Asymptotic frequency of shapes in supercritical branching trees

TL;DR

This paper develops a mathematical framework to determine the long-term frequency of specific shapes in supercritical branching trees, with applications to disease transmission modeling.

Contribution

It introduces a new class of characteristic functions and a formula for asymptotic shape frequencies in general branching processes.

Findings

Derived a formula for shape frequency limits in general trees

Applied the formula to birth-death models with constant and variable rates

Used Jumping Chronological Contour Process to address computational challenges

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 non-constant death rate.