 Research
 Open Access
Numerical solution of diffusive HBV model in a fractional medium
 Kolade M. Owolabi^{1, 2}Email author
 Received: 30 May 2016
 Accepted: 11 September 2016
 Published: 22 September 2016
Abstract
Evolution systems containing fractional derivatives can result to suitable mathematical models for describing better and important physical phenomena. In this paper, we consider a multicomponents nonlinear fractionalinspace reaction–diffusion equations consisting of an improved deterministic model which describe the spread of hepatitis B virus disease in areas of high endemic communities. The model is analyzed. We give some useful biological results to show that the diseasefree equilibrium is both locally and globally asymptotically stable when the basic reproduction number is less than unity. Our findings of this paper strongly recommend a combination of effective treatment and vaccination as a good control measure, is important to record the success of HBV disease control through a careful choice of parameters. Some simulation results are presented to support the analytical findings.
Keywords
 Disease free equilibrium
 Fourier spectral method
 Exponential integrator
 Fractional reaction–diffusion
 Nonlinear PDEs
 Numerical simulations
 Reproduction number
Mathematics Subject Classification
 34A34
 35A05
 35K57
 65L05
 65M06
 93C10
Background

In the case of an infinite system, x ∈ (−∞,∞), here R is a subset of (−∞,∞).

\(x \in [0,L],\frac{{\partial u_{i} }}{\partial x}(0,t) = \frac{{\partial u_{i} }}{\partial x}(L,t) = 0,i = 1,2, \ldots ,n,\) noflux or Neumann boundary condition for a finite system, and

\(x \in [0,L],u(0,t) = u(L,t) = u_{\alpha } ,i = 1,2, \ldots ,n,\) called the Dirichlet or fixed concentration boundary condition, also for a fixed system where \(u(x,t) \in R^{n} ,F_{i} :R^{n} \to R\).
 1.
In \(R_{ + }^{n}\) = {(u _{1} , u _{2}, … , u _{ n })u _{ i } > 0}, the vector field (F _{1}(u), F _{2}(u), … , F _{ n }(u)) has unstable equilibrium state at 0 = (0, 0, … , 0) and asymptotically stable at A = (A _{1}, A _{2}, … , A _{ n }) with A _{ i } > 0 for i = 1, 2, … , n.
 2.The coefficientsmust be finite.$$r_{i} = c_{ii} (0) = \mathop {\sup }\limits_{{0 \le u_{i} \le A_{i} ;\forall i = 1,2, \ldots ,n}} [c  ii(u)]$$
Fractional differential equations known as the differentiation and integration of noninteger order of the form (1) are becoming increasingly used as a modelling tool for diffusive processes associated with subdiffusion, diffusion and superdiffusion scenarios, and have been applied to an increasing number of situations in biochemistry (Yuste et al. 2004), control (Wang and Zhou 2011; Wang et al. 2012), medicine (Erturk et al. 2011; Hall and Barrick 2008; Otte et al. 2016) and mathematical biology or physics (Atangana 2015; Barkari et al. 2000; Tomovski et al. 2012; Wang and Du 2013). In literature, many definitions of fractional derivatives have been given among which are the definitions by Caputo, Grnwald–Letnikov, Hadamard, Marchaud and Riemann–Liouville are regarded as the most useful definitions (Haghighi et al. 2014; Podlubny et al. 2009; Polyanin and Zaitsev 2004; Yang et al. 2010; Zeng et al. 2014, 2015; Zheng et al. 2015; Zhou 2014). Quite Unfortunate that, similar usages of different definitions give rise to different results (Podlubny 1999).
Over the years, several numerical and analytical methods of solution have been adopted to solve both linear and nonlinear equations. Among which are the homotopy analysis method (Alomari et al. 2009), homotopy perturbation method (He 2005; Yildirim and Sezer 2010), Adomian decomposition method (Hassan 2008; Ray 2009), multistep differential transform method (Li Ding and LinJiang 2013; Erturk et al. 2011; Jiang et al. 2012), Kansa method (Chen et al. 2010), finite difference and tau methods (Celik and Duman 2012; Pang and Sun 2012; Saadatmandi and Dehghan 2011; Sausa 2009; Su et al. 2010) to mention a few. In the context of this paper, the spatial complexity of the domain imposes geometric constraints on the transport processes on all length scales, which can be measured as temporal correlations on all the time scales. Hence, we propose the study of fractional hepatitis B Virus (HBV) reaction–diffusion system in subdiffusive (0 < α < 1), diffusive (α = 1) and superdiffusive (1 < α < 2) scenarios, using the Fourier spectral method in space to remove the stiffness issue associated with the spatial fractional derivative in a finite but large domain size L. For the temporal discretization, we employ higherorder exponential time differencing scheme.
In “Basic definitions and numerical techniques for fractional diffusion problems” section, we provide some required basic information and definitions of fractional calculus, we also formulate Fourier spectral method for multicomponents fractionalinspace reactiondiffusion system. We present fractional HBV model, and examine the main system for both local and global stability in “Main model” section. Numerical experiment of the full nontrivial result is provided in “Numerical simulations” section. The final section concludes the paper.
Basic definitions and numerical techniques for fractional diffusion problems
Two important tasks are done in this section. The first one is to present some of the basic required information and definitions that guide the solution of fractional diffusion equations. The second task is centered on the introduction of Fourier spectral methods as an efficient and easytoimplement for the integration of fractionalinspace reaction–diffusion systems.
Basics definitions of fractional calculus
Definition 1
Definition 2
Definition 3
Definition 4
Primarily, the Caputo fractional differential equation computes an ordinary differential equation (ODE), followed by a fractional integral to obtain the desired order of fractional derivative, but in the reverse order we compute the Riemann–Liouville fractional differential system. The Caputo fractional differential equation permits the inclusion of traditional initial and boundary conditions in the problem formulation. These two operators coincide in the case of homogeneous initial condition. For details geometric and physical interpretation for fractional calculus for both the Riemann–Liouville and Caputo types, readers are referred to the classical books (Kilbas et al. 2005; Podlubny 1999).
Theorem 1
Proof
Numerical techniques for fractional reaction–diffusion equations
This implies that we work entirely in the spectral domain and invert a transform to recover u. It should be mentioned that once the stiffness is removed, one can rapidly and accurately advance in time with any explicit higherorder timesolvers, see for instance (Kassam and Trefethen 2005; Owolabi and Patidar 2014) for details.
At this point, we need to check the performance of the fractional Fourier transform technique in conjunction with both the classical fourthorder Runge–Kutta (Owolabi and Patidar 2014) and the fourthorder exponential timedifferencing Runge–Kutta schemes. Here we consider (1) in one component and set the reaction term to zero, so that the given multicomponents system reduces to fractional diffusion equation.
The relative error for fractional diffusion equation for various values of discretization, at D = 0.5 and t = 1
N  N = 64  N = 128  N = 256  N = 512 

RK4 ETDRK4  1.960e−06 1.6391e−08  4.9539e−06 2.0899e−08  7.5637e−06 2.9017e−08  1.5528e−05 4.7984e−08 
Ratio  72.9669  237.0400  260.6644  323.6079 
Main model
Over the years, mathematical modelling has become an important tool in the application areas medical and life sciences to address some of the health challenging problems that are not approachable experimentally. Hepatitis B is commonly referred to as a life threatening infectious disease caused by the hepatitis B virus (HBV) which leads to inflammation or causing serious damage to the liver. According to World Health Organization’s (WHO) 2002 data report, over 2000 million people have been effected, more than 350 million individuals remain chronically infected and carriers of the virus. An estimated population of 4 million people are considered as acute clinical cases of the virus.
Research have shown that children and adolescents are most vulnerable to the disease than adults due to exposure which may show some clinical symptom and have a higher percentage chance of being acutely infected. Based on report, about a quarter of chronic infected individuals die of liver cancer annually. As a result, hepatitis B is known to be one of the most common viral source of cancer in the world nowadays. Hence, HBV infection is a disease of global health and its prevalence varies from one region to the other.
A lot of researchers have worked on HBV in the past, among which are the notable papers of Medley et al. (2001) where compartmentalized model was used to describe the spread of the disease. Almost a decade later, Zou et al. (2010) worked on the modified version of the model (Medley et al. 2001). They develop a model to explore the impact of vaccination and other controlling measures of HBV infection. Their model has simple dynamical behavior which has a globally asymptotically stable diseasefree equilibrium when the basic reproduction number R _{0} < 1, and a globally asymptotically stable endemic equilibrium when R _{0} > 1. In the year 2014, Kimbir et al. (2014) give an extension to the earlier report in Zou et al. (2010) by including the treatment of chronically infected HBV carriers, it was also suggested in their report that the acute infected individuals are not subjected to antiviral treatment due to natural recovery. Wiah et al. (2015) employed a nonlinear extended deterministic model to address the impact of immigration on the population spread of HBV infection with acute and chronic infected carriers.
Stability analysis of the disease free equilibrium (DFE) point
In this section, we analyze the local stability of the diseasefree equilibrium (DFE). It is the stability of at DFE that can guarantee a biologically meaningful results. Here we assume that the disease variables U _{2} = U _{3} = U _{4} = 0. If otherwise, the disease will persist and put the whole population of susceptible individuals into serious danger.
Theorem 2
The disease free equilibrium point \(\hat{E} = (\gamma /(\tau + \phi ),0,0,0,\tau \phi /(\tau + \phi )\tau )\) is locally asymptotically stable for the spatially homogeneous stationary solution of model (17) with kinetics (20) if R _{0} < 1, and unstable if otherwise.
Proof
Global stability of the diseases free equilibrium point
Here, we are concerned with the global stability of DFE point. We adopt a similar technique to the proof of Theorem 2.
Theorem 3
The disease free equilibrium point \(\hat{E} = (\gamma /(\tau + \phi ),0,0,0,\tau \phi /(\tau + \phi )\tau )\) is globally asymptotically stable for the spatially homogeneous stationary solution of model (17) with kinetics (20), if R _{0} < 1.
Proof
Numerical simulations
In this section, we start to simulate numerically the solution of the spatially homogeneous system (17) with kinetics (20) using the numerical techniques formulated in “Basic definitions and numerical techniques for fractional diffusion problems” section above to substantiate our analytical findings.
Nondiffusive example
For the set of ecological parameters, we realized that the conditions given in Theorems 2 and 3 for nondiffusive system are satisfied for the disease freeequilibrium state.
Fractional reaction–diffusion example
Conclusion
In this paper, a mathematical model for investigating the hepatitis B virus disease in fractional medium is derived. The model disease free equilibrium state is analyzed. We established via theorems that the model diseasefree equilibrium is both locally and globally asymptotically stable, if the basic reproduction number is less than unity. Our aim is to examine the behaviour of diffusive fractional reaction–diffusion model in subdiffusive and superdiffusive scenarios, derive efficient and reliable numerical techniques. By the computer experiment of the fractional reaction–diffusion system we have given enough evidence that numerical solution in the diffusive (fractional) scenario, at 0 < α < 2 is practicably the same as in the case of nondiffusive case when applied to model Hepatitis B virus system. Our findings in this work strongly recommend a combination of effective treatment and vaccination as a good control measure is important to record the success of HBV disease control. It should be noted that the methodology presented in this paper can be applied to model other physical phenomena in higher dimensions.
Declarations
Competing interests
The author declares that he has no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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