Global stability of SAIRS epidemic models

TL;DR

This paper analyzes the global stability of an SAIRS epidemic model with vaccination, determining conditions for disease eradication or persistence, and validates findings through numerical simulations.

Contribution

It provides a comprehensive global stability analysis of the SAIRS model with vaccination, including an open problem solution and stability results at the threshold case.

Findings

Disease-free equilibrium is globally stable if R0<1

Endemic equilibrium exists when R0>1 and is globally stable under certain conditions

Numerical simulations confirm analytical stability results

Abstract

We study an SAIRS-type epidemic model with vaccination, where the role of asymptomatic and symptomatic infectious individuals are explicitly considered in the transmission patterns of the disease. We provide a global stability analysis for the model. We determine the value of the basic reproduction number $\mathcal{R}_0$ and prove that the disease-free equilibrium is globally asymptotically stable if $\mathcal{R}_0<1$ and unstable if $\mathcal{R}_0>1$, condition under which a positive endemic equilibrium exists. We investigate the global stability of the endemic equilibrium for some variations of the original model under study and answer to an open problem proposed in Ansumali et al. \cite{ansumali2020modelling}. In the case of the SAIRS model without vaccination, we prove the global asymptotic stability of the disease-free equilibrium also when $\mathcal{R}_0=1$. We provide a thorough numerical exploration of our model, to validate our analytical results.