In a population of 200,000, a contagious disease has basic reproduction number $R_0=6$. A vaccine with 70 % efficacy is given to 80 % of the population. Assuming random mixing, calculate the additional percentage of the population that must already be immune (through prior infection or other means) to achieve herd immunity.

Step-by-step solution

1. Calculate herd immunity threshold: $1-\frac{1}{R_0}=1-\frac{1}{6}=\frac{5}{6}\approx 0.8333$.
2. Determine immunity from vaccination: $0.8\times 0.7=0.56$ of the population.
3. Find additional immunity required: $0.8333-0.56=0.2733$, i.e. 27.33 %.
4. Round to nearest percent: 27 %.

Concept tested: Herd immunity calculation

Published 13 October 2026 · How this solution was prepared and checked