The same first integral, evaluated initially and after the epidemic when $I(\infty)=0$, gives
$$
\log\frac{S(0)}{S(\infty)}
=\frac\beta\gamma R(\infty).
$$
When $I(0)$ is negligible and $R(0)=0$, <conservation of population> gives $S(\infty)\simeq S(0)-R(\infty)$. Since $\beta/\gamma=R_0/S(0)$, this is equivalent to the <final size relation for an epidemic>
$$
R(\infty)\simeq
\frac{S(0)}{R_0}
\log\frac{S(0)}{S(0)-R(\infty)}.
$$

Solution

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

 Solution

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