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

Authors: Moussa Bachraoui; Khalid Hattaf; Noura Yousfi
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## 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.


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## Body

## 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: (1)∂tαU=σ−δUx,t−FUx,t,Vx,tVx,t+ɛIx,t,∂tαI=FUx,t,Vx,tVx,t−ρ+ɛIx,t,∂tαC=dCΔC+κIx,t−ρ+ηCx,t,∂tαV=dVΔV+ηCx,t−νVx,t.The state variablesUx,t, Ix,t, Cx,t, and Vx,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 FU,VV. The parameters ɛ, ρ,κ, η, and ν 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 dC and dV are the diffusion coefficients of capsids and virus. Δ=∑i=1n∂2/∂xi2 is the Laplacian operator. The incidence function of (1) is described by Hattaf–Yousfi functional response [17] of the form FU,V=bU/α0+α1U+α2V+α3UV, where the nonnegative constants αi, i=0,1,2,3, measure the saturation, inhibitory, or psychological effects and the positive constant b 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, ∂tα is the Caputo fractional derivative of order α∈0,1. 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 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.Throughout this paper, we consider system (1) with initial conditions(2)Ux,0=U0x≥0,Ix,0=I0x≥0,Vx,0=V0x≥0,Cx,0=C0x≥0,∀x∈Ω¯,and zero-flux boundary conditions (3)∂C∂n⟶=∂V∂n⟶=0,on∂Ω×0,+∞,where Ω is a bounded domain in ℝn with smooth boundary ∂Ω and ∂/∂n⟶ denotes the outward normal derivative on ∂Ω. From the biological point of view, these boundary conditions indicate that the capsids and free viral particles do not move across the boundary ∂Ω.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 P0U0,0,0,0, where U0=σ/δ. Then, the basic reproduction number of (1) is given by(4)ℛ0=κηFU0,0νη+ρɛ+ρ.The other steady states verify the following system:(5)σ−δU−FU,VV+ɛI=0,(6)FU,VV−ɛ+ρI=0,(7)κI−η+ρC=0,(8)ηC−νV=0.From (5)–(8), we obtain I=σ−δU/ρ, C=κσ−δU/ρη+ρ, V=ηκσ−δU/νρη+ρ, and (9)FU,ηκσ−δUνρη+ρ=νη+ρρ+ɛκη.I=σ−δU/ρ≥0 implies that U≤σ/δ. So, we consider the function G defined on interval 0,σ/δ by (10)GU=FU,ηκσ−δUνρη+ρ−νη+ρρ+ɛκη.We haveG0=−νη+ρρ+ɛ/κη<0, Gσ/δ=νη+ρρ+ɛ/κηℛ0−1, and (11)G′U=∂F∂U−κηδνρη+ρ∂F∂V>0.Ifℛ0>1, we deduce that system (1) admits a unique infection equilibrium P1U1,I1,C1,V1 with U1∈0,σ/δ, I1=σ−δU1/ρ, C1=κσ−δU1/ρη+ρ, and V1=ηκσ−δU1/νρη+ρ.We summarize the above discussions in the following result.Theorem 1.
(i)
Whenℛ0≤1, the FPDE model (1) has one infection-free steady state P0U0,0,0,0, where U0=σ/δ(ii)
Whenℛ0>1, the FPDE model (1) has uniquely one chronic infection steady state P1U1,I1,C1,V1, where U1∈0,σ/δ, I1=σ−δU1/ρ, C1=κσ−δU1/ρη+ρ, and V1=ηκσ−δU1/νρη+ρ


## 3. Global Dynamics

This section analyzes the global dynamics of the FPDE model (1).Theorem 2.
The infection-free steady stateP0 is globally asymptotically stable if ℛ0≤1.Proof.
Let(12)L0t=∫Ωα0α0+α1U0U0ΦUU0+α0εUx,t−U0+Ix,t22ρ+δα0+α1U0U0+Ix,t+ɛ+ρκCx,t+ɛ+ρη+ρκηVx,tdx,where Φx=x−1−lnx for x>0. According to [22], we obtain(13)DαL0t≤∫Ωα0α0+α1U01−U0U∂tαU+∂tαI+α0ɛU−U0+I∂tαU+∂tαIρ+δα0+α1U0U0ɛ+ρκ∂tαC+ɛ+ρη+ρκη∂tαVdx.
Byσ=δU0, we have (14)DαL0t≤∫Ω−1U+ɛρ+δU0α0δU−U02α0+α1U0−α0ɛρI2ρ+δα0+α1U0U0−α0ɛIU−U02α0+α1U0U0U+νɛ+ρη+ρκηℛ0FU,VFU0,0−1Vdx,≤−∫Ω1U+ɛρ+δU0α0δU−U02α0+α1U0+α0ɛρI2ρ+δα0+α1U0U0+α0ɛIU−U02α0+α1U0U0U−νɛ+ρη+ρκηℛ0−1Vdx.
Then,DαL0t≤0 when ℛ0≤1. In addition, P0 is the largest invariant set in U,I,C,V|DαL0t=0. By LaSalle’s invariance principle [23], P0 is globally asymptotically stable if ℛ0≤1.Theorem 3.
The chronic infection steady stateP1 is globally asymptotically stable when ℛ0>1 and (15)ℛ0≤1+δρνη+ρ+α2δσκηɛ+ρ+α3ɛηκσ2ɛνρɛ+ρα0δ+α1σ.Proof.
Let(16)L1t=∫Ωα0+α2V1α0+α1U1+α2V1+α3U1V1U1ΦUU1+ɛα0+α2V1Ux,t−U1+Ix,t−I122ρ+δα0+α1U1+α2V1+α3U1V1U1+I1ΦII1+ɛ+ρκC1ΦCC1+ɛ+ρη+ρκηV1ΦVV1dx.
Then,(17)DαL1t≤∫Ωα0+α2V1α0+α1U1+α2V1+α3U1V11−U1U∂tαU+ɛα0+α2V1U−U1+I−I1∂tαU+∂tαIρ+δα0+α1U1+α2V1+α3U1V1U1+1−I1I∂tαI+ɛ+ρκ1−C1C∂tαC+ɛ+ρη+ρκη1−V1V∂tαVdx.
Sinceσ=δU1+ρI1, FU1,V1V1=ɛ+ρI1, κI1=η+ρC1, and 1−FU1,V1/FU,V1=α0+α2V1/α0+α1U1+α2V1+α3U1V11−U1/U, we have (18)DαL1t≤∫Ω−δα0+α2V1U−U12α0+α1U1+α2V1+α3U1V1U+ɛα0+α2V1U−U1I−I1α0+α1U1+α2V1+α3U1V1U+FU,V1V11−FU1,V1FU,V1+FU,VV1FU,V1V1+FU1,V1V11−IC1I1C+FU1,V1V11−FU,VVI1FU1,V1IV1+FU1,V1V11−VV1−CV1C1V−ɛα0+α2V1δU−U12+ρI−I12+δ+ρU−U1I−I1α0+α1U1+α2V1+α3U1V1δ+ρU1dx−ɛ+ρκdCC1∫Ω▽C2C2dx−ɛ+ρη+ρκηdVV1∫Ω▽V2V2dx.
Hence,(19)DαL1t≤−∫Ωα0+α2V1U−U12α0+α1U1+α2V1+α3U1V1UU1δU1−ɛI1+δɛUρ+δ+ɛI+ɛρα0+α2V1I−I12α0+α1U1+α2V1+α3U1V1ρ+δU1−FU1,V1V15−FU1,V1FU,V1−C1ICI1−FU,VFU1,V1VI1V1I−CV1C1V−FU,V1FU,V+FU1,V1V1α0+α1Uα2+α3UV−V12α0+α1U+α2V1+α3UV1α0+α1U+α2V+α3UVV1dx−ɛ+ρκdCC1∫Ω▽C2C2dx−ɛ+ρη+ρκηdVV1∫Ω▽V2V2dx.
Since5−FU1,V1/FU,V1−C1I/CI1−FU,V/FU1,V1VI1/V1I−CV1/C1V−FU,V1/FU,V≤0, we have DαL1t≤0 if ℛ0>1 and ɛI1≤δU1. The last condition is equivalent to (20)ℛ0≤1+δρνη+ρ+α2δσκηɛ+ρ+α3ɛηκσ2ɛνρɛ+ρα0δ+α1σ.
Also,P1 is the largest invariant set in U,I,C,V|DαL1t=0. According to LaSalle’s invariance principle, we deduce that P1 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.LetΔt be the time step size, Ω=xmin,xmax, and Δx=xmax−xmin/N be the space step size, where N is a positive integer. The grid points for the space are xi=xmin+iΔx for i∈0,…,N and for time are tm=mΔt for m∈IN. From the Grünwald–Letnikov method [24], the Caputo fractional derivative is approximated as follows: (21)∂Ctαlxi,tm≈1Δtα∑j=0mβjαlxi,tm−j−l˜m,where l˜m=lxi,0tm−α/Γ1−α and βjα are the fractional binomial coefficients αj with the recursion formula:(22)βjα=1−1+αjβj−1α,β0α=1.LetUim,Iim,Cim,Vim be the approximations of the solution U,I,C,V of (1) at the discretized point xi,tm. Then, by applying (21), we obtain(23)1ΔtαUim+1+∑j=1m+1βjiαUm+1−j−U˜im+1=σ−δUim−FUim,VimVim+ɛIim,1ΔtαIim+1+∑j=1m+1βjαIim+1−j−I˜im+1=FUim,VimVim−ɛ+ρIim,1ΔtαCim+1+∑j=1m+1βjαCim+1−j−C˜im+1=dCCi+1m−2Cim+Ci−1mΔx2+κIim−η+ρCim,1ΔtαVim+1+∑j=1m+1βjαVim+1−j−V˜im+1=dVVi+1m−2Vim+Vi−1mΔx2+ηCim−νVim.Hence,(24)Uim+1=−∑j=1m+1βjαUim+1−j+ΔtαU˜im+1+σ−δHim−FUim,VimVim+εIim,Iim+1=−∑j=1m+1βjαIim+1−j+ΔtαI˜im+1+FUim,VimVim−ɛ+ρIim,Cim+1=−∑j=1m+1βjαCim+1−j+ΔtαC˜im+1+dCCi+1m−2Cim+Ci−1mΔx2+κIim−η+ρCim,Vim+1=−∑j=1m+1βjαVim+1−j+ΔtαV˜im+1+dVVi+1m−2Vim+Vi−1mΔx2+ηCim−νVim.For numerical simulations, we chooseΩ=0,1, dC=0.1, dV=0.1, σ=50400, b=3.6×10−6, δ=0.039, ρ=0.0693, κ=150, η=0.01, ɛ=0.01, α=0.8, α0=0.1, α1=0.1, α2=0.01, and α3=0.000001. By simple calculation, we find ℛ0=0.0128<1. According to Theorem 2, the infection-free steady state P01.2923×106,0,0,0 is globally asymptotically stable which means that the virus will disappear and the patient will be completely cured. Figure 1 confirms this result.Figure 1
Stability of the infection-free steady stateP0.To numerically illustrate the global stability of the second steady state of model (1), we take b=0.0018 without changing the values of the other parameters. In this case, ℛ0=6.4083>1. From Theorem 1, FPDE model (1) has the unique chronic infection steady state P11.244×106,1.891×104,3.562×107,5.314×105. In addition, we have(25)1+δρνη+ρ+α2δσκηε+ρ+α3ɛηκσ2ɛνρε+ρα0δ+α1σ=218.9236,which implies that (15) holds. By Theorem 3, P1 is globally asymptotically stable. Figure 2 validates this result.Figure 2
Stability of the chronic infection steady stateP1.


## 5. Conclusions

In this article, we have presented a fractional reaction-diffusion HBV model that takes into account the HBV DNA-containing capsids and the cure of infected liver cells. The 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 byℛ0. More concretely, the infection-free steady state P0 is globally asymptotically stable if ℛ0≤1, which biologically means that the HBV is cleared. However, the chronic infection steady state P1 is globally asymptotically stable when ℛ0>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 example, the trajectories quickly converge towards the equilibria of the model for higher values of the fractional derivative order (see, Figure3). On the other hand, the models and results presented in [8, 9, 12–16] are improved and extended.Figure 3
The state variableVx,t with different values of α.



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