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

For $\beta >0$ and positive initial populations, summing the coordinates of the attracting equilibrium point gives the steady total populationIts minimum is $1/(1+\beta )$ at $d=\beta /(1+\beta )$. Small mortality leaves a fully infected population with only modest demographic loss. Raising mortality first reduces that population; after the threshold it suppresses infection sufficiently for the uninfected population to recover. Beyond the invasion threshold the disease cannot persist.