The spatially heterogeneous diffusive rabies model and its shadow system

Abstract

<p style='text-indent:20px;'>In this paper, we consider a class of spatially heterogeneous reaction diffusion rabies model which was used to describe population dynamics of the rabies epidemic disease observed in Europe. The dynamics of both the original non-degenerate reaction-diffusion system and its corresponding shadow system are investigated in great details. Firstly, we prove that under certain conditions, the in-time solutions of both the original non-degenerate reaction-diffusion system and its shadow system exist globally and remain uniformly bounded. Secondly, we are capable of showing that the shadow system is the nice approximations for the original non-degenerate reaction-diffusion system when the diffusion rate <inline-formula><tex-math id="M1">\begin{document}$ d_R $\end{document}</tex-math></inline-formula> of the infectious rabid individuals (R) is sufficiently large. This implies that the dynamics of the shadow system can say as much as possible about the dynamics of the original system when <inline-formula><tex-math id="M2">\begin{document}$ d_R $\end{document}</tex-math></inline-formula> is sufficiently large. Finally, we characterize the basic reproduction number for the shadow system, and study the stability/instability of the disease-free steady state.