Herd immunity on the back of an envelope

This post presents a back-of-the-envelope calculation regarding COVID herd immunity in the US. Every input to the calculation is only roughly known, and I’m going to make simplifying assumptions left and right. So take this all with a grain of salt.

According to a recent article, about 26 million Americans have been vaccinated against COVID, about 26 million Americans have been infected, and 1.34 million a day are being vaccinated, all as of February 1, 2021.

Somewhere around half the US population was immune to SARS-COV-2 before the pandemic began, due to immunity acquired from previous coronavirus exposure. The proportion isn’t known accurately, but has been estimated as somewhere between 40 and 60 percent.

Let’s say that as of February 1, that 184 million Americans had immunity, either through pre-existing immunity, infection, or vaccination. There is some overlap between the three categories, but we’re taking the lowest estimate of pre-existing immunity, so maybe it sorta balances out.

The vaccines are said to be 90% effective. That’s probably optimistic—treatments often don’t perform as well in the wild as they do in clinical trials—but let’s assume 90% anyway. Furthermore, let’s assume that half the people being vaccinated already have immunity, due to pre-existing immunity or infection.

Then the number of people gaining immunity each day is 0.5*0.9*1,340,000, which is about 600,000 per day. This assumes nobody develops immunity through infection from here on out, though of course some will.

There’s no consensus on how much of the population needs to have immunity before you have herd immunity, but I’ve seen numbers like 70% tossed around, so let’s say 70%.

We assumed we had 184 M with immunity on February 1, and we need 231 M (70% of a US population of 330M) to have herd immunity, so we need 47 M more people. If we’re gaining 600,000 per day through vaccination, this would take 78 days from February 1, which would be April 20.

So, the bottom line of this very crude calculation is that we should have herd immunity by the end of April.

I’ve pointed out several caveats. There are more, but I’ll only mention one, and that is that herd immunity is not an objective state. Viruses never completely go away; only one human virus—smallpox—has ever been eradicated, and that took two centuries after the development of a vaccine.

Every number in this post is arguable, and so the result should be taken with a grain of salt, as I said from the beginning. Certainly you shouldn’t put April 20 on your calendar as the day the pandemic is over. But this calculation does suggest that we should see a substantial drop in infections long before most of the population has been vaccinated.

Update: A few things have changed since this was written. For one thing, we’re vaccinating more people per day. See an update post with code you can update (or just carry out by hand) as numbers change.

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