Logistic solution of the SI model
= Logistic solution of the SI model
{c}
If $S(0)=N\theta/(1+\theta)$ and $I(0)=N/(1+\theta)$, then
$$
I(t)=\frac{N}{1+\theta e^{-\beta t}},
\qquad
\dot I(t)=\frac{N\beta\theta e^{-\beta t}}{(1+\theta e^{-\beta t})^2}.
$$
After centering time at the incidence peak $t_*=\beta^{-1}\log\theta$, the incidence curve is $N\beta/[4\cosh^2\{\beta(t-t_*)/2\}]$.