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The distribution of general final state random variables for stochastic epidemic models

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Frank Ball  and
Philip O'Neill
Frank Ball*
Affiliation:
University of Nottingham
Philip O'Neill*
Affiliation:
University of Bradford
*
∗Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
∗∗Current address: Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK. Email address: [email protected].
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Abstract

In this paper we introduce the notion of general final state random variables for generalized epidemic models. These random variables are defined as sums over all ultimately infected individuals of random quantities of interest associated with an individual; examples include final severity. By exploiting a construction originally due to Sellke (1983), exact results concerning the final size and general final state random variables are obtained in terms of Gontcharoff polynomials. In particular, our approach highlights the way in which these polynomials arise via simple probabilistic arguments. For ease of exposition we focus initially upon the single-population case before extending our arguments to multi-population epidemics and other variants of our basic model.

Keywords

Epidemicfinal sizefinal severityGontcharoff polynomial

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60K99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 473 - 491
Copyright
Copyright © Applied Probability Trust 1999 

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References

Addy, C. L., Longini, J. M., and Haber, M. (1991). A generalised stochastic model for the analysis of infectious disease final size data. Biometrics 47, 961–974.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Ball, F. G. (1996). Threshold behaviour in stochastic epidemics among households. In Athens Conference on Applied Probability and Time Series, Vol. I, eds. Heyde, C. C., Prohorov, Y. V., Pyke, R. and Rachev, S. T. (Lecture Notes in Statist. 114), Springer, New York, pp. 253–266.CrossRefGoogle Scholar
Ball, F. G., and Clancy, D. (1993). The final size and severity of a generalised stochastic multitype epidemic model. Adv. Appl. Prob. 25, 721–736.CrossRefGoogle Scholar
Ball, F. G., Mollison, D., and Scalia-Tomba, G. (1997). Epidemics with two levels of mixing. Ann. Appl. Prob. 7, 46–89.Google Scholar
Becker, N. G., and Dietz, K. (1995). The effect of the household distribution on transmission and control of highly infectious diseases. Math. Biosci. 127, 207–219.CrossRefGoogle ScholarPubMed
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
Downton, F. (1968). The ultimate size of carrier-borne epidemics. Biometrika 55, 277–289.Google Scholar
Downton, F. (1972). A correction to ‘The area under the infectives trajectory of the general stochastic epidemic’. J. Appl. Prob. 9, 873–876.Google Scholar
Gani, J., and Jerwood, D. (1972). The cost of a general stochastic epidemic. J. Appl. Prob. 9, 257–269.Google Scholar
Lefèvre, C. (1990). Stochastic epidemic models for SIR infectious diseases: a brief survey of the recent general theory. In Stochastic Processes in Epidemic Theory, eds. Gabriel, J.-P., Lefèvre, C. and Picard, P. (Lecture Notes in Biomath. 86), Springer, New York, pp. 1–12.CrossRefGoogle Scholar
Lefèvre, C., and Picard, P. (1990). A non-standard family of polynomials and the final size distribution of Reed–Frost epidemic processes. Adv. Appl. Prob. 22, 25–48.CrossRefGoogle Scholar
Picard, P. 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
Scalia-Tomba, G. (1985). Asymptotic final-size distribution for some chain-binomial processes. Adv. Appl. Prob. 17, 477–495.Google Scholar
Sellke, T. (1983). On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390–394.Google Scholar
Whittle, P. (1955). The outcome of a stochastic epidemic – a note on Bailey's paper. Biometrika 42, 116–122.Google Scholar