Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 207 / 3 / c / Solution

Whenever $I>0$, divide the first SIR model equation by the recovery equation:The SIR susceptible-recovered identity follows by integrating from $R=0,S=N$, giving susceptibles as a function of recovered individuals:Population conservation then gives the infectious count:These identities extend continuously to the limiting state where the infectious count vanishes.