R₀, Herd Immunity Threshold & Vaccination Coverage Calculator

How it works

HIT = 1 − 1/R₀; coverage = HIT ÷ vaccine effectiveness; R_eff = R₀ × (1 − immune fraction); final size z = 1 − e^(−R₀ z); R ≈ 1 + (ln 2 ÷ doubling time) × serial interval

The basic reproduction number is the average number of people one case infects in a fully susceptible population; an epidemic grows while each case replaces itself more than once, so transmission stops being self-sustaining once a fraction 1 − 1/R₀ of contacts are immune — the herd immunity threshold, 50% for R₀ = 2 and 94% for measles. Because vaccines are imperfect, the coverage required is the threshold divided by effectiveness, which is why measles needs 95% two-dose coverage and why a 50%-effective vaccine cannot by itself stop a virus with R₀ of 3. The effective reproduction number applies the same logic to a partly immune population, and the final-size equation shows why unchecked epidemics overshoot the threshold: infections continue while susceptibles remain, so an R₀ of 3 infects 94% of a population, not 67%.

Worked example

A disease with R₀ = 6 has a herd immunity threshold of 83%; with a 90%-effective vaccine that needs 93% coverage. At 75% coverage plus 5% naturally immune, about 69% of the population is protected and R_eff ≈ 1.85 — still above 1, with an expected attack rate around 72% among those remaining susceptible. Unchecked, R₀ = 6 would infect about 99.7% of a naive population.

Frequently asked questions