Endemic equilibrium of the SIR model with waning immunity

When $\beta N>\nu$ and $f>0$, the positive equilibrium isWriting $a=\beta I^{\ast }>0$, the reduced $(S,I)$ Jacobian has characteristic polynomialBoth 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.