Herd Immunity Threshold

The threshold is $H_c = 1 - 1/R_0$, and the reasoning is one line. If each infection produces $R_0$ new infections in a fully susceptible population, and only a fraction $s$ of contacts remain susceptible, then each infection produces $R_0 s$ new infections. Transmission fades when that drops below one, which happens once $s < 1/R_0$ — so the immune fraction must exceed $1 - 1/R_0$. At $R_0 = 12$, roughly measles, that is 91.7 %; at $R_0 = 3$ it is 67 %.

The attribution deserves correcting, because a specific wrong version of it circulates widely. Kermack and McKendrick's 1927 paper is the right citation for the threshold theorem — they proved that an epidemic requires the density of susceptibles to exceed a critical value, which is the mathematical ancestor of everything on this page. They did not write $R_0$, did not use the symbol, and did not define it; the formulation "$R_0 = \beta/\gamma$, Kermack and McKendrick 1927" is an anachronism attaching a later concept to an earlier paper. Delamater and colleagues' 2019 review in Emerging Infectious Diseases traces the actual provenance of $R_0$, which arrives through demography — Alfred Lotka's net reproduction rate — and reaches infectious disease work considerably later, with George MacDonald's malaria models in the 1950s doing much of the transfer.

The same review makes the more important point: $R_0$ is not a biological constant of a pathogen. It is a product of transmissibility, contact rate and infectious duration, so it depends on the population as much as on the microbe, and published values are estimates from particular settings with particular models and particular assumptions. The wide ranges quoted for measles — anything from 12 to 18 depending on the source — are not measurement error but genuine variation between populations, and they move the threshold by several percentage points.

Three assumptions in the formula are worth naming, because all three fail in the field. It assumes immunity is complete and lasting, when many vaccines are partially effective and some immunity wanes. It assumes random mixing, when real populations cluster — and a community with locally low coverage sustains an outbreak while national coverage sits comfortably above threshold, which is how measles returns to countries that had eliminated it. And it treats the threshold as a line to be crossed rather than a level to be held, when what actually protects a population is coverage maintained across birth cohorts for years. The number is a floor, computed under conditions that never quite hold, and it should be read as the minimum rather than the target.