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The Early and Final States of an Epidemic in a Large Heterogeneous Population by a Small Initial Number of Infectives

Published online by Cambridge University Press:  01 July 2016

Steven M. Butler
Steven M. Butler*
Affiliation:
University of Kentucky
*
* Postal address: Department of Statistics, 871 Patterson Office Tower, University of Kentucky, Lexington, KY 40506-0027, USA.
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Abstract

This paper describes the early and final properties of a general S–I–R epidemic process in which the infectives behave independently, each infective has a random number of contacts with the others in the population, and individuals vary in their susceptibility to infection. For the case of a large initial number of susceptibles and a small (finite) initial number of infectives, we derive the threshold behavior and the limiting distribution for the final state of the epidemic. Also, we show strong convergence of the epidemic process over any finite time interval to a birth and death process, extending the results of Ball (1983). These complement some results due to Butler (1994), who considers the case of a large initial number of infectives.

Keywords

EPIDEMIC PROCESSFINAL EPIDEMIC SIZEHETEROGENEOUS POPULATION
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 3 , September 1994 , pp. 671 - 689
Copyright
Copyright © Applied Probability Trust 1994 

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References

Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications, Griffin, London.Google Scholar
Ball, F. G. (1983) The threshold behavior of epidemic models. J. Appl. Prob. 20, 227–240.CrossRefGoogle Scholar
Ball, F. G. (1985) Deterministic and stochastic epidemics with several kinds of susceptibles. Adv. Appl. Prob. 17, 1–22.Google Scholar
Becker, N. G. and Yip, P. (1989) Analysis of variations in an infection rate. Austral. J. Statist. 31, 42–52.CrossRefGoogle Scholar
Billingsley, P. (1986) Probability and Measure. Wiley, New York.Google Scholar
Butler, S. M. (1994) The final state of an epidemic in a large heterogeneous population with a large initial number of infectives. Adv. Appl. Prob. 26(3).Google Scholar
Kendall, D. G. (1956) Deterministic and stochastic epidemics in closed populations. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 4, 149–165.Google Scholar
Lefèvre, C. (1990) Stochastic epidemic models for S-I-R infectious diseases: a brief survey of the recent general theory. In Lecture Notes in Biomathematics 86, pp. 1–13. Springer-Verlag, Heidelberg.Google Scholar
Lefèvre, C. and Picard, Ph. (1989) On the formulation of discrete-time epidemic models. Math. Biosci. 95, 27–35.Google Scholar
Lefèvre, C. and Picard, Ph. (1990) The final size distribution of epidemics spread by infectives behaving independently. In Lecture Notes in Biomathematics 86, pp. 155–169. Springer-Verlag, Heidelberg.Google Scholar
Ludwig, D. (1975a) Final size distributions for epidemics. Math. Biosci. 23, 33–46.Google Scholar
Ludwig, D. (1975b) Qualitative behavior of stochastic epidemics. Math. Biosci. 23, 47–73.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
Picard, Ph. and Lefèvre, C. (1990) A unified analysis of the final size and severity distribution in collective Reed-Frost epidemic processes. Adv. Appl. Prob. 22, 269–294.Google Scholar
Picard, Ph. and Lefèvre, C. (1991) The dimension of Reed-Frost epidemic models with randomized susceptibility levels. Math. Biosci. 107, 225–233.CrossRefGoogle ScholarPubMed
Rajarshi, M. B. (1981) Simpler proofs of two threshold theorems for a general stochastic epidemic. J. Appl. Prob. 18, 721–724.Google Scholar
Scalia-Tomba, G. (1985) Asymptotic final size distribution for some chain-binomial processes. Adv. Appl. Prob. 17, 477–495.CrossRefGoogle Scholar
Scalia-Tomba, G. (1990) On the asymptotic final size distribution of epidemics in heterogeneous populations. In Lecture Notes in Biomathematics 86, pp. 189–196. Springer-Verlag, Heidelberg.Google Scholar
Von Bahr, B. and Martin-Löf, A. (1980) Threshold limit theorems for some epidemic processes. Adv. Appl. Prob. 12, 319–349.CrossRefGoogle Scholar
Williams, T. (1971) An algebraic proof of the threshold theorem for the general stochastic epidemic. Adv. Appl. Prob. 3, 223.Google Scholar