Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-18T10:49:40.578Z Has data issue: false hasContentIssue false
  • English
  • Français

Compound Poisson limits for household epidemics

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

Peter Neal
Peter Neal*
Affiliation:
University of Manchester
*
∗Postal address: School of Mathematics, University of Manchester, Sackville Street, Manchester M60 1QD, UK. Email address: p.neal-2@manchester.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider epidemics in populations that are partitioned into small groups known as households. Whilst infectious, a typical infective makes global and local contact with individuals chosen independently and uniformly from the whole population or their own household, as appropriate. Previously, the classical Poisson approximation for the number of survivors of a severe epidemic has been extended to the household model. However, in the current work we exploit a Sellke-type construction of the epidemic process, which enables the derivation of sufficient conditions for the existence of a compound Poisson limit theorem for the survivors of the epidemic. The results are specialised to the Reed-Frost and general stochastic epidemic models.

Keywords

Epidemic modelhouseholdcompound Poisson convergence

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 334 - 345
Copyright
© Applied Probability Trust 2005 

References

Andersson, H. (1999). Epidemic models and social networks. Math. Scientist 24, 128–147.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
Ball, F. G. (1996). Threshold behaviour in stochastic epidemics among households. In Athens Conference on Applied Probability and Time Series (Lecture Notes Statist. 114), Vol. 1, Applied Probability, eds Heyde, C. C., Prohorov, Y. V., Pyke, R. and Rachev, S. T., Springer, New York, pp. 253–266.Google Scholar
Ball, F. G. and Barbour, A. D. (1990). Poisson approximation for some epidemic models. J. Appl. Prob. 27, 479–490.Google 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
Ball, F. G. and Neal, P. J. (2004). Poisson approximations for epidemics with two levels of mixing. Ann. Prob. 32, 1168–1200.CrossRefGoogle 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.Google Scholar
Chow, Y. S. and Teicher, H. (1978). Probability Theory. Springer, New York.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
Lefèvre, C. and Utev, S. (1995). Poisson approximation for the final state of a generalised epidemic process. Ann. Prob. 23, 1139–1162.Google Scholar
Lefèvre, C. and Utev, S. (1996). Asymptotic behaviour of the final state of the generalised epidemic process. Siberian Math. J. 37, 753–763.Google Scholar
Lefèvre, C. and Utev, S. (1997). Mixed Poisson approximation in the collective epidemic model. Stoch. Process. Appl. 69, 217–246.Google Scholar
Sellke, T. (1983). On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390–394.CrossRefGoogle Scholar