# Analysis of a SIRI epidemic model with distributed delay and relapse

## TL;DR

This paper analyzes a SIRI epidemic model incorporating distributed delay and relapse, establishing conditions for disease extinction or persistence based on the basic reproduction number, using Lyapunov methods.

## Contribution

It introduces a SIRI model with distributed delay and relapse, providing rigorous proofs of global stability for disease-free and endemic states.

## Key findings

- Global stability of disease-free equilibrium when R0<1
- Global stability of endemic equilibrium when R0>1
- Explicit computation of the basic reproduction number R0

## Abstract

We investigate the global behaviour of a SIRI epidemic model with distributed delay and relapse. From the theory of functional differential equations with delay, we prove that the solution of the system is unique, bounded, and positive, for all time. The basic reproduction number $R_{0}$ for the model is computed. By means of the direct Lyapunov method and LaSalle invariance principle, we prove that the disease free equilibrium is globally asymptotically stable when $R_{0} < 1$. Moreover, we show that there is a unique endemic equilibrium, which is globally asymptotically stable, when $R_{0} > 1$.