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

TL;DR

This paper analyzes the global dynamics of a multi-patch epidemic model with saturation incidence, revealing conditions for disease eradication, complex bifurcation behavior, and effects of dispersal rates, supported by theoretical and numerical analysis.

Contribution

It provides a comprehensive analysis of the model's global dynamics, including bifurcation phenomena and asymptotic behaviors, which were not previously detailed in multi-patch epidemic models.

Findings

Solutions tend to disease-free equilibria when fatality rate is non-zero.

Saturation can cause backward bifurcation at R0=1.

Dispersal rates influence endemic equilibrium profiles.

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.