Approximating time to extinction for endemic infection models

TL;DR

This paper evaluates various methods for estimating the time to extinction in infection models, highlighting their accuracy, limitations, and applicability across different scenarios in stochastic epidemiology.

Contribution

It systematically compares multiple approximation techniques for infection extinction times, providing insights into their performance and suitability in different epidemiological contexts.

Findings

Branching process approximation works well for small initial outbreaks.

Hamiltonian methods accurately predict asymptotic behavior for large populations.

Diffusion approximations vary in accuracy and reliability.

Abstract

Approximating the time to extinction of infection is an important problem in infection modelling. A variety of different approaches have been proposed in the literature. We study the performance of a number of such methods, and characterize their performance in terms of simplicity, accuracy, and generality. To this end, we consider first the classic stochastic susceptible-infected-susceptible (SIS) model, and then a multi-dimensional generalization of this which allows for Erlang distributed infectious periods. We find that (i) for a below-threshold infection initiated by a small number of infected individuals, approximation via a linear branching process works well; (ii) for an above-threshold infection initiated at endemic equilibrium, methods from Hamiltonian statistical mechanics yield correct asymptotic behaviour as population size becomes large; (iii) the widely-used Ornstein-Uhlenbeck diffusion approximation gives a very poor approximation, but may retain some value for qualitative comparisons in certain cases; (iv) a more detailed diffusion approximation can give good numerical approximation in certain circumstances, but does not provide correct large population asymptotic behaviour, and cannot be relied upon without some form of external validation (eg simulation studies).