WKB calculation of an epidemic outbreak distribution

TL;DR

This paper applies WKB approximation to a stochastic SIR epidemic model to analyze how outbreak distributions evolve with population size, revealing the emergence of large outbreaks in smaller populations.

Contribution

It introduces an analytical WKB approach to quantify outbreak distributions in stochastic epidemic models, highlighting non-Gaussian behavior in small populations.

Findings

Distribution becomes highly non-Gaussian in small populations

Large outbreaks are more probable as population size decreases

Analytical results match simulations until fade-out dominates

Abstract

We calculate both the exponential and pre-factor contributions in a WKB approximation of the master equation for a stochastic SIR model with highly oscillatory dynamics. Fixing the basic parameters of the model we investigate how the outbreak distribution changes with the population size. We show that this distribution rapidly becomes highly non-Gaussian, acquiring large tails indicating the presence of rare, but large outbreaks, as the population is made smaller. The analytic results are found to be in excellent agreement with simulations until the systems become so small that the dynamics are dominated by fade-out of the disease.