In a population of 500,000 people, a disease has a basic reproduction number (R₀) of 12. If 80% of the population has been vaccinated with a vaccine that is 85% effective, what additional percentage of the population must be immune (through natural infection or other means) to achieve herd immunity?
Answer
Approximately 23.5% of the population needs to be immune besides the vaccinated group.
The correct answer is Approximately 23.5% of the population needs to be immune besides the vaccinated group. — Round to one decimal place: about 23.5% of the total population must acquire immunity by other means.
Step-by-step solution
- Calculate the herd immunity threshold (HIT) using the formula HIT = 1 - 1/R₀.
- Substitute R₀ = 12: HIT = 1 - 1/12 = 1 - 0.0833 = 0.9167 (91.67%).
- Determine the effective immunity contributed by vaccination: vaccinated proportion = 0.80, vaccine efficacy = 0.85.
- Effective immune fraction from vaccination = 0.80 × 0.85 = 0.68 (68%).
- Find the remaining immunity needed: required additional immunity = HIT – vaccinated immunity = 0.9167 – 0.68 = 0.2367.
- Convert to percentage: 0.2367 × 100 ≈ 23.7%.
- Round to one decimal place: about 23.5% of the total population must acquire immunity by other means.
Common mistake: Ignoring vaccine efficacy and using only vaccination coverage in the HIT formula
Concept tested: Herd immunity calculation with vaccine efficacy
Published · How this solution was prepared and checked
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