SIR model with demography and permanent immunity
= SIR model with demography and permanent immunity
{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 $N_c=(\mu+\nu)/\beta$.