# Evolving phylogenies of trait-dependent branching with mutation and competition. Part I: Existence

## TL;DR

This paper introduces a mathematical model for virus phylogenies considering trait-dependent branching, mutation, and competition, and establishes the existence of evolving phylogenies in a large population regime.

## Contribution

It develops a new type-dependent branching model incorporating mutation and competition, and proves the existence of its evolving phylogenies in the large population limit.

## Key findings

- Phylogenies are generally not ultra-metric due to heterogeneity.
- Established tightness in the space of marked metric measure spaces.
- Characterized the limit as a solution to a martingale problem.

## Abstract

We propose a type-dependent branching model with mutation and competition for modeling phylogenies of a virus population. The competition kernel depends for any two virus particles on the particles' types, the total mass of the population as well as genetic information available through the number of nucleotide substitutions separating the virus particles. We consider the evolving phylogenies of this individual based model in the huge population, short reproduction time and frequent mutation regime, and show tightness in the state space of marked metric measure spaces. Due to heterogeneity in the natural branching rates, the phylogenies are in general not ultra-metric. We therefore develop new techniques for verifying a compact containment condition. Finally, we characterize the limit as a solution of a martingale problem.