Institute of Mathematical Statistics Lecture Notes - Monograph Series

Critical scaling of stochastic epidemic models

Steven P. Lalley

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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

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

First available in Project Euclid: 4 December 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 60H30: Applications of stochastic analysis (to PDE, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

stochastic epidemic model spatial epidemic Feller diffusion branching random walk Dawson-Watanabe process critical scaling

Copyright © 2007, Institute of Mathematical Statistics


Lalley, Steven P. Critical scaling of stochastic epidemic models. Asymptotics: Particles, Processes and Inverse Problems, 167--178, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2007. doi:10.1214/074921707000000346.

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