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Poisson approximation for some epidemic models

Published online by Cambridge University Press:  14 July 2016

Frank Ball  and
A. D. Barbour
Frank Ball*
Affiliation:
University of Nottingham
A. D. Barbour*
Affiliation:
Universität Zurich
*
∗Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
∗∗Postal address: Institut für Angewandte Mathematik, Universität Zürich, Rämistrasse 74, CH-8001 Zürich, Switzerland.
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Abstract

The Daniels' Poisson limit theorem for the final size of a severe general stochastic epidemic is extended to the Martin-Löf epidemic, and an order of magnitude for the error in the approximation is also given. The argument consists largely of showing that the number of survivors of a severe epidemic is essentially the same as the number of isolated vertices in a random directed graph. Poisson approximation for the latter quantity is proved using the Stein–Chen method and a suitable coupling.

Keywords

STEIN–CHEN METHODEPIDEMIC SIZERANDOM GRAPH
Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 3 , September 1990 , pp. 479 - 490
Copyright
Copyright © Applied Probability Trust 1990 

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References

Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications, 2nd edn. Griffin, London.Google Scholar
Ball, F. G. (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
Barbour, A. D. (1982) Poisson convergence and random graphs. Math. Proc. Camb. Phil. Soc. 92, 349–359.CrossRefGoogle Scholar
Barbour, A. D., Holst, L. and Janson, S. (1990) Poisson Approximation. (to appear).Google Scholar
Cane, V. R. (1966) On the size of an epidemic and the number of people hearing a rumour. J. Roy. Statist. Soc. B 20, 487–490.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
En'Ko, P. D. (1889) The epidemic course of some infectious diseases. Vrac' 10, 1008–1010, 1039-1042, 1061-1063.Google Scholar
Erdos, P. and Renyi, A. (1960) On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–61.Google Scholar
Martin-Löf, A. (1986) Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Prob. 23, 265–282.Google Scholar
Sellke, T. (1983) On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390–394.CrossRefGoogle Scholar
Whittle, P. (1955) The outcome of a stochastic epidemic – a note on Bailey's paper. Biometrika 42, 116–122.Google Scholar