How many outbreaks before an epidemic?

TL;DR

This paper analyzes the finite-population behavior of the Reed-Frost epidemic model, identifying conditions for outbreak sizes and validating the model's applicability to COVID-19 through simulations.

Contribution

It provides exact and simulation-based analysis of outbreak size distributions, introducing a threshold-based approach to distinguish small and large outbreaks.

Findings

Distribution of outbreak sizes becomes bimodal above a critical threshold

Monte Carlo simulations complement exact calculations for stability

Reed-Frost model approximates COVID-19 outbreak dynamics well

Abstract

In this work, we study the finite-population behaviour of the Reed-Frost epidemic model. Our analysis relies on the exact expression for the final epidemic size, replaced by Monte Carlo simulations in cases where the exact formula becomes numerically unstable. When the initial reproduction number is greater than a critical threshold, the distribution of the final size becomes bimodal. We therefore define the probabilities of small and large outbreaks, providing an intuitive answer to the question posed in the title through simple arguments based on the geometric distribution. Finally, an agent-based simulation confirms that the Reed-Frost model offers a good approximation in the case of the COVID-19 outbreak.