Critical Behaviour of the Partner Model

TL;DR

This paper analyzes a stochastic infection model with partnership dynamics at the critical threshold R_0=1, showing the infection's extinction time scales as the square root of the population size, which is proven to be optimal.

Contribution

It provides a precise characterization of the infection's extinction time at the critical threshold R_0=1, extending previous results for subcritical and supercritical cases.

Findings

Infection dies out within O(√N) time at criticality.

The O(√N) estimate is sharp and cannot be improved.

Results extend understanding of infection dynamics at the critical point.

Abstract

We consider a stochastic model of infection spread incorporating monogamous partnership dynamics. In previous work a basic reproduction number $R_0$ is defined with the property that if $R_0<1$ the infection dies out within $O(\log N)$ units of time, while if $R_0>1$ the infection survives for at least $e^{\gamma N}$ units of time, for some $\gamma>0$. Here we consider the critical case $R_0=1$ and show that the infection dies out within $O(\sqrt{N})$ units of time, and moreover that this estimate is sharp.