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The disease-free equilibrium is $(S,I)=(N^{\ast },0)$. Forthe Jacobian matrix there isBecause this matrix is triangular, its eigenvalue are its diagonal entries:The second eigenvalue is positive exactly whenUnder this condition the linear stability analysis gives an unstable disease-free equilibrium, agreeing with the epidemic invasion threshold.

This is the SIR model with waning immunity. The mass-action interaction term $\beta IS$ transfers people from the susceptible compartment to the infected compartment, $\nu I$ transfers infectives to the recovered compartment, and $fR$ returns recovered people to susceptibility as immunity wanes. Adding the equations givesso the total population $N$ is conserved.

When $f=0$ and infection is rare, $S\simeq N$ andThus $I$ initially decays if $\beta N<\nu$. Equivalently, the basic reproduction number is ${\mathcal{R}}_0=\beta N/\nu <1$, so the epidemic invasion threshold is not crossed and no epidemic occurs.