Traveling waves for a two-group epidemic model with latent period and bilinear incidence in a patchy environment

Abstract

In this paper, we consider a two-group SIR epidemic model with bilinear incidence in a patchy environment. It is assumed that the infectious disease has a fixed latent period and spreads between two groups. Firstly, when the basic reproduction number \\begin{document}$ \\mathcal{R}_{0}>1 $\\end{document} and speed \\begin{document}$ c>c^{\\ast} $\\end{document} , we prove that the system admits a nontrivial traveling wave solution, where \\begin{document}$ c^{\\ast} $\\end{document} is the minimal wave speed. Next, when \\begin{document}$ \\mathcal{R}_{0}\\leq1 $\\end{document} and \\begin{document}$ c>0 $\\end{document} , or \\begin{document}$ \\mathcal{R}_{0}>1 $\\end{document} and \\begin{document}$ c\\in(0,c^{*}) $\\end{document} , we also show that there is no positive traveling wave solution, where \\begin{document}$ k = 1,2 $\\end{document} . Finally, we discuss and simulate the dependence of the minimum speed \\begin{document}$ c^{\\ast} $\\end{document} on the parameters.