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

The same first integral, evaluated initially and after the epidemic when $I(\infty )=0$, givesWhen $I(0)$ 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