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

This paper is concerned with the dynamic behavior of solutions to a multi-patch epidemic model incorporating a saturation incidence mechanism under two scenarios: when the disease-induced fatality rate is positive on at least one patch and when it is zero everywhere. In the first scenario, we establish that the disease will be eventually eradicated and the susceptible population will stabilize at a positive constant. In the second scenario, we provide a detailed analysis of the global dynamics of solutions in terms of the basic reproduction number $R_0$ under certain conditions. Notably, our findings show that the presence of the saturation incidence reduces transmission risk. However, the combination of saturation incidence, spatial heterogeneity among patches, and population movements can result in multiple endemic equilibria. Additionally, we investigate the asymptotic profiles of endemic equilibria as population dispersal rates tend to zero. In particular, our results reveal that limiting the dispersal rate of susceptible populations can significantly reduce disease impact if the epidemic network contains at least one patch of moderate or low risk or if the total population size falls below a critical threshold. These findings have important implications for disease control. Numerical simulations are provided to support and illustrate our theoretical results.