Dynamics of solutions to a multi-patch epidemic model with a saturation incidence mechanism

Abstract

This study examines the behavior of solutions in a multi-patch epidemic model that includes a saturation incidence mechanism. When the fatality rate due to the disease is not null, our findings show that the solutions of the model tend to stabilize at disease-free equilibria. Conversely, when the disease-induced fatality rate is null, the dynamics of the model become more intricate. Notably, in this scenario, while the saturation effect reduces the basic reproduction number ${\mathcal{R}}_0$, it can also lead to a backward bifurcation of the endemic equilibria curve at ${\mathcal{R}}_0=1$. Provided certain fundamental assumptions are satisfied, we offer a detailed analysis of the global dynamics of solutions based on the value of ${\mathcal{R}}_0$. Additionally, we investigate the asymptotic profiles of endemic equilibria as population dispersal rates tend to zero. To support and illustrate our theoretical findings, we conduct numerical simulations.

44 pages