Uniform sampling in a structured branching population

TL;DR

This paper studies the trait dynamics in a structured branching population, introducing an auxiliary process to describe trait evolution along a lineage, and demonstrates its role in sampling and modeling various growth-fragmentation processes.

Contribution

It explicitly characterizes the auxiliary process for trait dynamics, proves key formulas, and links the process to uniform sampling in large populations.

Findings

The auxiliary process precisely describes trait evolution along a lineage.

The paper establishes a Many-to-One formula and a forks version.

It applies the framework to three growth-fragmentation models.

Abstract

We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of the descendants at birth depends on the trait of the mother and on the number of descendants. In this article, we explicitly describe the penalized Markov process, named auxiliary process, corresponding to the dynamic of the trait along the spine by giving its associated infinitesimal generator. We prove a Many-to-One formula and a Many-to-One formula for forks. Furthermore, we prove that this auxiliary process characterizes exactly the process of the trait of a uniformly sampled individual in the large population approximation. We detail three examples of growth-fragmentation models: the linear growth model, the exponential growth model and the parasite infection model.