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Strong Convergence of Stochastic Epidemics

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

Frank Ball  and
Philip O'Neill
Frank Ball*
Affiliation:
University of Nottingham
Philip O'Neill*
Affiliation:
University of Nottingham
*
* Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
** Present address: Department of Mathematics, University of Bradford, Bradford, West Yorkshire BD7 1DP, UK.
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Abstract

This paper is concerned with a model for the spread of an epidemic in a closed, homogeneously mixing population in which new infections occur at rate f(x, y) and removals occur at rate g(x, y), where x and y are the numbers of susceptible and infective individuals, respectively, and f and g are arbitrary but specified positive real-valued functions. Sequences of such epidemics, indexed by the initial number of susceptibles n, are considered and conditions are derived under which the epidemic processes converge almost surely to a birth and death process as n tends to infinity. Thus a threshold theorem for such an epidemic model is obtained. The results are extended to models which incorporate immigration and emigration of susceptibles. The theory is illustrated by several examples of models taken from the epidemic literature. Generalizations to multipopulation epidemics are discussed briefly.

Keywords

EPIDEMICSSIZE OF EPIDEMICTHRESHOLD THEOREMSBIRTH AND DEATH PROCESSESCOUPLINGEPIDEMICS WITH IMMIGRATION AND EMIGRATION

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60F15: Strong theorems 60J27: Continuous-time Markov processes on discrete state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 3 , September 1994 , pp. 629 - 655
Copyright
Copyright © Applied Probability Trust 1994 

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