How long would it take a parasite that transmits via parent to child to spread through a population?

Question

So here is the basic setup.

The parasite lives in the cells of it hosts and can only spread via reproduction, and will spread 100% of the time to a child regardless if the infected was the mother or the father.

The infected are strongly encouraged to raise families with the uninfected. For the sake of simplicity, we shall assume all marriages are infected / uninfected matches.

The parasite lowers human fertility by about 50%, and culturally the infected are encouraged to keep family size small as well.

The parasite causes mutations that are highly advantageous at keeping the host alive, especially in regards to disease. To keep the maths simple, we should just assume death by disease before having children is 75% lower and death by violence and accidents is 50% lower.

A newly infected child is added to the general population pool about every 75 years or so. Sometimes two are added, but for the sake of number crunching we shall call it 1.25 children.

Now let’s assume we start with one infected person in a postion of high nobility in a setting similar to Norman England in 1066 and the setting follows the same broad trends as England in regards to population growth and mortality rates, how long is it going to take to spread?

Answer 1

The English will all be infected by the mid 1600s.

https://ourworldindata.org/grapher/population-of-england-millennium

Let us simplify assumptions. We will have 2 infected progeny replace their own 2 parents every 25 years. Assume each pair of parents is 1 infected 1 noninfected. The next generation their children are 2 infected. 2 infected the next generation become 4 infected etc. This takes into account lower reproductive rate and also better survival.

Thus the number of infected is 2^(# generations since 1066). I did it in excel to see when it would catch up with the English population.

The infected will be the entire population of England between 1616 (4194304 infected) and 1641 (8588608 infected).

This sidesteps emigration, the difficulty of finding noninfected mates in the 1600s, etc.

Answer 2

Assume the following: a generation is roughly 25 years (analysis shows it can vary, but this is a reasonable number). The number of ancestors one theoretically has at generation $n$ is $2^n$. Thus, at 100 years (4 generations previously) there's 16 ancestors, at 200 years 256, at 300 years 4,096, and so on.

Looking at it another way, for a given population of $P$ people, the number of generations you have to go back for the Most Recent Common Ancestor (MRCA) is $\log_2 P$. However, that has an issue in this scenario because it only says when the MRCA lived: there's no guarantee that the MRCA is the one who carried the parasite.

What you also want to look at is the Identical Ancestors Point (IAP): that's the generation where everyone either has no descendants alive now, or is an ancestor of everyone alive now. That sets the outer margin for how long ago the initial parasitized individual must have lived. So patient 0 had to be between the IAP and the generation which had the MRCA (which is the latest he could have lived to have everyone later be a carrier).

Chang (cited in the link above) calculated that in a population of size $P$, the IAP was $1.77\log_2 P$ generations ago. So, a spreadsheet easily works this out: we take the population of Great Britain at given points in history, and calculate, at that point, how long ago before that date the MRCA and IAP were. If 25 * the number of generations falls around 1066, then we know for each.

In 1600, the population of the island of Great Britain was about 5.2 million people. Calculating the MRCA ancestor puts that person being alive around 1042. That's actually pretty close to our 1066 date, and given the margins of error inherent in things like generation length and such, we can probably safely use that as an estimate. So the earliest everyone in Great Britain would carry the parasite is around 1600.

Now we're looking at the IAP. And here we run into an interesting issue: in 2000 the population of Great Britain was 59 million, but that only puts the IAP in 858. So, theoretically speaking, its possible that not everyone in Great Britain (not counting recent immigrants) would be infected yet. It would be highly unlikely, but possible.

However, given the point that infected will preferentially mate with uninfected, it's much more likely the point of 100% infection would fall a lot closer to the MRCA than it would the IAP.

All in all, you're safe saying that it would happen sometime in the 17th century. Now, this raises an interesting point because this is exactly around the time the British started going big into the colonization game around the world. The parasite will have certainly spread across most of Europe by this point, but it's going to explode across the entire planet in very short order after that.