Math error in herd immunity calculation from CNN epidemiology expert

Michael Weissman and Sander Greenland write:

Sanjay Gupta and Andrea Kane just ran an extensive front-page CNN article reporting that some residual T-cell immune responses cross-react with SARS-Cov-19, perhaps enough to provide many people with some protection. The article seemed straightforward and reasonable enough until it got to this strangely erroneous statement:

For herd immunity, if indeed we have a very large proportion of the population already being immune in one way or another, through these cellular responses, they can count towards the pool that you need to establish herd immunity. If you have 50% already in a way immune, because of these existing immune responses, then you don’t need 60 to 80%, you need 10 to 30%—you have covered the 50% already.

The source was Prof. John Ioannidis of Stanford Medical School, who since at least March has been making assertions (often challenged by statisticians and epidemiologists) minimizing the dangers of Covid-19.

What’s wrong with his analysis? Let’s look at a very simple and extreme case, just to clarify the logic of the problem, ignoring for now the practical issues, such as that immunity is only partial. Suppose 50% of the population is completely protected without infection, the other 50% is completely vulnerable, the initial reproductive number R0 (the average number of persons infected by each case as the epidemic begins) in a completely vulnerable population would be 4, infection always confers immunity, and our actual population is fully mixed. Then, since 50% of the people encountered by an infectious person would be immune, the R0 in our population would be 50% of 4, i.e. 2 (which is within the range of initial R0 estimates for Covid-19 in whole populations).

From that estimated R0 of 2, the standard estimate of the herd immunity threshold would be (1-1/R0) = 50%. The Ioannidis analysis would then say that, since 50% of the population is already immune, you should correct that estimate by subtracting that 50% from the estimated 50% herd immunity threshold. So you need to achieve only 0% more population immunity to reach herd immunity, meaning you start off with herd immunity and there isn’t an epidemic. With a slightly lower R0, Ioannidis’s analysis would give a negative herd immunity threshold.

That’s mathematically wrong. We got the R0=2 from watching the initial exponential growth in a mixed population, so there is an epidemic. Furthermore, within the vulnerable fraction of the population, the herd immunity threshold will only be achieved when 50% of that subpopulation has been infected. That 50% represents an additional 25% of the whole population beyond those who were immune to start. That’s a lot better than the 50% from applying R0=2 to the whole population, but it’s not 0%. With a R0=5 in the subpopulation, we get R0=2.5 in the whole population, and herd immunity requires 60% immunity in the subpopulation, which translates to 30% additional immune in the whole population, not 10% as in Ioannidis’s analysis. And so on.

In real life, things get a lot more complicated, among other things because of partial mixing of populations with a spectrum of susceptibilities, and because the reproductive number declines as preventive actions are taken and the disease spreads. But before dealing with those complications, one needs to develop equations that give sensible answers under simple assumptions.

In the immortal words of Barbie . . . Math class is tough!