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A non-standard family of polynomials and the final size distribution of Reed-Frost epidemic processes

Published online by Cambridge University Press:  01 July 2016

Claude Lefevre  and
Philippe Picard
Claude Lefevre*
Affiliation:
Université Libre de Bruxelles
Philippe Picard*
Affiliation:
Université de Lyon 1
*
∗Postal address: Institute de Statistique C. P. 210, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgique.
∗∗Postal address: Mathématiques Appliquées, Université de Lyon 1, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne, France.
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Abstract

This paper provides a global treatment of the final size distribution of Reed–Frost epidemic processes. Exact and asymptotic results are derived for both single and multipopulation situations. The key tool is a non-standard family of polynomials, introduced initially by Gontcharoff (1937) for one variable, revisited and extended here for several variables. The attractiveness of these polynomials will be enhanced in forthcoming works in the epidemic context as well as in other fields.

Keywords

MARTINGALELIMIT THEOREMS
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 1 , March 1990 , pp. 25 - 48
Copyright
Copyright © Applied Probability Trust 1990 

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Footnotes

Research partially supported by NATO Grant n° 0757/87.

Research partially supported by the Institut National de la Santé et de la Recherche Médicale under contract n° 878020

References

Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications. Griffin, London.Google Scholar
Ball, F. (1983). The threshold behavior of epidemic models. J. Appl. Prob. 20, 227–241.Google Scholar
Ball, F. (1986) A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models. Adv. Appl. Prob. 18, 289–310.Google Scholar
Daniels, H. E. (1967) The distribution of the total size of an epidemic. Proc. 5th Berkeley Symp. Math. Statist. Prob. 4, 281–293.Google Scholar
Gani, J. and Jerwood, D. (1972) The cost of a general stochastic epidemic. J. Appl. Prob. 9, 257–269.CrossRefGoogle Scholar
Gontcharoff, W. (1937) Détermination des Fonctions Entières par Interpolation. Hermann, Paris.Google Scholar
Ludwig, D. (1975) Final size distributions for epidemics. Math. Biosci. 23, 33–46.Google Scholar
Picard, Ph. (1980) Applications of martingale theory to some epidemic models. J. Appl. Prob. 17, 583–599.CrossRefGoogle Scholar
Picard, Ph. and Lefevre, Cl. (1990) A unified analysis of the final size and severity distribution in collective Reed–Frost epidemic processes. Adv. Appl. Prob. 22(2).Google Scholar
Scalia-Tomba, G. (1986) Asymptotic final size distribution of the multitype Reed–Frost process. J. Appl. Prob. 23, 563–584.CrossRefGoogle Scholar
Sellke, T. (1983) On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390–394.CrossRefGoogle Scholar
Von Bahr, B. and Martin-Löf, A. (1980) Threshold limit theorems for some epidemic processes. Adv. Appl. Prob. 12, 319–349.CrossRefGoogle Scholar