Herd Immunity Threshold Calculator

What Is Herd Immunity?

Herd immunity, also called population or community immunity, is reached when enough people are immune, through vaccination or prior infection, that transmission cannot sustain itself.
When enough people are immune, the disease can no longer spread efficiently and dies out, even protecting those who cannot be vaccinated (newborns, immunocompromised individuals).

The Basic Reproduction Number (R₀)

R₀ (pronounced “R-naught”) is the average number of people an infected person will infect in a completely susceptible population.

• R₀ < 1: The disease will die out on its own
• R₀ = 1: Each case replaces itself, so outbreaks neither grow nor shrink
• R₀ > 1: The disease will spread

Well-known R₀ values (approximate):

• Seasonal influenza: 1.2–1.4
• SARS-CoV-1: 2–3
• COVID-19 (original strain): 2–3; Omicron: around 8 in one review (Liu and Rocklöv, 2022), with wide uncertainty
• Polio: 5–7
• Mumps: 4–7
• Rubella: 5–7
• Smallpox: 5–7
• Measles: 12–18 (one of the highest known)

These are textbook ranges. Published estimates for any one disease spread much wider, because R₀ depends on how a population lives as much as on the germ: a 2017 review of measles studies (Guerra and colleagues) found values well outside 12 to 18 in both directions.

Herd Immunity Threshold (HIT) Formula

HIT = 1 − (1 / R₀)

This gives the minimum proportion of the population that must be immune to stop transmission.

Vaccination Coverage Needed

If the vaccine is not 100% effective, more people need to be vaccinated:
Vaccination coverage needed = HIT ÷ (vaccine efficacy / 100)

Efficacy here has to mean protection against catching and passing on the infection, not against getting seriously ill. The two can be far apart. The COVID-19 vaccines were above 90% effective against illness in their trials, and much less effective against infection once Omicron arrived, which is why they never produced herd immunity.

Worked Examples

An illustration with round numbers (R₀ = 5, a vaccine 90% effective against infection):
HIT = 1 − 1/5 = 80%
Coverage needed = 80% ÷ 0.90 = 88.9%

Measles (R₀ = 15, two MMR doses 97% effective, the CDC figure):
HIT = 1 − 1/15 = 93.3%
Coverage needed = 93.3% ÷ 0.97 = 96.2% (This is why the World Health Organization aims for 95% coverage with both doses)

Why the real number is always higher than this

This formula assumes a population that mixes at random and immunity that does not fade. Neither is true, so treat the result as a floor rather than a target:

• Immunity wanes over time for both vaccines and natural infection
• R₀ varies with behavior, population density, and variants
• Coverage has to be spread evenly. Clustered unvaccinated groups sustain local outbreaks even when the national figure looks fine, which is how measles keeps returning to countries that had eliminated it
• Some diseases have animal reservoirs, making true eradication impossible

The threshold is not the final toll

Reaching the threshold through infection instead of vaccination overshoots it. An epidemic already in full swing keeps infecting people after the threshold is crossed, so far more than 1 − 1/R₀ end up infected. The SEIR model linked below shows that overshoot.