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VOL. 55 | 2007 Critical scaling of stochastic epidemic models
Steven P. Lalley

Editor(s) Eric A. Cator, Geurt Jongbloed, Cor Kraaikamp, Hendrik P. Lopuhaä, Jon A. Wellner

## Abstract

In the simple mean-field \emph{SIS} and \emph{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.

## Information

Published: 1 January 2007
First available in Project Euclid: 4 December 2007

zbMATH: 1185.60104
MathSciNet: MR2459938

Digital Object Identifier: 10.1214/074921707000000346

Subjects:
Primary: 60H30, 60K30, 60K35