Since $S\le N$, the infected population satisfies $\dot I=(rS-a)I\le(rN-a)I$. Integration gives $I(t)\le I(0)e^{(rN-a)t}$. Hence $rN<a$ forces the infected population to decrease exponentially from the outset: \b[there is no epidemic outbreak].

Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 6B / a / Solution

 Solution (source code)