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Asymptotic frequency of shapes in supercritical branching trees

Part of: Markov processes

Published online by Cambridge University Press:  09 December 2016

Giacomo Plazzotta  and
Caroline Colijn
Giacomo Plazzotta*
Affiliation:
Imperial College London
Caroline Colijn*
Affiliation:
Imperial College London
*
* Postal address: Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK.
* Postal address: Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK.
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Abstract

The shapes of branching trees have been linked to disease transmission patterns. In this paper we use the general Crump‒Mode‒Jagers branching process to model an outbreak of an infectious disease under mild assumptions. Introducing a new class of characteristic functions, we are able to derive a formula for the limit of the frequency of the occurrences of a given shape in a general tree. The computational challenges concerning the evaluation of this formula are in part overcome using the jumping chronological contour process. We apply the formula to derive the limit of the frequency of cherries, pitchforks, and double cherries in the constant-rate birth‒death model, and the frequency of cherries under a nonconstant death rate.

Keywords

Branching processshape frequencybasic reproduction number

MSC classification

Primary: 60J85: Applications of branching processes
Secondary: 92D30: Epidemiology
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1143 - 1155
Copyright
Copyright © Applied Probability Trust 2016 

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