Aristocrats, mutations and diseases: the Galton–Watson process

This post was originally posted on my attempt at a science communication blog, Probably Interesting. I moved it over here in March 2026 so I could have everything I’ve blogged about in one place. It also explains why this wasn’t posted on a Sunday.

Imagine you’re a nineteenth-century noble, and, like Henry VIII before you, you’re obsessed with your bloodline. Specifically, you want your surname to carry on for many generations to come, but you know that that depends on having male heirs… and on those heirs having male heirs, and so on. Out of this deeply relatable problem was born an important bit of mathematical theory.

Sir Francis Galton (also known for devising the first weather map and, more problematically, for his work on eugenics) investigated the question, and, together with his correspondent, the Revd. Watson, devised a way to think about it. Translated into modern parlance, they assumed that each man has a random number of male children, and they made a few further assumptions to simplify the problem. Firstly, they assumed that each man has children independently, so unaffected by anyone else. Secondly, they assumed that generations were discrete: that all the children in the first generation are born before the first children in the second, and so on. Thirdly, they assumed that, over time, the chance of each man having a certain number of children did not change. (There are ways to address these simplifications, but the fundamentals remain the same.)

As you might expect, what’s going to make the difference to whether the name dies out is the distribution of the number of children, which means the probability of having zero sons, or one son, or two, etc. But it ends up being simpler than that: all that matters is the average number of male offspring. If the average is one, or less, then eventually the name will die out—it might take a while, but, given enough time, it will happen. If the average number of sons is more than one, the name might still die out: if they were really unlucky, everyone in the first generation could have no sons, and that’s that. But there’s some chance that the name will live on forever.

That’s great for the aristocrats, but why do we care? Well, we can think about things other than surnames that behave in the same way. Think about a genetic mutation, for instance—we could apply this to more serious things, but let’s say one that causes your hair to be a fetching shade of pink. That might pass on to a random proportion of your random number of children, and it’s whether that results in, on average, more than one pink-haired child that will determine whether this will be a brightly-coloured flash in the pan, or whether pink-haired legions are here to stay.

In fact, you may not realise it, but you’ve probably already heard about this in the news. You might have seen talk about something called R₀, which is the average number of people that each person with a virus infects. At least in the early stages of an outbreak—when most of the population remain uninfected—you can again think of this as a Galton–Watson process, and, yep, the only thing that makes a significant difference is whether or not R₀ is bigger than one. Push it below, and the virus eventually can’t sustain itself, though that might take a while.

Again, let’s imagine that people are infected in ordered “generations”, just to make everything easier to deal with. Then we can compute the number of people we’d expect to be infected at each generation—and, in fact, it’s the number of people we’d expect in the previous generation, multiplied by R₀. If R₀ is bigger than one, this gives us exponential growth: an increase that gets faster and faster. If R₀ is less than one, we get exponential decay instead, so that we tail off to zero (relatively) quickly. This is why small changes in the number of people each person infects can make a big difference overall.

In fact, it turns out that one of the things that the Galton–Watson process isn’t very good at analysing is whether surnames die out, because surnames can change for all sorts of other reasons—like marriage, or migration. Nineteenth-century nobles will have to find another way to solve their problems.