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If populations are highly vaccinated, we& #39;d expect a higher proportion of future cases to have been previously vaccinated (because by definition, there aren& #39;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 ( https://en.wikipedia.org/wiki/Bayes%27_theorem)">https://en.wikipedia.org/wiki/Baye... to work out the proportion of cases that we& #39;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& #39;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& #39;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& #39;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& #39;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& #39;s intuitive response is to ask & #39;surely the vaccine can& #39;t be that effective?& #39; The answer: it may well be effective, just in a highly vaccinated population e.g. https://twitter.com/AdamJKucharski/status/1200329736544759808?s=20">https://twitter.com/AdamJKuch... 7/
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=20">https://twitter.com/AdamJKuch... 8/
https://twitter.com/AdamJKucharski/status/1382006630997323776?s=20">https://twitter.com/AdamJKuch... 8/
In short: Bayes rule is very useful, and case/vaccine patterns in highly vaccinated populations don& #39;t always do what you may assume. 9/9
