 Cooperative SIS coinfection threshold

For two otherwise symmetric SIS models with recovery rate $\gamma$, primary transmission $\beta$ and secondary susceptibility multiplier $q$, split hosts into uninfected, singly infected and doubly infected classes. The displayed conditions give stable disease-free and joint endemic states, with an intermediate saddle: a strong Allee effect. The endemic susceptible fraction solves $(q-1)S^2-qS+{\mathcal{R}}_0^{-2}=0$. The smaller root is the stable high-prevalence state. Coinfected hosts recover each infection independently, so recovery from coinfection leads to a single-infection class rather than directly to complete susceptibility.