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In a country of 20 mil. people, 25% of the inhabitants are fully vaccinated anti Covid-19 and 75% not vaccinated at all. It was noticed that, in the last weeks, 82% of the people found positive, after being tested, never received the vaccine and only 9% of those who died due of Covid-19 had received the vaccine.

In short:

  • 25% vaccinated,
  • 75% not-vaccinated,
  • 82% of the infected subjects are not-vaccinated,
  • 9% of the deceased people are vaccinated.

Question: What are the two efficiencies of the vaccine: (1) regarding its power to prevent an infection, (2) concerning its ability to prevent death.

If is self evident that, had the vaccine had no effect (had it been just a placebo) 75% of the people who would have died, and 75% of those infected, would have not been vaccinated and the rest of 25% would have been vaccinated. In this case the efficiencies of the vaccine would have been zero.

It is also evident that if the only people who died or got infected had been not-vaccinated, than the efficiencies of the vaccine would have been both 100%.

From this point forward, I don't really know how to continue the reasoning for finding the two efficiencies.

(Remark: You can make any simplifying assumption you would like.)

Robert Werner
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  • Since this is phrased as a mathematical question, it would help to provide the definition of "vaccine efficiency." Following Wikipedia, this is apparently computed from the attack rates $ARV,ARU$ for vaccinated/unvaccinated populations as $$\text{efficiency} = \frac{ARU-ARV}{ARU}\times 100%.$$ So if you're 10 times more likely to be infected while unvaccinated than vaccinated, this would give an efficiency of 90%. If that matches your usage, you should update your question accordingly to reflect this. – Semiclassical Aug 12 '21 at 22:28
  • Should I conclude that the efficiency of the vaccine in the country I presented is in fact unknown, it can not be determined based on the available data? – Robert Werner Aug 13 '21 at 02:05
  • If the definition you're using for efficiency is as above, then no: the information given is entirely sufficient to compute the vaccine efficiency. (If it is, I'd suggest updating the question accordingly so that the body of the question includes the full context.) – Semiclassical Aug 13 '21 at 02:17

1 Answers1

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Here are some ideas to get started. Following the notation of Wikipedia, let the (unknown) attack rates of the infection be $ARU$ and $ARV$ in the vaccinated and unvaccinated populations respectively. (What follows is directed to part (1), but the idea applies to part (2) as well.) For convenience, I'll use the notation of the Wikipedia article I linked in comments to the OP.

Since the efficiency may be written as $$\text{efficiency} = \frac{ARU-ARV}{ARU} = 1-\frac{ARV}{ARU},$$ it suffices to compute the ratio of these attack rates. Next, to make the computation a bit more concrete we may focus on a representative sample---say, 400 persons. (This specific number doesn't actually matter but it's convenient enough.) Of these 400, 100 are vaccinated and the remaining 300 are not. From this and the definition of attack rate, we may express symbolically:

  • how many of the 100 vaccinated persons will be infected
  • how many of the 300 unvaccinated persons will be infected
  • how many people out of the 400 are infected in total

From these, we can express (symbolically) what fraction of those infected in this 400-person sample did or did not receive the vaccine. This will give a condition on the attack rates, and from this there is enough info to find the efficiency.

I will note that the above is not the most direct calculation possible. (Use of Bayes' theorem reduces it to one line, for instance.) But for now I wanted to provide a more intuitive starting point.

Semiclassical
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  • In that country of 20 mil. people, it is known that 5 mil. (25%) are vaccinated and the remaining 15 mil. (75%) unvaccinated. It is also known that, out of the group of infected people discovered in the past weeks (which is small in comparison to the total population), 82% were not vaccinated. However, the total number of infected people during the same few weeks is unknown. It is not known how many of those 5 mil vaccinated got the virus. – Robert Werner Aug 14 '21 at 04:58
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    @RobertWerner I note that you withdrew your original comment. I didn't have a chance to look at it in detail, but it seemed like it was on the right track. Note that the whole point of the efficiency definition is that "how many vaccinated/unvaccinated people were infected" is irrelevant: what matters for the efficiency is the ratio of these populations, not their values. – Semiclassical Aug 14 '21 at 05:01
  • My original comment was this: "I got these results: (1) Effectiveness against an infection = 1-(18%/25%)/(82%/75%)=34.1%, (2) Effectiveness against death = 1-(9%/25%)/(91%/75%)=70.3%. They are not encouraging at all. I still hope I made a mistake." I deleted it because I do not believe the two efficiencies are correct. – Robert Werner Aug 14 '21 at 08:16
  • My problem is that the data I have about the people tested in the past weeks is only about the infected ones (82% of them are unvaccinated and 18% vaccinated). I do not know anything about the people tested but found healthy. – Robert Werner Aug 14 '21 at 08:34
  • I would assume that "tests as healthy" is treated the same as "uninfected" for the purpose of vaccine efficiency. The definition doesn't account for false positives/negatives, for instance. I also want to stress the interpretation of efficiency: your probability of getting infected has gone down by 34%. Suppose for instance that the unvaxed infection rate was 100%. Then the vaxed rate would be 66%, so 66 of the 100 patients in my example above would still be infected. By contrast, all 300 unvaxed patients would be infected, i.e., 300/366 = 82% of the infectees aren't vaccinated. – Semiclassical Aug 14 '21 at 08:47
  • If you're not convinced by that last case, pick a different "unvaxed infection rate", e.g., 50%, and run the numbers again to see how many infectees you get. (You may want to make the initial sample size larger in that case, e.g., 10000 instead of 400.) Also, part of the effect here is that there's significantly more unvaxed people and therefore they'll tend to represent a large fraction of the infectees even when the vaccine isn't particularly effective. (After all, if the vaccine wasn't effective at all, you'd expect 75% instead of 82%. So the increase is not that dramatic.) – Semiclassical Aug 14 '21 at 08:58
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    Robert, you said, "I got these results: (1) Effectiveness against an infection = 1-(18%/25%)/(82%/75%)=34.1%, (2) Effectiveness against death = 1-(9%/25%)/(91%/75%)=70.3%." They are correct. See my answer to this question: https://math.stackexchange.com/questions/4262345/80-of-population-is-vaccinated-and-25-of-infected-had-received-the-vaccine-wh – Derek Jennings Nov 07 '21 at 14:49
  • @RobertWerner I too got the same 34% and 70%. My explanation is next to Derek's in the above page that he linked to. – ryang Feb 08 '22 at 06:12