The herd immunity threshold is 66.7%, so why does the epidemic infect 94.1%?
Two numbers from the same simulation: the outbreak turns the corner at 66.7% and finishes at 94.1%. Everything interesting is in the gap.
Run the SIR model at its defaults: transmission β = 0.3, recovery γ = 0.1, ten thousand people, ten of them infectious on day one. The panel reports R₀ = β/γ = 3.00 and a herd immunity threshold of 1 − 1/R₀ = 66.7%.
It also reports the outcome. Peak infected: 3,017 people, 30.2% of the population ill simultaneously. Total infected: 9,412 — 94.1%.
Those last two numbers are the ones worth stopping on. The threshold said 66.7%, and the epidemic went 27.4 percentage points past it. That is 2,745 people out of ten thousand who were infected after the outbreak had already begun to shrink.
What the threshold actually claims
The threshold is a statement about a derivative, not about an ending, and reading it as an ending is the whole mistake.
Each infectious person, in a fully susceptible population, passes the infection to R₀ others. If a fraction of the population is already immune, only the susceptible share of those contacts can be infected, so the effective reproduction number is
Reff = R₀ · S/N
and the case count stops growing when Reff = 1, which happens when the susceptible fraction S/N falls to 1/R₀, one third here. Equivalently, when 66.7% are no longer susceptible.
Notice what does not happen at that moment. Transmission does not stop. Reff = 1 means each infectious person infects exactly one more, so the epidemic holds steady at its peak, and this is why the peak and the threshold are the same event. At the defaults, 3,017 people are infectious at that instant. Every one of them goes on to infect roughly one more person, each of whom infects slightly fewer, and the tail plays out from a very large starting stock. The outbreak decays from the peak rather than halting at it, and everything infected during that decay is the overshoot.
The tail is where the extra 27 points live, and you can catch it mid-flow. Set the days control to 60: the peak still reads 3,017, so it has already been and gone, while the total reads 8,122. Another 1,290 infections are still to come, every one of them after the outbreak began shrinking.
Why the final size is not a choice
The endpoint is not tunable by anything except R₀. Kermack and McKendrick derived the relation in 1927: if f is the fraction eventually infected, then
f = 1 − e−R₀·f
an implicit equation with a single solution above zero whenever R₀ > 1. At R₀ = 3 it gives f ≈ 0.94, matching the panel's 94.1%. The equation contains no term for population size, no term for how fast the epidemic ran, and nothing about when it started. Doubling the population doubles the casualties and changes the fraction not at all.
The overshoot grows with R₀, and it grows unkindly:
- R₀ = 1.5, the flu preset — threshold 33.3%, final size 5,845 people, or 58.5%. Overshoot 25 points.
- R₀ = 3 — threshold 66.7%, final size 94.1%. Overshoot 27 points.
- R₀ = 15, the measles preset — threshold 93.3%, and the panel prints a final size of 10,000 out of 10,000. Everyone. Overshoot around 7 points.
The overshoot is largest in the middle. Below R₀ = 1 there is no epidemic; at very high R₀ the threshold is already so close to everyone that there is little room left to overshoot into. The moderately transmissible pathogens, the ones it feels plausible to "let run", have the widest gap between the point of no more growth and the point of no more transmission.
The consequence that matters
Both routes to the threshold end with 66.7% immune. Only one of them costs 94.1% infected.
If immunity is already in place before the pathogen arrives — vaccination, or prior exposure to something cross-protective — then Reff starts below 1, there is no growth phase, no peak, and no overshoot. Introductions produce small clusters that die out. The 27.4 points are not the price of immunity; they are the price of acquiring it during an epidemic instead of before one.
This is also the honest answer to the argument that emerged repeatedly during 2020: that a population could be walked up to the threshold deliberately and stopped there. The model says the stopping is the part you do not control. By the time the data shows you have arrived at the threshold, several thousand people are already infectious, and their infections are already committed. The peak is not a brake; it is the moment the brake begins to be applied, with the vehicle at full speed.
Two limits worth keeping in view, because this is a model with four dials and reality has thousands. It assumes everyone mixes with everyone at random, which no population does. Clustered contact structures lower the final size and can produce local burnout well below the naive threshold. And it assumes immunity is permanent, which for many pathogens it is not; where immunity wanes, "the threshold" is not a level you reach and keep but one you have to keep paying for. Both corrections change the numbers. Neither changes the shape of the argument: the point where growth stops and the point where transmission stops are different points, and the distance between them is measured in people.
References (1)
- the model, and the threshold theorem, in its original form Kermack & McKendrick (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society A 115(772).