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Simpler proofs of two threshold theorems for a general stochastic epidemic

Published online by Cambridge University Press:  14 July 2016

M. B. Rajarshi
M. B. Rajarshi*
Affiliation:
University of Poona
*
∗Postal address: Department of Statistics, University of Poona, Ganeshkhind, Pune–411 007, India.
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Abstract

We give here fairly elementary proofs for the threshold theorems due to Williams (1971) and Whittle (1955). Our proofs are based on an application of the reflection principle through the ballot problem and the exact distribution of the size of the epidemic as derived by Foster (1955). Williams's threshold theorem is extended to an epidemic with multiple introduction of cases.

Keywords

GENERAL STOCHASTIC EPIDEMICTHRESHOLD THEOREMSSIZE OF THE EPIDEMICREFLECTION PRINCIPLE
Type
Short Communications
Information
Journal of Applied Probability , Volume 18 , Issue 3 , September 1981 , pp. 721 - 724
Copyright
Copyright © Applied Probability Trust 

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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
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Volume I, 3rd edn. Wiley, New York.Google Scholar
Foster, F. G. (1955) A note on Bailey's and Whittle's treatment of a general stochastic epidemic. Biometrika 42, 123–125.Google Scholar
Whittle, P. (1955), The outcome of a stochastic epidemic — a note on Bailey's paper. Biometrika 42, 116–122.Google Scholar
Williams, T. (1971) An algebraic proof of the threshold theorem for the general stochastic epidemic (abstract). Adv. Appl. Prob. 3, 223.Google Scholar