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On the Waiting Time to Escape

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Maria Conceição Serra
Maria Conceição Serra*
Affiliation:
Chalmers University of Technology
*
∗ Postal address: Department of Mathematical Statistics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden. Email address: mcserra@math.chalmers.se
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Abstract

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The mathematical model we consider here is a decomposable Galton-Watson process with individuals of two types, 0 and 1. Individuals of type 0 are supercritical and can only produce individuals of type 0, whereas individuals of type 1 are subcritical and can produce individuals of both types. The aim of this paper is to study the properties of the waiting time to escape, i.e. the time it takes to produce a type-0 individual that escapes extinction when the process starts with a type-1 individual. With a view towards applications, we provide examples of populations in biological and medical contexts that can be suitably modeled by such processes.

Keywords

Decomposable Galton-Watson branching processprobability generating function

MSC classification

Primary: 60J85: Applications of branching processes
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Short Communications
Information
Journal of Applied Probability , Volume 43 , Issue 1 , March 2006 , pp. 296 - 302
Copyright
© Applied Probability Trust 2006 

References

Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Axelrod, D. E. and Kimmel, M. (2002). Branching Processes in Biology. Springer, New York.Google Scholar
Bruss, F. T. and Slavtchova-Bojkova, M. (1999). On waiting times to populate an environment and a question of statistical inference. J. Appl. Prob. 36, 261–267.CrossRefGoogle Scholar
Haccou, P., Jagers, P. and Vatutin, V. A. (2005). Branching Processes: Variation, Growth and Extinction of Populations. Cambridge University Press.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
Iwasa, Y., Michor, F. and Nowak, M. (2003). Evolutionary dynamics of escape from biomedical intervention. Proc. R. Soc. London B 270, 2573–2578.CrossRefGoogle ScholarPubMed
Iwasa, Y., Michor, F. and Nowak, M. (2004). Evolutionary dynamics of invasion and escape. J. Theoret. Biol. 226, 205–214.Google Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.Google Scholar
Mode, C. (1971). Multitype Branching Processes: Theory and Applications. Elsevier, New York.Google Scholar