Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution

TL;DR

This paper introduces an efficient, error-controlled algorithm for calculating finite-time transition probabilities in general birth-death processes, with applications across ecology, genetics, and evolution, improving upon existing methods.

Contribution

The authors develop a novel, robust algorithm based on continued fractions to accurately compute transition probabilities in general birth-death processes, addressing a key computational challenge.

Findings

Algorithm outperforms previous methods in accuracy and efficiency

Method agrees with known solutions in benchmark cases

Applications demonstrate relevance in ecology, genetics, and evolution

Abstract

A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with $n$ current particles, a new particle is born with instantaneous rate $\lambda_n$ and a particle dies with instantaneous rate $\mu_n$. Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth-death process with arbitrary birth and death rates. In this paper, we first revisit the theory of continued fractions to obtain expressions for the Laplace transforms of these transition probabilities and make explicit an important derivation connecting transition probabilities and continued fractions. We then develop an efficient algorithm for computing these probabilities that analyzes the error associated with approximations in the method. We demonstrate that this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities. Finally, we apply our novel method to several important problems in ecology, evolution, and genetics.