SI model

The SI model divides a closed population into susceptible and permanently infectious compartments. Under frequency-dependent homogeneous mixing,and conservation of population gives $S+I=N$.

If $S(0)=N\theta /(1+\theta )$ and $I(0)=N/(1+\theta )$, thenAfter centering time at the incidence peak $t_{\ast }={\beta }^{-1}\log \theta$, the incidence curve is $N\beta /[4{\cosh }^2\{\beta (t-t_{\ast })/2\}]$.