## Journal of Applied Probability

### Asymptotic frequency of shapes in supercritical branching trees

#### 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.

#### Article information

Source
J. Appl. Probab., Volume 53, Number 4 (2016), 1143-1155.

Dates
First available in Project Euclid: 7 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.jap/1481132842

Mathematical Reviews number (MathSciNet)
MR3581247

Zentralblatt MATH identifier
1356.60134

Subjects
Primary: 60J85: Applications of branching processes [See also 92Dxx]
Secondary: 92D30: Epidemiology

#### Citation

Plazzotta, Giacomo; Colijn, Caroline. Asymptotic frequency of shapes in supercritical branching trees. J. Appl. Probab. 53 (2016), no. 4, 1143--1155. https://projecteuclid.org/euclid.jap/1481132842

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