A cross-infection model with diffusion and incubation period

Abstract

<p style='text-indent:20px;'>In this paper, we study a cross-infection model with diffusion and incubation period. Firstly, we prove the global attractivity of the infection-free equilibrium and infected equilibrium for the spatially homogeneous system. Secondly, we establish the threshold dynamics for the spatially heterogeneous system in terms of the basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula>. It turns out that the infection-free steady state is globally attractive if <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{R}_0&lt;1 $\end{document}</tex-math></inline-formula>; and the system is uniformly persistent if <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{R}_0&gt;1 $\end{document}</tex-math></inline-formula>. Finally, we explore the influence of different diffusion coefficients, spatial heterogeneity of the disease transmission rate and the incubation period on <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula>. Our numerical results show that <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula> are decreasing functions of the diffusion coefficients and the incubation period, respectively, while it is increasing with respect to the spatial heterogeneity.</p>