Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 6B (Mathematical Biology)

Adding the three equations of the SIR model with waning immunity givesHence conservation of population holds: $N=S+I+R$ is an integral of motion.

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.

Eliminate $R=N-S-I$ to obtain the two-dimensional autonomous differential equationIts Jacobian matrix at the endemic equilibrium isSince $I^{\ast }>0$,The trace-determinant stability criterion shows that both eigenvalues have negative real part. The Endemic equilibrium of the SIR model with waning immunity is therefore locally asymptotically stable.