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
$$
\Lambda(t)=\dot I(t)
=\frac{\theta N\beta e^{-\beta t}}
{(1+\theta e^{-\beta t})^2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-207/3/c/solution