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In this paper, we employ the nonstandard finite difference method to discretize a nonlinear initial boundary value problem that describes the global dynamics of hepatitis B virus infection with adaptive immunity. The model considered incorporates intracellular HBV DNA-containing capsids, a general incidence function and two categories of infected hepatocytes: exposed infected hepatocytes and productively infected hepatocytes. Adaptive immunity cells are recruited to attack all infected hepatocytes and free virus particles. The dynamics of fundamental properties of both discrete and continuous models are thoroughly studied, and it is shown that the discrete system is dynamically consistent with the continuous model. These properties include existence, nonnegativity and boundedness of solutions as well as the global stability of spatially homogeneous equilibria. We carry out some numerical simulations to support the theoretical findings and illustrate the behaviour of the solution in space and time. [ABSTRACT FROM AUTHOR]

In this work, we propose and investigate a delay cell population model of hepatitis B virus (HBV) infection. We suppose spatial diffusion of free HBV particles, and use a Beddington-DeAngelis incidence function to describe viral infection. The model takes into account the exposed hepatocytes and the usually neglected humoral immune response. Moreover, a time delay is introduced to account for the transformation processes necessary for actual HBV production. We naturally find two threshold parameters, namely the basic reproduction number R 0 and the humoral immune response reproduction number R 1 which completely determine the global stability of the spatially homogeneous equilibria of the model obtained. By constructing appropriate Lyapunov functionals and using LaSalle's invariance principle we show that, if R 0 ≤ 1 , the disease-free equilibrium is globally asymptotically stable. Furthermore, we prove that the endemic equilibrium without humoral immune response and the endemic equilibrium with humoral immune response are globally asymptotically stable if R 1 ≤ 1 < R 0 and R 1 > 1 , respectively. Finally, in one dimensional space, we perform some numerical simulations to illustrate the theoretical results obtained. [ABSTRACT FROM AUTHOR]

In this paper, two non-standard finite difference (NSFD) schemes are proposed for a mathematical model of hepatitis B virus (HBV) infection with spatial dependence. The dynamic properties of the obtained discretized systems are completely analyzed. Relying on the theory of M-matrix, we prove that the proposed NSFD schemes is unconditionally positive. Furthermore, we establish that the NSFD method used preserves all constant steady states of the corresponding continuous initial boundary value problem (IBVP) model. We prove that the conditions for those equilibria to be asymptotically stable are consistent with the continuous IBVP model independently of the numerical grid size. The global asymptotical properties of the HBV-free equilibrium of the proposed NSFD schemes are derived via the construction of a suitable discrete Lyapunov function, and coincides with the continuous system. This confirms that the discretized models are dynamically consistent since they maintain essential properties of the corresponding continuous IBVP model. Finally, numerical simulations are performed from which it is demonstrated that the proposed NSFD method is advantageous over the standard finite difference (SFD) method. [ABSTRACT FROM AUTHOR]

Abstract In this work, we investigate the hepatitis B virus infection. We first derive a nonlinear PDE model for the studied biological phenomenon. The obtained initial boundary value problem is completely analyzed. To begin with the analysis of the model, we use Lou and Zhao Lemma concerning globally attractive steady states to prove boundedness of potential solutions. Then we prove global existence, uniqueness and positivity of the solution by a variational method combined with semigroups theory and some other useful tools from functional analysis. Moreover, the basic reproduction number R 0 determining the extinction or the persistence of the HBV infection is thoroughly computed via a new method based on spectral properties of differential operators, the residue Theorem and some arguments from numerical analysis. Also, the global asymptotical properties of the HBV-free equilibrium of the model are derived via a skillful construction of a suitable Lyapunov function. Local stability of the endemic equilibrium of the model is studied as well. Finally, numerical simulations are performed to support the theoretical results obtained. [ABSTRACT FROM AUTHOR]