A non-standard finite difference method for a hepatitis B virus infection model with spatial diffusion

A non-standard finite difference method is proposed for an epidemic model which describes the hepatitis B virus infection with spatial dependence. Using the theory of M-matrix, it is shown that the proposed method is unconditionally positive. Moreover, this method preserves all constant steady-state solutions of the corresponding continuous system. Through constructing discrete Lyapunov functions, the globally asymptotical stabilities of the steady-state solutions are fully determined by the basic reproduction number R-0 independent of the time and space step sizes, which coincides with the continuous system. Finally, numerical experiments are provided to illustrate the theoretical results.