★★☆ Moderate

💉 Herd Immunity

When enough people are immune, an infectious disease can no longer spread — even unvaccinated individuals are protected. The herd immunity threshold is p_c = 1 − 1/R₀. Higher R₀ demands higher vaccine coverage. Explore how vaccination halts epidemics with the SIRV model.

❌ Herd immunity NOT reached — epidemic possible

R₀ (basic reproduction): 3.0 Vaccination %: 50% Infectious days: 7

R₀: 3.0 Threshold p_c: 66.7% Vaccinated: 50% Peak infected: —% Final immune: —%

💉 How Herd Immunity Works

Each infected person infects on average R₀ others in a fully susceptible population. If a fraction p is already immune, the effective reproduction number drops to Rₑ = R₀(1−p). The epidemic shrinks when Rₑ < 1:

p_c = 1 − 1/R₀ (herd immunity threshold)

Measles (R₀ ≈ 12–18) requires ~95% coverage. COVID-19 (R₀ ≈ 2–4) needs ~50–75%. Influenza (R₀ ≈ 1.3–2) needs only ~25–50%. The SIRV model splits the population into Susceptible, Infected, Recovered, and Vaccinated compartments, tracking flows between them over time.