Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 3 / 11E / b / Solution

At the disease-free equilibrium point $(1,0)$, the linearization of a dynamical system of the infected equation is $\dot{y}=(\beta -d)y$. Thus exponential invasion occurs exactly when $\boxed{\beta >d}$. At equality there is no positive linear growth rate; the nonlinear boundary dynamics described below do not give sustained invasion.