December 2016 Asymptotic frequency of shapes in supercritical branching trees
Giacomo Plazzotta, Caroline Colijn
Author Affiliations +
J. Appl. Probab. 53(4): 1143-1155 (December 2016).

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.

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Giacomo Plazzotta. Caroline Colijn. "Asymptotic frequency of shapes in supercritical branching trees." J. Appl. Probab. 53 (4) 1143 - 1155, December 2016.

Information

Published: December 2016
First available in Project Euclid: 7 December 2016

zbMATH: 1356.60134
MathSciNet: MR3581247

Subjects:
Primary: 60J85
Secondary: 92D30

Keywords: Basic reproduction number , branching process , shape frequency

Rights: Copyright © 2016 Applied Probability Trust

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Vol.53 • No. 4 • December 2016
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