 Solution

Let $i=I/N$ be the infectious proportion in a closed population. With homogeneous frequency-dependent transmission rate $\beta$ and recovery rate $\gamma >0$, the SIS model isThe basic reproduction number is the expected number of secondary infections produced by a typical newly infected individual in an otherwise susceptible population under the specified contact conditions. Here the infectious duration has mean $1/\gamma$, so $R_0=\beta /\gamma$.

For $R_0>1$ the attracting endemic equilibrium has $i_{\ast }=1-\gamma /\beta$. The homogeneous SIS susceptible fraction is thereforeFor $R_0\le 1$ the physical equilibrium is disease-free, $i_{\ast }=0$, $s_{\ast }=1$. At equality approach to zero is algebraic. Thus $s_{\ast }=1/R_0$ is an endemic relation, not a formula demanding more than one susceptible person per person below the invasion threshold.