Critical case stochastic phylogenetic tree model via the Laplace transform

TL;DR

This paper studies a critical birth-and-death model for phylogenetic trees, using Laplace transforms to analyze its asymptotic behavior and correct previous proofs in the critical case.

Contribution

It introduces a corrected analysis of the critical case in a birth-and-death phylogenetic model using Laplace transforms, addressing gaps in prior work.

Findings

Corrected the proof of the critical case in the model

Analyzed asymptotic behavior using Laplace transform

Validated the model's applicability to influenza and HIV trees

Abstract

Birth-and-death models are now a common mathematical tool to describe branching patterns observed in real-world phylogenetic trees. Liggett and Schinazi (2009) is one such example. The authors propose a simple birth-and-death model that is compatible with phylogenetic trees of both influenza and HIV, depending on the birth rate parameter. An interesting special case of this model is the critical case where the birth rate equals the death rate. This is a non-trivial situation and to study its asymptotic behaviour we employed the Laplace transform. With this we correct the proof of Liggett and Schinazi (2009) in the critical case.