In a city of 400 000 people, a disease has a basic reproduction number $R_0=8$. A vaccine that is 90 % effective is given to 65 % 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. Step 1: Calculate the effective vaccination rate: $0.65\times 0.90=0.585$ (58.5 % of the population is effectively immunised).
2. Step 2: Determine the herd immunity threshold: $1-\frac{1}{R_0}=1-\frac{1}{8}=0.875$ (87.5 % of the population must be immune).
3. Step 3: Subtract the effective vaccination immunity from the threshold: $0.875-0.585=0.290$.
4. Step 4: Convert to percentage: $0.290\times 100=29$. This is the additional immunity required.

Concept tested: Herd immunity calculation with vaccine efficacy

Published 28 September 2026 · How this solution was prepared and checked