= Solution
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.
Back to article page
