SIRS and SEIRS models
SIRS and SEIRS models extend SIR and SEIR models by allowing immunity to wane.
\[S\to I\to R\to S,\qquad S\to E\to I\to R\to S.\]Waning immunity
If immunity is lost at per-capita rate \(\omega\), a term \(\omega R\) transfers individuals from the recovered class back to susceptibility.
For an SIRS model,
\[\frac{dR}{dt}=\gamma I-\omega R,\qquad \frac{dS}{dt}=-\beta\frac{SI}{N}+\omega R.\]Biological consequence
Because susceptible individuals are replenished, infection may persist or recur rather than producing only one epidemic wave.
Key idea. SIRS-type models are useful when immunity is temporary rather than permanent.