In a city of 250,000 people, a measles outbreak has a basic reproduction number $R_0=18$. A vaccine with 95 % efficacy is given to 85 % of the population. Assuming random mixing, calculate the additional percentage of the population that must already be immune (through prior infection or other measures) to achieve herd immunity.

Step-by-step solution

1. Calculate herd immunity threshold: $H=1-\frac{1}{R_0}=1-\frac{1}{18}=\frac{17}{18}\approx 94.44$.
2. Determine effective vaccination coverage: $V_{eff}=0.85\times 0.95=0.8075,(80.75$.
3. Subtract to get additional immunity needed: $94.44$.
4. Thus about 13.7 % of the population must already be immune.

Concept tested: Herd immunity calculation

Published 4 October 2026 · How this solution was prepared and checked