Critical scaling of stochastic epidemic models

TL;DR

This paper investigates the critical behavior of stochastic epidemic models, both mean-field and spatial, revealing thresholds for initial infections that determine whether epidemics die out or grow, with detailed scaling laws at criticality.

Contribution

It introduces scaling laws for critical epidemic models, including spatial variants, and identifies thresholds for initial infections influencing epidemic outcomes.

Findings

Existence of a critical threshold for initial infections.

Epidemics below threshold resemble branching processes.

At threshold, epidemics follow a size-dependent drift process.

Abstract

In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent $p-$coin tosses. Spatial variants of these models are proposed, in which finite populations of size $N$ are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for both the mean-field and spatial models are given when the infection parameter $p$ is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a size-dependent drift.