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

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: there is no epidemic outbreak.