In a town of 300 000 residents, a viral disease has a basic reproduction number $R_0=12$. A vaccine that is 80 % effective has been administered to 60 % of the population. Assuming random mixing, calculate the additional percentage of people that must already be immune (through prior infection or other means) to achieve herd immunity.

Step-by-step solution

1. Compute the herd‑immunity threshold: $H=1-\frac{1}{R_0}=1-\frac{1}{12}=\frac{11}{12}\approx 0.9167$ (91.67%).
2. Calculate the effective immune fraction contributed by vaccination: $V=\text{coverage}\times \text{efficacy}=0.60\times 0.80=0.48$ (48%).
3. Let $x$ be the fraction already immune. The total immune fraction needed is $V+x\ge H$.
4. Set up the inequality: $0.48+x\ge 0.9167$ and solve for $x$: $x\ge 0.9167-0.48=0.4367$.
5. Convert to percentage: $0.4367\times 100\approx 43.7\%$, i.e., about 44 % of the population must already be immune.

Concept tested: Herd immunity calculation with vaccine efficacy

Published 7 October 2026 · How this solution was prepared and checked