An epidemiologist models a city’s vaccination rollout: 60% of the population must be immune to achieve herd immunity. The city has 2.4 million people. Vaccination efficiency is 90% per dose, meaning only 90% of administered doses confer immunity. How many doses must be administered to reach herd immunity? - Malaeb

Understanding the Context

Herd immunity occurs when enough of a population is immune to an infectious disease—either through vaccination or prior infection—so the spread becomes unsustainable. For a given disease, the required percentage of immune individuals depends on the pathogen’s transmissibility, measured by the basic reproduction number (R₀). However, in vaccination planning, public health experts often assume a target herd immunity threshold of 60%.

This means 60% of the total population must be immune to significantly reduce community transmission.

Adjusting for Vaccine Efficiency

In real-world vaccination campaigns, vaccines are not 100% effective. Here, the model considers a vaccine efficacy (or efficiency) of 90% per dose. This means that only 90% of people who receive the vaccine actually develop lasting immunity.

Key Insights

Let’s define the population:
City population = 2,400,000 people
Herd immunity threshold = 60% → Required immune individuals = 0.60 × 2,400,000 = 1,440,000

Because each dose is only 90% effective, the number of people who gain immunity from a single administered dose is reduced.

Calculating Total Doses Required

Let x be the total number of vaccine doses to be administered.

Only 90% of these doses lead to immunity:
Immunity from x doses = 0.90 × x

Final Thoughts

We set this equal to the 1,440,000 immune individuals needed:
0.90x = 1,440,000

Solving for x:
x = 1,440,000 / 0.90
x = 1,600,000

Conclusion

To achieve herd immunity in this city—where 60% of the 2.4 million residents must be immune—1.6 million vaccine doses must be administered, accounting for a 90% vaccination efficacy rate.

This modeling approach helps public health officials plan logistics, allocate resources, and set realistic goals for immunization campaigns, ensuring efficient use of available vaccines and ultimately protecting vulnerable populations from outbreaks.