= Endemic equilibrium of the SIR model with waning immunity
{c}
When $\beta N>\nu$ and $f>0$, the positive equilibrium is
$$
S^*=\frac\nu\beta,
\qquad
I^*=\frac{f(\beta N-\nu)}{\beta(f+\nu)},
\qquad
R^*=\frac{\nu(\beta N-\nu)}{\beta(f+\nu)}.
$$
Writing $a=\beta I^*>0$, the reduced $(S,I)$ Jacobian has characteristic polynomial
$$
\lambda^2+(f+a)\lambda+a(f+\nu).
$$
Both roots have negative real part. Its discriminant is negative for sufficiently small $f$ and positive for sufficiently large $f$, giving respectively a stable focus and a stable node.
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