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On mutations in the branching model for multitype populations

Part of: Markov processes

Published online by Cambridge University Press:  26 July 2018

Loïc Chaumont  and
Thi Ngoc Anh Nguyen
Loïc Chaumont*
Affiliation:
Université d'Angers
Thi Ngoc Anh Nguyen*
Affiliation:
Université d'Angers
*
* Postal address: LAREMA – UMR CNRS 6093, Université d'Angers, 2 bd Lavoisier, 49045 Angers cedex 01, France.
* Postal address: LAREMA – UMR CNRS 6093, Université d'Angers, 2 bd Lavoisier, 49045 Angers cedex 01, France.
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Abstract

The forest of mutations associated to a multitype branching forest is obtained by merging together all vertices in each of its clusters and by preserving connections between them. (Here, by cluster, we mean a maximal connected component of the forest in which all vertices have the same type.) We first show that the forest of mutations of any multitype branching forest is itself a branching forest. Then we give its progeny distribution and we describe some of its crucial properties in terms of the initial progeny distribution. We also obtain the limiting behaviour of the number of mutations both when the total number of individuals tends to ∞ and when the number of roots tends to ∞. The continuous-time case is then investigated by considering multitype branching forests with edge lengths. When mutations are nonreversible, we give a representation of their emergence times which allows us to describe the asymptotic behaviour of the latter, under certain conditions on the mutation rates. These results have potential relevance for emergence of mutations in population cells, particularly for genetic evolution of cancer or development of infectious diseases.

Keywords

Multitype branching forestmutationforest of mutationsemergence time

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 2 , June 2018 , pp. 543 - 564
Copyright
Copyright © Applied Probability Trust 2018 

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