A geometric analysis of the SIRS model with secondary infections

TL;DR

This paper introduces a geometric analysis of a compartmental SIRS model with secondary infections, exploring stability, equilibria, and multi-scale dynamics through analytical and numerical methods.

Contribution

It provides a novel geometric framework for analyzing multi-scale dynamics and stability in SIRS models with secondary infections, including bifurcation analysis.

Findings

Existence of two endemic equilibria under certain conditions.

Conditions for system evolution across three time scales.

Numerical simulations confirming analytical predictions.

Abstract

We propose a compartmental model for a disease with temporary immunity and secondary infections. From our assumptions on the parameters involved in the model, the system naturally evolves in three time scales. We characterize the equilibria of the system and analyze their stability. We find conditions for the existence of two endemic equilibria, for some cases in which $\mathcal{R}_0 < 1$. Then, we unravel the interplay of the three time scales, providing conditions to foresee whether the system evolves in all three scales, or only in the fast and the intermediate ones. We conclude with numerical simulations and bifurcation analysis, to complement our analytical results.