## Institute of Mathematical Statistics Lecture Notes - Monograph Series

### Critical scaling of stochastic epidemic models

Steven P. Lalley

#### 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.

#### Chapter information

Source
Eric A. Cator, Geurt Jongbloed, Cor Kraaikamp, Hendrik P. Lopuhaä, Jon A. Wellner, eds., Asymptotics: Particles, Processes and Inverse Problems: Festschrift for Piet Groeneboom (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2007), 167-178

Dates
First available in Project Euclid: 4 December 2007

https://projecteuclid.org/euclid.lnms/1196797075

Digital Object Identifier
doi:10.1214/074921707000000346

Mathematical Reviews number (MathSciNet)
MR2459938

Zentralblatt MATH identifier
1185.60104

Rights