SIRS model — waning immunity drives oscillations toward endemic state
The SIRS model adds waning immunity (R→S at rate ω) to SIR: dS/dt = −βSI + ωR; dI/dt = βSI − γI; dR/dt = γI − ωR. Endemic equilibrium: I* = (β/γ−1)·ω/(β+ω) when R₀ = β/γ > 1. The system oscillates toward I* with damped spirals in phase space, driven by the interplay between transmission, recovery, and immunity loss.