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 R0 independent of the time and space step sizes, which coincides with the continuous system. Finally, numerical experiments are provided to illustrate the theoretical results.