The Herd Immunity Threshold
Herd immunity is the epidemiological tipping point where enough individuals are immune that an infectious disease can no longer sustain transmission. The critical threshold is elegantly simple: p_c = 1 - 1/R₀. For measles with R₀ around 15, this means 93% of the population must be immune — explaining why even small drops in vaccination coverage trigger outbreaks. For less transmissible diseases, the bar is lower but still demanding.
Vaccine Efficacy as a Multiplier
No vaccine is 100% effective. When vaccine efficacy (VE) is less than perfect, the required vaccination coverage increases proportionally. If a disease needs 67% immunity and the vaccine is 90% effective, you must vaccinate 67%/0.9 = 74% of the population. This amplification effect means that modest differences in vaccine efficacy translate into large differences in the logistics and cost of achieving herd immunity.
Waning Immunity and Booster Campaigns
The simple herd immunity calculation assumes permanent immunity, but many vaccines and natural infections provide protection that wanes over months to years. When immunity wanes, the effective immune fraction decreases and R_effective rises back above 1. This creates a dynamic equilibrium where continuous vaccination or periodic booster campaigns are needed to maintain the immune fraction above the herd immunity threshold — as seen with pertussis, influenza, and COVID-19.
Interactive Threshold Explorer
This simulation visualizes the herd immunity calculation across a range of diseases. Adjust R₀ to move between influenza (R₀ ~ 1.5), COVID-19 (R₀ ~ 3), and measles (R₀ ~ 15). Reduce vaccine efficacy to see coverage requirements jump. The R_effective gauge shows whether the current coverage achieves herd protection, with a clear red/green indicator for outbreak risk.