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The distribution of general final state random variables for stochastic epidemic models

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Frank Ball  and
Philip O'Neill
Frank Ball*
Affiliation:
University of Nottingham
Philip O'Neill*
Affiliation:
University of Bradford
*
∗Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
∗∗Current address: Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK. Email address: [email protected].
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Abstract

In this paper we introduce the notion of general final state random variables for generalized epidemic models. These random variables are defined as sums over all ultimately infected individuals of random quantities of interest associated with an individual; examples include final severity. By exploiting a construction originally due to Sellke (1983), exact results concerning the final size and general final state random variables are obtained in terms of Gontcharoff polynomials. In particular, our approach highlights the way in which these polynomials arise via simple probabilistic arguments. For ease of exposition we focus initially upon the single-population case before extending our arguments to multi-population epidemics and other variants of our basic model.

Keywords

Epidemicfinal sizefinal severityGontcharoff polynomial

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60K99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 473 - 491
Copyright
Copyright © Applied Probability Trust 1999 

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