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The threshold behaviour of epidemic models

Published online by Cambridge University Press:  14 July 2016

Frank Ball
Frank Ball*
Affiliation:
University of Sussex
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Abstract

We provide a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptibles N, from a time-homogeneous birth-and-death process. The construction is used to show strong convergence of the general stochastic epidemic to a birth-and-death process, over any finite time interval [0, t], and almost sure convergence of the total size of the general stochastic epidemic to that of a birth-and-death process. The latter result furnishes us with a new proof of the threshold theorem of Williams (1971). These methods are quite general and in the remainder of the paper we develop similar results for a wide variety of epidemics, including chain-binomial, host-vector and geographical spread models.

Keywords

EPIDEMICSSIZE OF EPIDEMICTHRESHOLD THEOREMSBIRTH-AND-DEATH PROCESSESSTRONG CONVERGENCEBRANCHING PROCESSESGEOGRAPHICAL SPREAD

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 2 , June 1983 , pp. 227 - 241
Copyright
Copyright © Applied Probability Trust 1983 

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