This is a public service announcement… with
What does it mean when we see headlines about how half of the infections in Israel’s delta outbreak are among the vaccinated? Are the vaccines that ineffective? Should we comfort ourselves with the knowledge that these cases are generally not severe?
No! to the former. And sure, if you want to!, to the latter. The truth is that in a country as highly vaccinated as Israel, this is not only expected, but should be celebrated as evidence of vaccine efficacy. To see why, let’s do a little brunchtime back-of-the-envelope math.
85% of Israeli adults are vaccinated. So let’s imagine a well-attended birthday party with 100 people. 85 are vaccinated, and 15 are not. This party is poorly ventilated and turns into a superspreader event. Every unvaccinated attendee gets sick. Oh no! Even worse, 15 of the vaccinated get sick, too. How effective is the vaccine? Since we’re assuming a 100% exposure rate, and that the unvaccinated population is 100% susceptible, the math is actually very simple. With rate of vaccination r, vaccine “vulnerability” is (1-r)/r. So in this toy example, it’s (1 – 0.85)/0.85 = 0.17, or 83% effectiveness against symptomatic disease. Not quite as high as the effectiveness against wild-type, but hardly something to panic over.
Obviously this is a toy example. This outbreak started with large clusters in schools, and only half those infected are adults, so the population isn’t anywhere near homogeneous. Our toy’s 100% attack rate is also a bit silly. But even this simple calculation gets us admirably close to the real figures. A study by Public Health England found that the two-shot treatment was 88% effective at preventing symptomatic disease from the delta variant. Pfizer claims 90%. (And remember, these breakthrough cases are generally asymptomatic or mild.)
This stuff isn’t always intuitive, so I thought I’d share a little word problem that can help to reconcile these two pieces of news. Get vaccinated!