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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{R}_{0}>1 $\end{document}</tex-math></inline-formula> and speed <inline-formula><tex-math id="M2">\begin{document}$ c>c^{\ast} $\end{document}</tex-math></inline-formula>, we prove that the system admits a nontrivial traveling wave solution, where <inline-formula><tex-math id="M3">\begin{document}$ c^{\ast} $\end{document}</tex-math></inline-formula> is the minimal wave speed. Next, when <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R}_{0}\leq1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ c>0 $\end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{R}_{0}>1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ c\in(0,c^{*}) $\end{document}</tex-math></inline-formula>, we also show that there is no positive traveling wave solution, where <inline-formula><tex-math id="M8">\begin{document}$ k = 1,2 $\end{document}</tex-math></inline-formula>. Finally, we discuss and simulate the dependence of the minimum speed <inline-formula><tex-math id="M9">\begin{document}$ c^{\ast} $\end{document}</tex-math></inline-formula> on the parameters.</p>