Conclusions
In this article, we have presented a fractional reaction-diffusion HBV model that takes into account the HBV DNAcontaining capsids and the cure of infected liver cells. e spatial diffusion is considered in capsids and virions, and the incidence of infection is described by Hattaf-Yousfi functional response that includes various forms existing in the literature. We have shown that the global dynamics of the FPDE model is fully determined by a threshold parameter called the basic reproduction number and labeled by R 0 . More concretely, the infection-free steady state P 0 is globally asymptotically stable if R 0 ≤ 1, which biologically means that the HBV is cleared. However, the chronic infection steady state P 1 is globally asymptotically stable when R 0 > 1 and condition (15) holds. In this case, the HBV persists in the liver and the infection becomes chronic. According to the above analytic results and the numerical simulations, we deduce that the diffusion and the order of fractional derivative in sense of Caputo have no effects on the stability of both steady states, but they can affect the time for arriving to these steady states. For Discrete Dynamics in Nature and Society example, the trajectories quickly converge towards the equilibria of the model for higher values of the fractional derivative order (see, Figure 3). On the other hand, the models and results presented in [8,9,[12][13][14][15][16] are improved and extended.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.