If populations are highly vaccinated, we'd expect a higher proportion of future cases to have been previously vaccinated (because by definition, there aren't as many non-vaccinated people around to be infected). But what sort of numbers should we expect? A short thread... 1/
In above question, there are a lot of things happening conditional on other things happening (e.g. probability cases have been vaccinated), which means we can use Bayes rule (en.wikipedia.org/wiki/Bayes%27_…) to work out the proportion of cases that we'd expect to have been vaccinated. 2/
If we want to know the probability of event A given event B, or P(A|B) for short, we can calculate this as
P(A|B) = P(B|A) P(A)/ P(B)
There are a couple more mathsy tweets coming up, so hold on as then we'll get back to the real-life implications. 3/
P(A|B) = P(B|A) P(A)/ P(B)
There are a couple more mathsy tweets coming up, so hold on as then we'll get back to the real-life implications. 3/
Applying the above to our vaccine question, we therefore have:
P(vaccinated | case) = P(case | vaccinated) x P(vaccinated) / P(case)
which is equivalent to
P(vaccinated | case) = (1–V) x P(vaccinated)/P(not protected)
where V is vaccine effectiveness. 4/
P(vaccinated | case) = P(case | vaccinated) x P(vaccinated) / P(case)
which is equivalent to
P(vaccinated | case) = (1–V) x P(vaccinated)/P(not protected)
where V is vaccine effectiveness. 4/
If we write out the ways in which we could get P(not protected), we end up with below equation (I've labelled the terms on the bottom of the fraction to make it clearer where these come from): 5/ 
Now we have something we can apply to real-life situations, because can measure many of these things. For example, if 60% of a population have been vaccinated, and vaccine is 80% effective, above means we'd expect (1-0.8)x0.6/(1-0.6x0.8)= 23% of cases to have been vaccinated. 6/
This is an important result, because if cases appear among vaccinated individuals, many people's intuitive response is to ask 'surely the vaccine can't be that effective?' The answer: it may well be effective, just in a highly vaccinated population e.g.
https://twitter.com/AdamJKucharski/status/1200329736544759808?s=207/
We can also flip the above equation around, which allows us to use data on % cases vaccinated and % vaccinated to get a rough estimate of vaccine effectiveness:
https://twitter.com/AdamJKucharski/status/1382006630997323776?s=208/
In short: Bayes rule is very useful, and case/vaccine patterns in highly vaccinated populations don't always do what you may assume. 9/9
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