Dynamics of a Nonlocal Diffusive and Infectious SIR Epidemic Model with Double Free Boundaries

This paper concerns a nonlocal diffusive SIR epidemic model with nonlocal infectious and double free boundaries, which can be used to describe the spreading of infectious diseases. This model is a strongly coupled nonlocal diffusion system in some sense. We mainly study criteria for spreading and vanishing, and the long time behaviors. In addition to the usual Basic Reproduction NumberR0=k(a-beta)b(a+gamma)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0=\frac{k(a-\beta )}{b(a+\gamma )}$$\end{document}, we also discover another number R0ka-beta b,<middle dot>,(-h0,h0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0\left( k\frac{a-\beta }{b}, \cdot , (-h_0,h_0)\right) $$\end{document}, and find that these two numbers play a crucial role in determining both spreading and vanishing.