“A Vaccine Is a Wanted Poster for a Germ” ended with your immune system as a detective, holding a photograph of a criminal it has never met. A vaccine protects the person who receives it. But it also does something stranger and, for a society, far more powerful: it protects people who never saw the poster at all. To see how, you need one number and one line of algebra.
The number is called R₀, pronounced “R nought”, the basic reproduction number. It’s the average number of people that one infected person passes a disease on to, in a population where nobody is immune. If R₀ is 2, one case leads to 2, which lead to 4, then 8, 16, 32, and after just ten rounds of passing it on, more than a thousand people are infected. That’s exponential growth, and it’s why outbreaks can explode so quickly. If R₀ were below 1, the opposite would happen: each generation of cases would be smaller than the last, and the disease would fizzle out on its own.
R₀ depends on the disease. For seasonal flu it is usually somewhere between 1 and 2. For polio, perhaps 5 to 7. For measles, one of the most contagious diseases known, it is around 12 to 18. One child with measles, in a classroom where nobody is immune, can infect most of the class, because the virus hangs in the air of a room for up to two hours after the sick person has left.
Now add immunity. Suppose some fraction of people are immune, either because they’ve been vaccinated or because they’ve had the disease. An infected person still meets the same number of people, but some of those meetings are now wasted, because the germ can’t take hold. If half the population is immune, half the meetings are wasted, and the disease spreads to only half as many as before. The real rate of spread, which epidemiologists call R, is R₀ multiplied by the fraction of people who can still catch it.
For an outbreak to shrink, we need R to fall below 1. So R₀ times the susceptible fraction must be less than 1, which means the susceptible fraction must be less than 1 divided by R₀. Turn that round and you get the share of people who must be immune: 1 minus 1 over R₀. That’s the herd immunity threshold, and it’s one of the most useful formulas in public health.
Try it. For a disease with R₀ of 2, you need 1 − 1/2, or half the population, to be immune. For polio, with R₀ around 6, it’s about 83 per cent. For measles, with R₀ around 15, it’s 1 − 1/15, about 93 per cent, and to be safe across the range, public health experts aim for 95. Because measles is so contagious, almost everyone has to be immune before the chains of infection break.
Once you’re above that threshold, something remarkable happens. Even the people who aren’t immune are protected, because the disease can no longer find a chain of people to travel along. A baby too young to be vaccinated, a child having cancer treatment whose immune system is weakened, an elderly person whose vaccine didn’t take: all of them are shielded by everyone around them. The wanted poster works for the whole street, not just the houses that put it up.
The formula also explains why small drops in vaccination matter so much for a disease like measles. If coverage falls from 95 per cent to 85, it doesn’t just mean ten per cent more risk. It can take the community from below the threshold to well above it, from a disease that fizzles out to one that can spread. That’s why measles outbreaks often appear in places where coverage has slipped by only a few points.
Smallpox showed a smarter version of the same idea. In its final years, health workers didn’t try to vaccinate entire countries. They searched for every case, then vaccinated everyone around it, family, neighbours and contacts, in a ring. Each ring pushed R below 1 exactly where the virus was, and one by one the chains broke. The last natural case was in 1977. India’s polio campaign, which reached more than a hundred and seventy million children in a single round, worked by pushing immunity so high, so fast, that the virus ran out of people to infect.
So when you get a vaccine, you are doing two things at once. You’re giving your own detectives a photograph. And you’re becoming one of the wasted meetings in a germ’s chain, a dead end, which is a much more generous thing than it sounds.