Herd immunity: the maths behind population protection thresholds

Herd immunity is one of those concepts that gets used loosely in everyday conversation, but is fairly precise as a piece of population biology. The idea: if enough people in a community are immune (through vaccination or past infection), the disease cannot find enough susceptible people to keep itself going, and even the unvaccinated are protected. Calculating “enough” comes down to one number per disease, the basic reproduction number, or R0.

R0 in one paragraph

R0 is the average number of new infections one infected person would generate in a fully susceptible population. R0 is a property of the pathogen and the contact patterns of the population, it’s not fixed across cultures or climates, but you can quote a typical range for most well-studied diseases.

• Measles: R0 of 12–18 (often the highest of any human disease).
• Pertussis: 12–17.
• Mumps: 10–12.
• Polio: 5–7.
• Rubella: 5–7.
• Diphtheria: 6–7.
• Seasonal influenza: 1.3–2.0.
• COVID-19 (original strain): 2.5–3.5; later variants higher.

The critical immunisation threshold

The minimum fraction of the population that needs to be immune to prevent sustained transmission is given by a simple formula:

Hc = 1 − 1/R0

• Measles (R0=15): 1 − 1/15 = 93%, which is why the public-health target for two-dose MMR coverage is around 95%, allowing for vaccine effectiveness less than 100%.
• Polio (R0=6): 1 − 1/6 = 83%.
• Rubella (R0=6): 1 − 1/6 = 83%, again allowing for less-than-perfect effectiveness.
• Diphtheria (R0=7): 1 − 1/7 = 86%.
• Seasonal flu (R0=1.5): 1 − 1/1.5 = 33%, much easier in principle, much harder in practice given waning immunity and strain mismatch.

Two important caveats. The formula assumes homogeneous mixing, that everyone has equal contact with everyone else. In real populations, contact patterns are clustered, by household, school, workplace, geography, social network. Pockets of low coverage can sustain outbreaks even when national averages exceed the threshold. The second caveat: vaccines are not 100% effective, so the operational coverage target is always higher than the bare mathematical threshold.

Where Australia sits

NCIRS publishes the annual immunisation coverage report. Recent national headline figures:

• Fully vaccinated at 12 months: around 93–94%.
• Fully vaccinated at 24 months: around 90–92%.
• Fully vaccinated at 5 years: around 93–94%.
• Aboriginal and Torres Strait Islander children at 5 years: similar to general population, though gaps remain in specific cohorts.

These figures hide local variation. Some local government areas in Victoria and NSW have under-vaccinated pockets in the 85–88% range for two-dose MMR, well below the measles threshold. This is why imported measles cases occasionally seed local outbreaks even when national coverage looks high.

What “eliminated” actually means

Australia is classed as having eliminated endemic measles and rubella transmission. Elimination is defined as the absence of sustained local transmission of the same chain of virus for 12 months or more. It does not mean no cases; importations from countries with active transmission continue to occur. Elimination is sustained by community immunity; if coverage slips, elimination status is at risk.

Polio in Australia has been certified as eliminated since 2000. Smallpox was eradicated globally in 1980. Rinderpest (cattle) in 2011. These remain the only two diseases ever fully eradicated.

What this means for individual decisions

Two takeaways. First, individual vaccination is not just for individual protection, it is the contribution you make to community immunity for the small group who genuinely cannot be vaccinated (infants too young, severely immunocompromised people, those with allergy contraindications). Second, herd immunity isn’t a stable steady state: as new births enter the population, new arrivals join the community, and immunity wanes in older adults, coverage has to be maintained every year or it drifts.