When I first heard friends getting excited about T-cell immunity to COVID-19, I was non-plussed.

“This means the disease is less contagious than we thought!,” they said.

And I replied, “You’re double-counting!  I If some people are immune, that will already be reflected in existing estimates of R0.”

As it turns out, however, my friends were right for the wrong reason.  While immunity doesn’t matter for initial estimates of R0, it is crucial for estimating the path of R0.  This in turn is crucial for ascertaining when the pandemic will end.  David Friedman explains everything with admirable clarity:

Suppose, for simplicity, that half the population consists of people vulnerable to the disease and half, for behavioral or biological reasons, invulnerable. Observing the early spread of the disease, we find that, on average, each infected person passes the disease on to two others. We conclude that we will only reach herd immunity when half the population have had the disease and become immune as a result.

But the relevant figure is not what fraction of the population has become immune but what fraction of the vulnerable population has. In my simple model, half the vulnerable population is only a quarter of the total population, so we reach herd immunity much earlier than the simple calculation implies.

Semi-experts are often quick to say that you can’t reach herd immunity until 60-70% of the population gets infected.  If half of the population is immune, however, this is plainly wrong.  If 50% of the population is immune, you’ll never hit 60-70% infection rates!

So what’s the correct story?  Consider the classic logistic contagion graph:

The key question is: Where’s the asymptote for the infected share of the population?  If everyone is vulnerable, you probably won’t hit the dashed line until most people get sick.  As the immune share of the population goes up up up, though, the dashed line shifts down down down.

The real question, then, is not whether individual immunity helps, but how much.  Many countries now look like they’ve hit herd immunity.  But as Scott Sumner emphasizes, “herd immunity with the help of drastic behavioral changes” is a far cry from “herd immunity with normal behavior.”

Unfortunately, both versions of herd immunity superficially look the same.  To tell them apart, you have to relax behavior and restrictions and see if the pandemic returns.  So while my improved understanding of the mathematics of immunity makes me more hopeful in the long-run, in the medium-run I remain diffident.  I won’t be surprised if the pandemic is over in every U.S. state by Christmas.  I won’t be surprised if the pandemic is worse in every U.S. state by Christmas.

Closing thought: The extreme rarity of public bets on the path of the pandemic tells me that even the best-informed experts remain about as confused as I am.  They’re just too high on their own punditry to admit that they know little indeed about what’s going to happen.  I wish they would read Tetlock’s Superforecasting before they speak another overconfident word, but I know they won’t.