Large-deviations of the SIR model around the epidemic threshold

TL;DR

This study uses large-deviation techniques to analyze the probability of rare epidemic outcomes in the SIR model on small-world networks, revealing three distinct dynamical regimes and confirming the large-deviation property.

Contribution

It introduces a large-deviation approach to quantify rare events in the SIR model, providing detailed probability densities and insights into epidemic dynamics near the threshold.

Findings

Identified three regimes: mild, severe, and sustained outbreaks.

Confirmed the large-deviation property for the SIR model.

Quantified correlations with outbreak duration and peak infection levels.

Abstract

We numerically study the dynamics of the SIR disease model on small-world networks by using a large-deviation approach. This allows us to obtain the probability density function of the total fraction of infected nodes and of the maximum fraction of simultaneously infected nodes down to very small probability densities like $10^{-2500}$. We analyze the structure of the disease dynamics and observed three regimes in all probability density functions, which correspond to quick mild, quick extremely severe and sustained severe dynamical evolutions, respectively. Furthermore, the mathematical rate functions of the densities are investigated. The results indicate that the so called large-deviation property hold for the SIR model. Finally, we measured correlations with other quantities like the duration of an outbreak or the peak position of the fraction of infections, also in the rare regions which are not accessible by standard simulation techniques.