How many people need to be immune?

A common picture of herd immunity is a finish line: once enough people are immune, the disease stops, and until then vaccination doesn’t do much. Both halves of that picture are wrong. There is a real threshold, and it comes from one simple idea. But an epidemic that is left alone runs straight past it, and vaccination protects people long before the threshold is reached.

One case makes how many?

Picture one person with an infection, walking into a town where nobody has ever had it. While they are infectious they pass it to some number of people. The average of that number, in a population where everyone can catch it, is called the basic reproduction number, written R₀ (“R-nought”). Seasonal flu has an R₀ of about 1.3. Measles, one of the most contagious diseases known, is usually quoted at 12 to 18.

Now suppose some people are already immune, through vaccination or an earlier infection. The infectious person meets just as many people, but only the susceptible ones — those who can still catch it — become new cases. The average number of new cases per case is now the effective reproduction number:

R_eff = R₀ × (share of people still susceptible)

If R_eff is above 1, each generation of cases is bigger than the last, and the outbreak grows. If it is below 1, each generation is smaller, and the outbreak dies out.

Two things stand out. First, even well above 1, many chains fizzle by luck: with R_eff = 2, about one chain in five dies out by chance. That is why a single imported case sometimes leads nowhere. Second, below 1 no chain survives for long, however lucky. A chain can still infect a handful of people before it ends, so small outbreaks still happen, but they can’t take hold and grow.

The threshold

The tipping point is where R_eff = 1. Setting R₀ × (susceptible share) = 1 says the susceptible share must be at most 1/R₀. So the share that must be immune, the herd immunity threshold, is:

threshold = 1 − 1/R₀

For R₀ = 2 that is 1 − ½ = 50%. For R₀ = 4 it is 75%. For measles at R₀ = 15 it is 1 − 1/15 ≈ 93%.

Here is a way to feel why it climbs so steeply. Each case would meet enough people to infect R₀ of them. To stop growth, on average only one of those R₀ may be susceptible. With R₀ = 2, one person in two may be susceptible; with R₀ = 15, only one in fifteen. Going from R₀ = 2 to 15 only moves the threshold from 50% to 93%, but it shrinks the share allowed to remain susceptible from 50% to under 7% — more than seven times smaller.

An epidemic, day by day

The chains show single generations. To see a whole epidemic unfold we need a model of a whole population. The classic one is the SIR model, which sorts everyone into three groups: Susceptible (can catch it), Infected (have it and can pass it on) and Recovered (had it and are now immune). Infections happen at a rate proportional to how many susceptible and infected people there are, as if everyone mixed at random. Here each person is infectious for 7 days on average, and the epidemic starts with 1 person in 10,000 infected. A share of people can be vaccinated beforehand; in this model they are simply immune.

The peak falls where it does for the reason in Figure 1. The number of people ill at once grows as long as each case is replaced by more than one new case, that is, while the susceptible share is above 1/R₀. The moment it drops below, each case is replaced by fewer than one, and the number ill starts to fall. The top of the curve is the herd immunity threshold being crossed.

Now look at the vaccination results. With R₀ = 3 the threshold is 67%. Vaccinating 50% is below it, so an epidemic still happens — but it is slower and smaller. The peak drops from 30% of people ill at once to about 3%, and it arrives around day 103 instead of day 35. Of the unvaccinated half, 58% catch it instead of 94%. People who never got the vaccine are protected by the people who did. That indirect protection is what “herd immunity” most usefully means, and it starts with the first vaccinated person, not at the threshold.

Why epidemics overshoot

In Figure 2, with no vaccination and R₀ = 3, the peak comes when 67% of people are immune — but by the end, 94% have been infected. When the susceptible share drops through 1/R₀, a large number of people are infected at that very moment. Each of them still passes it on to nearly one more person, and those to nearly one more. The epidemic declines, but it keeps infecting people all the way down.

How many people are infected in the end, if nothing is done? The SIR model has a neat answer. The final share infected, z, satisfies the final-size equation:

z = 1 − e^−R₀·z

In words: the share who escape, 1 − z, equals e^−R₀z, the chance of dodging every one of the infections that the whole epidemic sent your way. It has no neat solution, but a computer can solve it in a moment. The difference between this final share and the threshold is the overshoot: people infected after the epidemic had already passed its peak.

For R₀ = 2, the threshold is 50%, but an unchecked epidemic infects about 80%. The overshoot is largest, 29.8% of the whole population, at R₀ ≈ 2.15; that maximum has been proved for this model. So “let it spread until we reach herd immunity” doesn’t stop at the threshold: it infects enough people to reach the threshold, and then up to 30% of the whole population on top. Vaccination gets to the threshold without the overshoot.

Real vaccines, real diseases

Two more facts turn the threshold into a vaccination target. Vaccines are not 100% effective: two doses of the measles vaccine protect about 97% of people, one dose about 93%. If a share E of those vaccinated become immune, vaccinating a share c makes c × E immune. So the coverage needed is:

coverage needed = (1 − 1/R₀) ÷ E

And R₀ is not a constant of nature. It depends on how people live, meet and gather, so studies in different places and years give different values. The bars under the chart below show ranges, not single numbers.

At R₀ = 15 and 97% effectiveness, 96% of people need to be vaccinated. With a 93% effective vaccine (one dose) the answer is just over 100%: impossible, however hard you try. One dose falls short whenever R₀ is above about 14.3, because that is where 1 − 1/R₀ passes 0.93. Below that it is possible, but only just: at R₀ = 12 one dose would need almost 99% of people vaccinated. That is one reason measles vaccination uses two doses and aims for about 95% coverage, and why measles returns quickly when coverage slips by just a few percent. For an infection with R₀ near 1.3, the threshold is only about 23%, which is small in the model — but flu viruses change from year to year and immunity to them fades, which the model leaves out.

So the rule to take away: the share that must be immune is 1 − 1/R₀, because the share left susceptible can be at most 1/R₀. Getting there by infection overshoots it; getting there by vaccination doesn’t; and every step towards it, even short of it, means fewer people ill.