Herd Immunity Threshold Calculator
Find the share of a population that has to be immune for spread to shrink, using the R0 value for a disease.
What it does
Herd immunity, more precisely herd immunity threshold, is the point at which a pathogen cannot keep spreading because too few people are left to catch it. The arithmetic is short. If each infected person passes the disease to R0 others on average, and the susceptible share is s, then the effective reproduction number is R0 times s. Spread shrinks once that product drops below one.
Rearranging gives the threshold: the immune share has to reach 1 minus 1 divided by R0. A WHO hosted article on the epidemiology of COVID-19 in Egypt states the formula in exactly this form, and papers on the 2020 search for herd immunity work through the same model with its assumptions spelled out.
The threshold is not a property of the virus alone. R0 depends on how contagious a disease is, how long people stay infectious, how often people meet each other and how much of the population is susceptible in the first place. The CDC notes in Emerging Infectious Diseases that R0 is not a biological constant, and the published ranges for the same disease differ between studies and places.
The threshold is also not the same as the vaccination target. Immunity from a vaccine is rarely complete, and it also protects against infection less well than against severe disease. When you enter a vaccine effectiveness, the coverage figure divides the threshold by that value, which is why some diseases need more than 100% coverage and cannot reach the threshold that way at all.
Treat this as an educational model with even mixing, full immunity and no other interventions. Real campaigns add waning protection, uneven contact patterns, a variant with a different R0 and measures such as masking or testing that change transmission on their own.
How to use
- Enter the reproduction number R0 for the disease you are looking at.
- Enter the population if you want a head count as well as a percentage.
- Enter the share that is already immune.
- Enter the vaccine effectiveness if you want the coverage figure.
- Read the threshold, the number of people and the effective reproduction number.
- Compare the effective number with 1: above 1 means the disease still grows, below 1 means it shrinks.
Formula
- R0
- Average number of people infected by one case in a fully susceptible population.
- HIT
- Herd immunity threshold, the immune share needed for R to reach 1.
- S/N
- Share of the population that is still susceptible.
- VE
- Vaccine effectiveness against infection, as a fraction.
At exactly the threshold the effective reproduction number is 1, which means the case count holds steady rather than falling, so the share has to be above the threshold for spread to shrink. When R0 is 1 or lower the model gives no positive threshold, because the disease already dies out on its own. The coverage figure can exceed 100%, which is the signal that the vaccine is not effective enough to reach the threshold by vaccination alone.
Worked example
Example: R0 of 5 in a population of one million
Defaults: R0 of 5, a population of 1,000,000, no immune share yet, and no vaccine effectiveness entered.
- Threshold: 1 - (1 / 5) = 1 - 0.2 = 0.8.
- As a percentage that is 80.00% of the population.
- Head count: 1,000,000 x 0.8 = 800,000 people.
- Effective reproduction number at 0% immune: 5 x 1 = 5, so the disease still grows.
- If 80% were immune, the effective number would be 5 x 0.2 = 1, which is the boundary.
- At 95% vaccine effectiveness the coverage needed would be 0.8 / 0.95 = 84.2%.
- The tool shows 80.00%, 800,000 people, an effective R of 5 and a coverage line that asks for an effectiveness.
Reference table
| Disease | R0 in the source | Threshold from those values |
|---|---|---|
| Measles | 12 to 18 | 91.67% to 94.44% |
| Pertussis | 15 to 17 | 93.33% to 94.12% |
| Chickenpox | 8.5 | 88.24% |
| Seasonal influenza | 1.6 to 3 | 37.50% to 66.67% |
Limits
- Assumes even mixing and that immune people never transmit the disease, which is rarely exact.
- Ignores waning immunity, so a threshold reached years ago may no longer hold.
- Uses one R0 value, while published estimates for the same disease differ by place, season and strain.
- Says nothing about individual risk, and it is not a reason to skip a vaccine that protects against severe disease.
Common mistakes
- Entering the effective reproduction number instead of the basic one, which double counts the immune share.
- Treating the threshold as the vaccination target and ignoring how imperfect the vaccine is.
- Reading the threshold as the point where cases stop completely, since it is where the case count stops growing.
- Using a single R0 for a disease whose value changes with the season or the population.
- Ignoring that other measures lower transmission, which lowers the threshold as well.
Try it yourself
Quick checks. Answers sit tucked under each question.
What is the herd immunity threshold when R0 is 2?
1 - 1/2 = 0.5, so 50.00% of the population has to be immune.
What is the threshold when R0 is 12, as published for measles?
1 - 1/12 = 0.91667, so the threshold is 91.67%. That is why measles programmes aim so high.
With R0 of 5 and 80% of the population already immune, what is the effective reproduction number?
R = 5 x 0.2 = 1. That is exactly the boundary, so the case count would hold steady rather than fall. It has to be below 1 for the outbreak to shrink.
Questions
What is R0?
The average number of people one infected person passes a disease to, in a population where everyone is susceptible. It depends on the disease, on how people move around and on the measures in place.
Why does the threshold go up so fast for measles?
Because measles is so contagious, with R0 in the range of 12 to 18, the formula 1 minus 1 over R0 lands above 91%. That is why measles programmes push coverage into the nineties.
Is the herd immunity threshold the same as herd immunity?
It is the boundary that herd immunity theory points at. Below it spread fades, above it the disease keeps growing. In practice the level of immunity a population reaches rarely sits exactly on the line.
Can a disease have a threshold of zero?
In this model, yes. If R0 is 1 or lower the disease cannot sustain itself even with no immunity, so no threshold is needed.
What about R at the threshold?
Exactly 1, which means the number of cases stays level. That is why the immune share has to be above the threshold rather than at it before the outbreak turns around.
Does this tell me whether I should be vaccinated?
No. It is a population level model, and vaccines are recommended for the protection they give you individually, which includes protection against severe illness. Ask a clinician about your own situation.
Sources
- World Health Organization Eastern Mediterranean Region: Epidemiology of SARS-CoV-2 in Egypt
- CDC Emerging Infectious Diseases: Complexity of the Basic Reproduction Number
- Nature Immunology: COVID-19 herd immunity, where are we?
- World Health Organization: measles and rubella elimination session, R0 of 12 to 18