Today the Wall Street Journal has the following alarming news. In the recent surge of cases in Israel involving the Delta variant, half of all newly infected adults had been fully vaccinated. I almost cried when I read that. wsj.com/articles/vaccinated-…
Delta Variant Outbreak in Israel Infects Some Vaccinated Adults
Preliminary findings by Israeli health officials prompted the government to reimpose an indoor mask requirement and other measures to contain the highly transmissible strain of Covid-19.wsj.com
But hang on. Isn't almost everyone fully vaccinated in Israel? Thought experiment. Suppose I told you there was a superspreader party attended by 100 people and half of those infected had been fully vaccinated. Does that make you worry about vaccine effectiveness?
What if I then told you that in fact 99 of the 100 people at that party had been fully vaccinated. Well these facts together tell you that 1 person was unvaccinated and got infected and 1 of the 99 vaccinated people got infected. In other words the vaccine was 99% effective.
In general if you know that a fraction r of a population is vaccinated and from that population equal numbers of vaccinated and unvaccinated people have gotten infected from some event, then (1-r)/r is the relative probability of infection vaccinated versus unvaccinated.
Why is that? Let q be the fraction of individuals exposed to the virus by the event, and let p_v and p_u be the probability of becoming infected conditional on exposure for vaccinated and unvaccinated respectively. Then we have q*p_v*r = q*p_u*(1-r) i.e. p_v/p_u = (1-r)/r
Remember that my 83% was just an estimate. But more importantly, another detail from this outbreak in Israel is that it started with some large clusters in schools. Certainly the fraction of adults in a school context who are vaccinated is much larger than the baseline.