Critical behaviour of the partner model

Abstract

We consider a stochastic model of infection spread incorporating monogamous partnership dynamics. In [Ann. Appl. Probab. 26 (2016) 1297–1328], 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.