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Herd immunity occurs when a sufficient fraction of a population is immune that an infectious agent can no longer sustain transmission, protecting even unimmunized individuals. The herd immunity threshold (HIT) equals 1 − 1/R₀, where R₀ is the basic reproduction number—the mean number of secondary cases one infected person generates in a fully susceptible population. Highly contagious pathogens (e.g., measles, R₀ ≈ 12–18) require >90% coverage to achieve herd protection. Vaccination programs must account for vaccine efficacy, coverage heterogeneity, waning immunity, and population subgroups with lower uptake that can sustain pockets of transmission.
Calculate the HIT for several pathogens with known R₀ values and compare to actual vaccination coverage rates. Discuss how heterogeneous mixing (clustering of unvaccinated individuals) can allow outbreaks below the theoretical HIT.
To understand herd immunity, start with the basic reproduction number R₀—a concept from your population ecology prerequisites. R₀ measures how many new infections one case generates in a fully susceptible population. If R₀ = 3 (as with polio), each case infects three others on average, and the disease spreads exponentially. But if a fraction of the population is already immune, some of those three potential contacts cannot be infected. When the fraction immune is large enough, the average infected person generates fewer than one new case—meaning chains of transmission die out rather than propagate.
The herd immunity threshold formula HIT = 1 − 1/R₀ captures this precisely. For polio (R₀ ≈ 3), you need roughly 67% immune; for measles (R₀ ≈ 15), you need roughly 93%. The math explains why measles vaccination programs are so unforgiving of coverage gaps: even a few percentage points below 93% leaves enough susceptibles for outbreaks to sustain themselves. This is also why the goal of vaccination programs is not just to protect individuals but to push effective reproduction number Rₑ below 1 across the whole population—at which point even unvaccinated individuals (the immunocompromised, newborns, those with medical contraindications) are protected by the barrier of immune people around them.
Real populations, however, do not mix randomly. The HIT formula assumes that every susceptible person has an equal probability of encountering any infected person—a homogeneous mixing assumption that is rarely true. Unvaccinated individuals often cluster: in communities with shared vaccine skepticism, in close-knit religious groups, in geographic areas with poor healthcare access. These clusters can sustain local outbreaks even when national coverage exceeds the theoretical HIT. This is why public health surveillance tracks not just aggregate coverage but its distribution.
Vaccine programs must also contend with imperfect vaccines (efficacy < 100%), waning immunity over time, and the difference between infection-blocking and disease-blocking protection. A vaccine that is 90% efficacious requires higher population coverage than the HIT formula implies, because only 90% of vaccinated individuals become immune. These considerations explain why achieving durable herd protection requires not just reaching a coverage target once, but sustaining it across birth cohorts and maintaining booster programs for vaccines with waning immunity.