# Moderate deviations and extinction of an epidemic

## TL;DR

This paper investigates how stochastic fluctuations influence the extinction time of an epidemic in large populations, using probabilistic theories like the Central Limit Theorem and Moderate Deviations to estimate extinction times.

## Contribution

It introduces a novel approach applying Moderate and Large Deviations principles to estimate epidemic extinction times in stochastic models near deterministic equilibria.

## Key findings

- Estimates of epidemic extinction times depend on population size.
- Moderate deviations provide precise asymptotic estimates.
- Large deviations help understand rare extinction events.

## Abstract

Consider an epidemic model with a constant flux of susceptibles, in a situation where the corresponding deterministic epidemic model has a unique stable endemic equilibrium. For the associated stochastic model, whose law of large numbers limit is the deterministic model, the disease free equilibrium is an absorbing state, which is reached soon or later by the process. However, for a large population size, i.e. when the stochastic model is close to its deterministic limit, the time needed for the stochastic perturbations to stop the epidemic may be enormous. In this paper, we discuss how the Central Limit Theorem, Moderate and Large Deviations allow us to give estimates of the extinction time of the epidemic, depending upon the size of the population.