Critical Behaviour of the Partner Model

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.