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Conditional Distributions and Waiting Times in Multitype Branching Processes

Part of: Stochastic processes

Published online by Cambridge University Press:  04 January 2016

H. K. Alexander
H. K. Alexander*
Affiliation:
Queen's University and ETH Zürich
*
∗ Postal address: Institute for Integrative Biology, ETH Zürich, CHN H.74, Universitätsstrasse 16, Zürich, CH-8092, Switzerland. Email address: helen.alexander@env.ethz.ch
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Abstract

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In this paper we present novel results for discrete-time and Markovian continuous-time multitype branching processes. As a population develops, we are interested in the waiting time until a particular type of interest (such as an escape mutant) appears, and in how the distribution of individuals depends on whether this type has yet appeared. Specifically, both forward and backward equations for the distribution of type-specific population sizes over time, conditioned on the nonappearance of one or more particular types, are derived. In tandem, equations for the probability that one or more particular types have not yet appeared are also derived. Brief examples illustrate numerical methods and potential applications of these results in evolutionary biology and epidemiology.

Keywords

Multitype branching processprobability generating functionwaiting timeescape mutantevolutionary biologyepidemiology

MSC classification

Primary: 60G99: None of the above, but in this section
Secondary: 92D15: Problems related to evolution 92D30: Epidemiology
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 3 , September 2013 , pp. 692 - 718
Copyright
© Applied Probability Trust 

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