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The infection term $rIS$ is a mass-action contact rate: each infection removes one susceptible individual and adds one infected individual. Recovery at rate $aI$ transfers individuals from the infected class to the recovered class. Loss of immunity at rate $pR$ transfers recovered individuals back to the susceptible class. Thus these equations describe the SIR model with waning immunity, also called the SIRS model. Adding the equations gives $\dot{S}+\dot{I}+\dot{R}=0$, so $S+I+R=N$. The nonnegative population region is invariant because the derivative of any population on its zero boundary is nonnegative.

Adding the three equations of the SIR model with waning immunity givesHence conservation of population holds: $N=S+I+R$ is an integral of motion.

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.