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The Probability of Containment for Multitype Branching Process Models for Emerging Epidemics

Published online by Cambridge University Press:  14 July 2016

Simon E. F. Spencer*
Affiliation:
University of Warwick
Philip D. O‘Neill*
Affiliation:
University of Nottingham
*
Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: s.e.f.spencer@warwick.ac.uk
∗∗Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK. Email address: philip.oneill@nottingham.ac.uk
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Abstract

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This paper is concerned with the definition and calculation of containment probabilities for emerging disease epidemics. A general multitype branching process is used to model an emerging infectious disease in a population of households. It is shown that the containment probability satisfies a certain fixed point equation which has a unique solution under certain conditions; the case of multiple solutions is also described. The extinction probability of the branching process is shown to be a special case of the containment probability. It is shown that Laplace transform ordering of the severity distributions of households in different epidemics yields an ordering on the containment probabilities. The results are illustrated with both standard epidemic models and a specific model for an emerging strain of influenza.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Ball, F. (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, 289310.Google Scholar
[2] Ball, F., Mollison, D. and Scalia-Tomba, G. (1997). Epidemics with two levels of mixing. Ann. Appl. Prob. 7, 4689.Google Scholar
[3] Ball, F. and Lyne, O. D. (2001). Stochastic multitype SIR epidemic among a population partitioned into households. Adv. Appl. Prob. 33, 99123.CrossRefGoogle Scholar
[4] Daley, D. J. (1990). The size of epidemics with variable infectious periods. Tech. Rep. SMS–012–90, Australian National University.Google Scholar
[5] Ferguson, N. M. et al. (2005). Strategies for containing an emerging influenza pandemic in Southeast Asia. Nature 437, 209214.CrossRefGoogle ScholarPubMed
[6] Lefèvre, C. P. and Picard, P. (1993). An unusual stochastic order relation with some applications in sampling and epidemic theory. Adv. Appl. Prob. 25, 6381.Google Scholar
[7] Longini, I. M. et al. (2005). Containing pandemic influenza at the source. Science 309, 10831087.Google Scholar
[8] Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.Google Scholar
[9] Mode, C. J. (1971). Multitype Branching Processes. Theory and Applications. American Elsevier, New York.Google Scholar
[10] Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, Boston, MA.Google Scholar
[11] Spencer, S. E. F. (2007). Stochastic epidemic models for emerging diseases. , University of Nottingham.Google Scholar