Dynamics of a reaction–diffusion SIRS model with general incidence rate in a heterogeneous environment

In this paper, we study a diffusive SIRS-type epidemic model with transfer from the infectious to the susceptible class. Our model includes a general nonlinear incidence rate and spatially heterogeneous diffusion coefficients. We compute 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} of our model and establish the global stability of the disease-free steady state when 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}. Furthermore, we study the uniform persistence when 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} and perform a bifurcation analysis for a special case of our model. Some numerical simulations are presented to illustrate the dynamics of the solutions as the model parameters are varied.