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Stochastic multi-type SIR epidemics among a population partitioned into households

Part of: Limit theorems Special processes

Published online by Cambridge University Press:  01 July 2016

Frank Ball  and
Owen D. Lyne
Frank Ball*
Affiliation:
University of Nottingham
Owen D. Lyne*
Affiliation:
University of Nottingham
*
∗ Postal address: School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK.
∗∗ Email address: frank.ball@nottingham.ac.uk
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Abstract

We consider a stochastic model for the spread of an SIR (susceptible → infective → removed) epidemic among a closed, finite population that contains several types of individuals and is partitioned into households. The infection rate between two individuals depends on the types of the transmitting and receiving individuals and also on whether the infection is local (i.e., within a household) or global (i.e., between households). The exact distribution of the final outcome of the epidemic is outlined. A branching process approximation for the early stages of the epidemic is described and made fully rigorous, by considering a sequence of epidemics in which the number of households tends to infinity and using a coupling argument. This leads to a threshold theorem for the epidemic model. A central limit theorem for the final outcome of epidemics which take off is derived, by exploiting an embedding representation.

Keywords

epidemic processfinal outcomelocal and global contactsmultitype branching processcouplingthreshold theoremweak convergencecentral limit theoremsembedded process

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60F99: None of the above, but in this section 60K99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 1 , March 2001 , pp. 99 - 123
Copyright
Copyright © Applied Probability Trust 2001 

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