Herd-Immunity Threshold
Minimum fraction of the population that must be immune to prevent sustained spread.
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Example
R₀ = 5 → H = 1 − 1/5 = 80 % must be immune
How it works
This formula gives the smallest proportion of people who need to be immune so that each infected person, on average, passes the disease to less than one other person. It comes from the idea that immunity reduces the pool of susceptible contacts, and once the effective reproduction number drops below 1, transmission cannot keep growing steadily. A larger basic reproduction number means the pathogen spreads more easily, so a larger share of the population must be immune to stop sustained outbreaks.
When to use it
Use this when estimating how much of a population must be protected by vaccination or prior infection to interrupt ongoing spread of an infectious disease. It is especially useful in public-health planning, outbreak modeling, and comparing how contagious different diseases are.
Worked example, step by step
Suppose a disease has a basic reproduction number of R₀ = 4. Use the herd-immunity formula: H = 1 - 1/R₀ Substitute the value: H = 1 - 1/4 Convert to a decimal: H = 1 - 0.25 Calculate: H = 0.75 Convert to a percentage: H = 75% So, about 75% of the population would need to be immune to stop sustained spread under idealized assumptions.
Common mistakes to avoid
- Forgetting to convert the result into a percentage when reporting the threshold
- Using this formula for a disease with changing R₀ without adjusting for real-world conditions
- Assuming herd immunity means zero cases instead of just preventing sustained transmission
Did You Know?
For measles (R₀ ≈ 15), about 93–95 % of the population must be immunised — one of the hardest thresholds to maintain.
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More in Epidemiology & Public Health
View allIncidence Rate
BasicRate at which new cases of a disease appear in a population over a given period.
Prevalence
BasicProportion of a population that has a particular condition at a point in time.
Basic Reproduction Number
IntermediateAverage number of secondary infections produced by one infected individual in a fully susceptible population.
Effective Reproduction Number
IntermediateActual average infections per case when part of the population is already immune.
SIR Model — Susceptible
AdvancedRate of change of the susceptible population in the SIR compartmental model.
SIR Model — Infected
AdvancedRate of change of the infected population — new infections minus recoveries.