Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T22:09:35.813Z Has data issue: false hasContentIssue false
  • English
  • Français

The distribution of time to extinction in subcritical branching processes: applications to outbreaks of infectious disease

Part of: Inference from stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

C. P. Farrington  and
A. D. Grant
C. P. Farrington*
Affiliation:
Open University
A. D. Grant*
Affiliation:
Public Health Laboratory Service Statistics Unit, London
*
∗Postal address: Statistics department, Faculty of Mathematics and Computing, The Open University, Walton Hall, Milton Keynes, MK7 6AA. Email address: c.p.farrington@open.ac.uk
∗∗Postal address: Statistics Unit, PHLS, 61 Colindale Avenue, London, NW9 5EQ.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the distribution of the number of generations to extinction in subcritical branching processes, with particular emphasis on applications to the spread of infectious diseases. We derive the generation distributions for processes with Bernoulli, geometric and Poisson offspring, and discuss some of their distributional and inferential properties. We present applications to the spread of infection in highly vaccinated populations, outbreaks of enteric fever, and person-to-person transmission of human monkeypox.

Keywords

Branching processpower series familyextinctiongeneration distributionmaximum likelihood estimationepidemic model

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 62M05: Markov processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 771 - 779
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Becker, N. (1974). On parametric estimation for mortal branching processes. Biometrika 61, 393–453.CrossRefGoogle Scholar
Braddick, M. R., and Sharp, J. C. M. (1993). Enteric fever in Scotland 1975–1990. Publ. Health 107, 193–198.CrossRefGoogle ScholarPubMed
Fine, P. E. M., Jezek, Z., Grab, B., and Dixon, H. (1988). The transmission potential of monkeypox virus in human populations. Int. J. Epidemiol. 17, 643–650.CrossRefGoogle ScholarPubMed
Gustafson, T. L., Lievens, A. W., Brunell, P. A., Moellenberg, R. G., Buttery, C. M. G., and Sehulster, L. M. (1987). Measles outbreak in a fully immunised secondary-school population. N. Engl. J. Med. 316, 771–774.CrossRefGoogle Scholar
Harris, T. E. (1948). Branching processes. Ann. Math. Statist. 19, 474–494.CrossRefGoogle Scholar
Karlin, S., and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Lattès, S. (1911). Sur les suites récurrentes non linéaires et sur les fonctions génératrices de ces suites. Ann. Fac. Sci. Univ. Toulouse 3, 73–124.CrossRefGoogle Scholar
Nkowane, B. M., Bart, S. W., Orenstein, W. A., and Baltier, M. (1987). Measles outbreak in a vaccinated school population: epidemiology, chains of transmission and the role of vaccine failures. Am. J. Publ. Health 77, 434–438.CrossRefGoogle Scholar
Sharp, J. C. M., and Heymann, C. S. (1976). Enteric fever in Scotland, 1967–1974. J. Hygiene, Camb. 76, 83–89.CrossRefGoogle ScholarPubMed