Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 6B / ii / Solution

At an endemic equilibrium, $I^{\ast }>0$, so $0=\dot{I}=I^{\ast }(\beta S^{\ast }-\gamma )$ forcesThe equation $\dot{R}=0$ gives $R^{\ast }=\gamma I^{\ast }/\alpha$. The conservation law $S^{\ast }+I^{\ast }+R^{\ast }=N$ therefore yieldsA positive endemic state exists exactly when $\beta N>\gamma$, equivalently when the basic reproduction number ${\mathcal{R}}_0=\beta N/\gamma$ exceeds one.