"Breakthrough infections" is a bad name

If you search Google News for “breakthrough infections” you’ll get a ton of results. Everyone seems to be wondering — do the vaccines work against these new variants? What would happen if everyone got vaccinated? And so on. 

Unfortunately, “breakthrough” infections — that is, Covid cases among people that have been fully vaccinated — implies that there was some impenetrable wall that has been surmounted by a new and more formidable enemy. This isn’t true and it isn’t helpful. 

As I dug into way back in March, the way that vaccines are shown to be effective is by preventing some portion of cases that would have otherwise happened. If a vaccine is stated to be 95% effective, that means that out of 100 cases that would have happened without a vaccine, only 5 cases did actually happen with the vaccine, meaning that 95 cases didn’t happen relative to this without-vaccine counterfactual. 

But 5 is not zero! And those 5 cases are not “breakthrough” cases, they are normal, expected cases that will sometimes happen even with a vaccine that is 95% effective. 

And the 5 cases that you expect is also a relative number. If there were 500 cases among people that were unvaccinated, you’d expect 5 * 5 = 25 cases among vaccinated folks. 

So how do these numbers compare with the Delta variant? 

Well, as Vox put it;

Unvaccinated people still make up the vast majority of cases, hospitalizations, and deaths. They’ve made up more than 94 percent of reported Covid-19 cases in states with available data, a report last week from the Kaiser Family Foundation found. They’ve also made up similar, or higher, shares of hospitalizations and deaths.

The key number I’m looking at is the 94% figure. That means that for every 94 cases that happened without a vaccine, only about 6 have happened with a vaccine. (This is of course putting aside confounding factors, so this will be rough justice, no more). That means that approximately 6/94 = 6.4% of Covid cases are not prevented by the vaccine, or that it has an efficacy of 1 - (6 / 94) = 93.6%. (!!!!!)

So, yes, the vaccines work. They work well! Very well!