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Nonstationarity and randomization in the Reed-Frost epidemic model

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Claude Lefèvre  and
Philippe Picard
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
Philippe Picard*
Affiliation:
Université de Lyon 1
*
∗Postal address: Institut de Statistique et de Recherche Opérationnelle, CP 210, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgium. Email address: [email protected]
∗∗Postal address: Institut de Science Financière et d'Assurances, Université de Lyon 1, 50 Avenue Tony Garnier, F-69007 Lyon, France.
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Abstract

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The purpose of this paper is to determine the exact distribution of the final size of an epidemic for a wide class of models of susceptible–infective–removed type. First, a nonstationary version of the classical Reed–Frost model is constructed that allows us to incorporate, in particular, random levels of resistance to infection in the susceptibles. Then, a randomized version of this nonstationary model is considered in order to take into account random levels of infectiousness in the infectives. It is shown that, in both cases, the distribution of the final number of infected individuals can be obtained in terms of Abel–Gontcharoff polynomials. The new methodology followed also provides a unified approach to a number of recent works in the literature.

Keywords

Epidemicfinal sizerandom susceptibilityvaccinationrandom infectivitymartingaleAbel–Gontcharoff polynomial

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K99: None of the above, but in this section
Secondary: 92D30: Epidemiology
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 4 , December 2005 , pp. 950 - 963
Copyright
© Applied Probability Trust 2005 

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