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The final size and severity of a generalised stochastic multitype epidemic model

Published online by Cambridge University Press:  01 July 2016

Frank Ball  and
Damian Clancy
Frank Ball
Affiliation:
University of Nottingham
Damian Clancy*
Affiliation:
University of Nottingham
*
* Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
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Abstract

We consider a stochastic model for the spread of an epidemic amongst a population split into m groups, in which infectives move among the groups and contact susceptibles at a rate which depends upon the infective's original group, its current group, and the group of the susceptible. The distributions of total size and total area under the trajectory of infectives for such epidemics are analysed. We derive exact results in terms of multivariate Gontcharoff polynomials by treating our model as a multitype collective Reed–Frost process and slightly adapting the results of Picard and Lefèvre (1990). We also derive asymptotic results, as each of the group sizes becomes large, by generalising the method of Scalia-Tomba (1985), (1990).

Keywords

DYNAMIC POPULATION EPIDEMICSCONTACT DISTRIBUTION EPIDEMICSGONTCHAROFF POLYNOMIALSGAUSSIAN LIMIT THEOREMSSIZE OF EPIDEMICAREA UNDER TRAJECTORY OF INFECTIVES

MSC classification

Primary: 92D30: Epidemiology
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 4 , December 1993 , pp. 721 - 736
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

Work carried out while Damian Clancy was in receipt of an SERC research studentship.

References

Andersson, H. (1992) A threshold limit theorem for a multitype epidemic model. Research report no. 165, Institute of Actuarial Mathematics and Mathematical Statistics, Stockholm University.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
Ball, F. G. (1983) The threshold behaviour of epidemic models. J. Appl. Prob. 20, 227–241.CrossRefGoogle 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. (1991) Dynamic population epidemic models. Math. Biosci. 107, 299–324.CrossRefGoogle ScholarPubMed
Ball, F. G. and Clancy, D. (1992) The final outcome of a generalised stochastic multitype epidemic model. Research report no. 92-4, Nottingham Statistics Group, Department of Mathematics, University of Nottingham.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Clancy, D. (1994) Some comparison results for multitype epidemic models. J. Appl. Prob. 31 (1).Google Scholar
Faddy, M. J. (1983) The outcome of a general spatial epidemic on the line. J. Appl. Prob. 20, 715–727.Google Scholar
Faddy, M. J. (1986) A note on the behaviour of deterministic spatial epidemics. Math. Biosci. 80, 19–22.Google Scholar
Faddy, M. J. and Slorach, I. H. (1980) Bounds on the velocity of spread of infection for a spatially connected epidemic process. J. Appl. Prob. 17, 839–845.CrossRefGoogle Scholar
Kurtz, T. G. (1981) Approximation of Population Processes. SIAM, Philadelphia, Pa.CrossRefGoogle Scholar
Lefevre, 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
Martin-Löf, A. (1986) Symmetric sampling procedures, general epidemic processes, and their threshold limit theorems. J. Appl. Prob. 23, 265–282.Google Scholar
Martin-Löf, A. (1990) Threshold limit theorems in the theory of rumors, snowball sampling and epidemics. In Stochastic Processes in Epidemic Theory, ed. Gabriel, J. P., Lefèvre, C. and Picard, P., Lecture Notes in Biomathematics 86, Springer, New York, 184–188.Google Scholar
Mode, C. J. (1971) Multitype Branching Processes. Elsevier, New York.Google Scholar
Mollison, D. (1977) Spatial contact models for ecological and epidemic spread. J. R. Statist. Soc. B 39, 283–326.Google Scholar
Picard, P. and Lefevre, 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.CrossRefGoogle Scholar
Scalia-Tomba, G. (1986) Asymptotic final size distribution of the multitype Reed-Frost process. J. Appl. Prob. 23, 563–584.Google Scholar
Scalia-Tomba, G. (1990) On the asymptotic final size distribution of epidemics in heterogeneous populations. In Stochastic Processes in Epidemic Theory, ed. Gabriel, J. P., Lefèvre, C. and Picard, P., Lecture Notes in Biomathematics 86, Springer, New York, 189–196.CrossRefGoogle Scholar
Sellke, T. (1983) On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390–394.Google Scholar
Watson, R. K. (1972) On an epidemic in a stratified population. J. Appl. Prob. 9, 659–666.Google Scholar