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Estimation in branching processes with restricted observations

Part of: Markov processes Applications Stochastic processes Parametric inference

Published online by Cambridge University Press:  08 September 2016

Ronald Meester  and
Pieter Trapman
Ronald Meester*
Affiliation:
Vrije Universiteit Amsterdam
Pieter Trapman*
Affiliation:
Utrecht University and Vrije Universiteit Amsterdam
*
∗ Postal address: Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands. Email address: rmeester@few.vu.nl
∗∗ Current address: Julius Center, University Medical Center Utrecht, PO Box 85500, 3508 GA Utrecht, The Netherlands. Email address: trapman@math.uu.nl
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Abstract

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We consider an epidemic model where the spread of the epidemic can be described by a discrete-time Galton-Watson branching process. Between times n and n + 1, any infected individual is detected with unknown probability π and the numbers of these detected individuals are the only observations we have. Detected individuals produce a reduced number of offspring in the time interval of detection, and no offspring at all thereafter. If only the generation sizes of a Galton-Watson process are observed, it is known that one can only estimate the first two moments of the offspring distribution consistently on the explosion set of the process (and, apart from some lattice parameters, no parameters that are not determined by those moments). Somewhat surprisingly, in our context, where we observe a binomially distributed subset of each generation, we are able to estimate three functions of the parameters consistently. In concrete situations, this often enables us to estimate π consistently, as well as the mean number of offspring. We apply the estimators to data for a real epidemic of classical swine fever.

Keywords

Galton-Watson branching processconsistent estimationpartial observationepidemicreal-time estimation

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 62F12: Asymptotic properties of estimators
Secondary: 62P10: Applications to biology and medical sciences 60G42: Martingales with discrete parameter
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 4 , December 2006 , pp. 1098 - 1115
Copyright
Copyright © Applied Probability Trust 2006 

References

Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and Their Statistical Analysis (Lecture Notes Statist. 151). Springer, New York.Google Scholar
Becker, N. G. and Hasofer, A. M. (1997). Estimation in epidemics with incomplete observations. J. R. Statist. Soc. B 59, 415–429.Google Scholar
Diekmann, O. and Heesterbeek, J. A. P. (2000). Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation. John Wiley, Chichester.Google Scholar
Guttorp, P. (1991). Statistical Inference for Branching Processes. John Wiley, New York.Google Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.Google Scholar
Jacob, C. and Peccoud, J. (1998). Estimation of the parameters of a branching process from migrating binomial observations. Adv. Appl. Prob. 30, 948–967.Google Scholar
Klinkenberg, D. (2003). Mathematical epidemiology and the control of classical swine fever virus. , Utrecht University. Available at http://igitur-archive.library.uu.nl/dissertations/2003-0828-125441/full.pdf.Google Scholar
Meester, R., de Koning, J., de Jong, M. C. M. and Diekmann, O. (2002). Modelling and prediction of classical swine fever epidemics. Biometrics 41, 178–184.Google Scholar
Trapman, P. (2006). On stochastic models for the spread of infections. , Vrije Universiteit Amsterdam. Available at http://dare.ubvu.vu.nl/bitstream/1871/10202/1/7546.pdf.Google Scholar
Trapman, P., Meester, R. and Heesterbeek, H. (2004). A branching model for the spread of infectious animal diseases in varying environments. J. Math. Biol. 49, 553–576.Google Scholar