Field notes

An infection spreads when each case, on average, produces more than one new case. That average is the effective reproduction number, Re = R₀·(1 − p), where p is the immune fraction. The moment Re drops below 1, every chain of transmission peters out.

Setting Re = 1 gives the herd-immunity threshold Hc = 1 − 1/R₀. A more transmissible pathogen — a bigger R₀ — demands a higher immune fraction to stop it. Measles, with R₀ near 15, needs about 93% immunity; there is no single "the" threshold.

Above the line, even the unvaccinated are protected: most of the contacts an infectious person makes land on immune people, and every immune contact is a dead end. Protection is a property of the crowd, not the individual.

This chamber assumes a perfect vaccine (efficacy E = 1), so the immune fraction equals the coverage you set and the target line is exactly Hc. Real vaccines have E < 1, so the coverage actually needed is higher: Vc = Hc / E. Clustered gaps — pockets of unvaccinated people — can also let an outbreak burn even when the overall average clears the line.