= Solution
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 gives
$$
\frac d{dt}(S+I+R)=0,
$$
so the total population $N$ is conserved.
When $f=0$ and infection is rare, $S\simeq N$ and
$$
I'=(\beta S-\nu)I\simeq(\beta N-\nu)I.
$$
\b[Thus $I$ 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.]
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