Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 6A / a / Solution

{c}

With susceptible, infective, and immune populations $X,Y,Z$, equal per-capita birth and death rate $\mu$, mass-action transmission coefficient $\beta$, and recovery rate $\nu$, the equations are
$$
X'=\mu N-\beta XY-\mu X,
\qquad Y'=\beta XY-(\mu+\nu)Y,
\qquad Z'=\nu Y-\mu Z.
$$
The conserved population is $N=X+Y+Z$, and the threshold population is $N_c=(\mu+\nu)/\beta$.


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 Incoming links: SIR model with demography and permanent immunity