Epidemic Vaccination Threshold & Herd Immunity

Explore how vaccination coverage prevents epidemic spread. The herd immunity threshold p_c = 1 − 1/R₀ shows the fraction that must be vaccinated to stop transmission.

Basic Reproduction R₀ 4.0

Vaccination Coverage v 30%

Recovery Rate γ (days⁻¹) 0.15

R₀ (basic)4.0

R_eff (effective)—

Herd threshold p_c—

Peak infected %—

Total attacked %—

Set parameters & run

SIR model: dS/dt = −βSI/N, dI/dt = βSI/N − γI, dR/dt = γI
where β = R₀·γ. With vaccination fraction v: S₀ = (1−v)N, so R_eff = R₀(1−v).

Herd immunity: p_c = 1 − 1/R₀. If v > p_c, epidemic dies out without spreading (R_eff < 1). Examples: measles R₀≈15 (threshold 93%), COVID R₀≈4–6 (threshold 75–83%), flu R₀≈1.3 (threshold 23%).