Simulate disease spread using the SIR compartmental model. Adjust R₀, vaccination coverage, and intervention timing to explore herd immunity thresholds and flatten the curve.
The SIR model divides population N into Susceptible (S), Infectious (I), and Recovered (R). The governing ODEs are: dS/dt = −β·S·I/N; dI/dt = β·S·I/N − γ·I; dR/dt = γ·I. The basic reproduction number R₀ = β/γ determines how many people one case infects on average. Herd immunity threshold is p_c = 1 − 1/R₀.
Vaccination reduces the susceptible pool: S(0) = (1−v)·N. When vaccination coverage v exceeds the herd immunity threshold p_c, the epidemic cannot grow. For COVID-19 (R₀≈3), ~67% vaccination is needed; for measles (R₀≈15), ~93% is required. The epidemic peaks when S = N/R₀.
Lockdown and social distancing reduce the transmission rate β (halving it cuts R₀ in half). The Intervention Day slider simulates non-pharmaceutical interventions starting at day d. Earlier interventions delay and reduce the peak but don't eliminate the epidemic unless combined with high vaccination.