Conjecture on extinction of the disease in the stochastic SIS model

Let $I(t)$ denote the infected population in the stochastic SIS model, with parameters $\beta$, $\mu$, $\gamma$, $N$, and noise intensity $\sigma$, and define the stochastic basic reproduction number

$$R_0^S:=\frac{\beta N}{\mu+\gamma}-\frac{\sigma^2N^2}{2(\mu+\gamma)}.$$

Extinction conjecture. If $R_0^S<1$ and

$$\frac{\beta^2}{2(\mu+\gamma)}\geqslant\sigma^2>\frac{\beta}{N},$$

then the disease will die out with probability one. This fills the parameter regime not covered by the previously established extinction conditions for the stochastic SIS model; the conjecture concerns almost-sure extinction in the intermediate noise-intensity range.