Solution
= Solution
Using $S=N-I$ reduces the system to the <logistic differential equation>
$$
\dot I=\beta I(1-I/N).
$$
Separation of variables and the initial values give
$$
\log\frac{I}{N-I}=\beta t-\log\theta,
\qquad
I(t)=\frac{N}{1+\theta e^{-\beta t}}.
$$
Differentiation yields the incidence
$$
\Lambda(t)=\dot I(t)
=\frac{\theta N\beta e^{-\beta t}}
{(1+\theta e^{-\beta t})^2}.
$$