Qualitative analysis on a spatial SIS epidemic model with linear source in advective environments: I standard incidence

This paper is concerned with a reaction–diffusion SIS epidemic model with standard incidence infection mechanism and linear source in advective heterogeneous environments. We have derived the threshold-type dynamics in terms of the basic reproduction number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_0$$\end{document}: the disease will be eliminated if R0≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_0\le 1$$\end{document} while it persists uniformly if R0>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_0>1$$\end{document}. The global asymptotic stability of the endemic equilibrium is discussed in a special case. We mainly investigate the effects of linear source, advection and diffusion on asymptotic profiles of the endemic equilibrium. It is shown that the linear source can enhance persistence of infectious disease, advection may induce the concentration phenomenon and small dispersal rate of infected individuals can eradicate the disease. These results may offer some implications on disease control and prediction.