Numerical analysis of a reaction–diffusion susceptible–infected–susceptible epidemic model

This paper presents the numerical properties of a reaction–diffusion susceptible–infected–susceptible epidemic model. Comparing with existing literature, our numerical scheme gains advantage in terms of preserving the biological meanings (such as positivity or invariance of total population) unconditionally. An implicit–explicit technique is implemented in the time integration, which ensures the numerical positivity without CFL conditions while reducing the computation complexity. The solvability, convergence in finite time and the long-time behaviors of numerical solutions are investigated. A threshold value R0Δx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{\Delta x}$$\end{document} for the long-time dynamics of numerical solutions is proposed, which is named as a numerical basic reproduction number. It is proved that the numerical disease-free equilibrium is locally asymptotically stable if R0Δx<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{\Delta x}<1$$\end{document} and unstable if R0Δx>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{\Delta x}>1$$\end{document}. It is presented that R0Δx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{\Delta x}$$\end{document} shares the same monotonicity and limits as the basic reproduction number of the underlying model and converges to the exact one. Some numerical experiments are given in the end to confirm the conclusions and explore the stability of the endemic equilibrium.