Multiplicity of endemic equilibria for a diffusive SIS epidemic model with mass-action

TL;DR

This paper investigates a diffusive SIS epidemic model with mass-action transmission, revealing conditions for multiple endemic equilibria and complex bifurcation behaviors, thereby addressing open questions on disease persistence and extinction.

Contribution

It demonstrates the existence of multiple endemic equilibria and complex bifurcation structures in a simple diffusive SIS model, advancing understanding of disease dynamics.

Findings

Multiple endemic equilibria can exist under certain parameters.

The model exhibits S-shaped and backward bifurcation curves.

Results clarify conditions for disease persistence when R0<1.

Abstract

We study a diffusive SIS epidemic model with the mass-action transmission mechanism and show, under appropriate assumptions on the parameters, the existence of multiple endemic equilibria (EE). Our results answer some open questions on previous studies related to disease extinction or persistence when $\mathcal{R}_0<1$ and the multiplicity of EE solutions when $\mathcal{R}_0>1$. Interestingly, even with such a simple nonlinearity induced by the mass-action, we show that the diffusive epidemic model may have an S-shaped or backward bifurcation curve of EE solutions.