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Abel-Gontcharoff pseudopolynomials and the exact final outcome of SIR epidemic models (III)

Part of: Stochastic processes

Published online by Cambridge University Press:  01 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, Université Libre de Bruxelles, C.P. 210, Boulevard du Triomphe, B-1050 Bruxelles, Belgique. Email address: clefevre@ulb.ac.be
∗∗ Postal address: Mathématiques Appliquées, Université de Lyon 1, 43 boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France. Email address: isfa@cismsun.univ-lyon1.fr
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Abstract

The paper is concerned with the final state and severity of a number of SIR epidemic models in finite populations. Two different classes of models are considered, namely the classical SIR Markovian models and the collective models introduced recently by the authors. First, by applying a simple martingale argument, it is shown that in both cases, there exists a common algebraic structure underlying the exact law of the final state and severity. Then, a unified approach to these statistics is developed by exploiting the theory of Abel-Gontcharoff pseudopolynomials (presented in a preceding paper).

Keywords

Final state and severityexact distributioncollective epidemicsMarkovian modelsmartingalesAbelian expansionsAbel-Gontcharoff pseudopolynomials

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 2 , June 1999 , pp. 532 - 550
Copyright
Copyright © Applied Probability Trust 1999 

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