SIRS Epidemic Oscillations

Waning immunity (S→I→R→S) creates sustained oscillations or damped spirals. The endemic equilibrium can be stable or undergo a Hopf bifurcation as parameters change.

S: —%

I: —%

R: —%

R₀: —

β (transmission)0.50

γ (recovery)0.10

δ (waning)0.050

ν (birth/death)0.010

Speed4

SIRS dynamics: dS/dt = νN − βSI/N + δR − νS, dI/dt = βSI/N − γI − νI, dR/dt = γI − δR − νR. The endemic equilibrium exists when R₀ = β/(γ+ν) > 1. Waning immunity (δ > 0) enables recurrent epidemic waves — a key mechanism behind seasonal influenza cycling.