Covid: The Invisible Elephant in the Room

Calculations of the requirement for herd immunity are based, as I understand it, on the implicit assumption that everyone is equally at risk, hence that the probability that one infected person will infect more than one other can be deduced from data on the early spread of the disease. That assumption is unlikely to be true, for both behavioral and biological reasons. Some people spend more time interacting at close range with those likely to be speaking loudly than others. And it seems likely that some can catch the disease more easily than others.

Suppose we drop that assumption. 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. 

The real world distribution of vulnerability will, of course, be much more complicated than that, but the qualitative conclusion still holds. Over time, the people most vulnerable will be most likely to get the disease, so the average vulnerability of those who have not yet gotten it will decline, lowering the average number that each infected person passes the disease to. Hence herd immunity will come sooner than the simple calculation implies.

So far I have only considered differences in how easily individuals can get the disease. There will be similar differences, at least for behavioral reasons, to how easily individuals can transmit it. The two will tend to correlate — someone who spends a lot of time in loud conversation with lots of others will be more likely than average to get the disease and, if he gets it, more likely to pass it on. So, over time, the probability of transmission will fall as those most likely to transmit are selectively removed from the pool of potential transmitters.

All of this tells us that herd immunity will come earlier than the simple calculation implies, but not how much earlier — that depends on the actual distribution of vulnerability, of probability of transmission, and the correlation between the two.

Has anyone done the research necessary to estimate these numbers and recalculated the requirement for herd immunity according?

A further problem with the simple calculation is that it ignores behavioral changes due to the pandemic itself. Presumably part of the point of lockdowns was to temporarily push the transmission rate below one, driving the virus to near extinction — far enough down so that it could be controlled by a test and trace approach. In most countries that imposed a lockdown that didn't happen, but even without a lockdown the existence of the pandemic changes behavior in ways that should reduce the transmission rate below its initial value, at least somewhat.

P.S. a commenter on another blog where I raised the post gave two links to relevant material. The first goes to the editors' blog of Science magazine. The most interesting bit is:

we were concerned that forces that want to downplay the severity of the pandemic as well as the need for social distancing would seize on the results to suggest that the situation was less urgent. We decided that the benefit of providing the model to the scientific community was worthwhile.

That implies that the editors believe that part of their job is filtering the scientific literature in order to bias the public perception in the direction they approve of, although in this case they decided not to. It follows that one cannot take the published scientific literature on any controversial issue as giving an unbiased picture of the actual science. That is disturbing, but not surprising.

The post contains a link to the paper, which appears to be simply a fancier version of my argument, without actual empirical data that could be used to figure out the size of the effect.

The second link goes to a paper which, judged by the abstract, does make an attempt at estimating real world numbers, and concludes that particularly hard hit areas, such as New York City, may already be close to herd immunity. I couldn't find the full text online.

But a commenter could.