| Abstract: |
| In this talk, we consider a class of spatially heterogeneous diffusive SIR (susceptible-infected but noninfectious-rabid and infectious) rabies model with the general incidence rates. The model 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 the original non-degenerate reaction-diffusion system exist globally and remain uniformly bounded; This makes it possible to derive the shadow system for the original non-degenerate reaction-diffusion system. Secondly, we are capable of showing that the shadow system will be the nice approximations for the original non-degenerate reaction-diffusion system when the diffusion rate $d_R$ 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 $d_R$ 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 |
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