Periodic forcing of transmission rate creates annual, biennial, and chaotic epidemic cycles
The SIR model with seasonal forcing: β(t) = β₀(1 + ε·cos(2πt)), where t is in years. The basic reproduction number R₀ = β/γ determines epidemic threshold. With seasonal forcing, the system exhibits resonance phenomena: annual outbreaks for moderate ε, period-doubling to biennial cycles, and ultimately chaos via a Feigenbaum route as seasonality increases. This explains measles/influenza dynamics — childhood diseases often show 2-year cycles (London measles data 1944–1966 is a classic case). The bifurcation structure depends on both R₀ and ε.