Thread by @gro_tsen

I should probably write a preventive thread about this, because I feel I'm going to get a few comments of the kind “more than 60% in place <X> have been infected by covid, and infections are still taking place! this proves that herd immunity DOES NOT WORK! ChEcKmAtE!!!”. •1/24

So yes, I've claimed a number of times, and I still do, that the trivial estimation of the collective immunity threshold given by the formula 1 − 1/R, which gives 60% for R=2.5, is pessimistic (but that it's hard to figure out the true value). •2/24

This is essentially because the reasoning behind this formula assumes a homogeneous population (everyone is equally likely to get infected) with perfect mixing (everyone is equally likely to infect anyone) and deviations from this lower the threshold. •3/24

Basically, any kind of heterogeneity within the population (geographical, socio-economic, individual, etc.) will tend to lower the collective immunity threshold. The details can be complex, but the general idea isn't complicated to undestand. {BEGIN SUMMARY} •4/24

‣ In the case of individual variations of social contacts: individuals who tend to have more contacts than others will be infected sooner, and by becoming immune they will contribute a larger share of reduction of disease infectiousness than predicted naïvely. •5/24

(I wrote a thread about this very early on in the pandemic: threadreaderapp.com/thread/1241745… — before I knew that the phenomenon had been studied and confirmed by others, so I didn't know the right terminology at that time.) •6/24

‣ In the case of heterogeneity between population groups: this is because the observed reproduction number for the population is essentially that of the subpopulation which has most infections, i.e., that in which it reproduces most. … •7/24

… But the collective immunity threshold, on the other hand, is an average. So you observe essentially the highest reproduction number of all subpopulation, but the one which matters to collective immunity is an average. This leads to an overestimation. •8/24

E.g., if you have R=2.5 in urban areas and R=1.5 in rural areas (random example), you will observe nearly 2.5 for the country as a whole, because urban areas are hit earlier, but herd immunity is attained by 60% in urban areas and 35% in rural areas: less than 60% overall. •9/24

And there's nothing specific about urban-vs-rural in the above example, it works with any kind of subgroups within the population, provided (more or less) that intra-group infections dominate inter-group infections. •10/24

(I've written about all of this many times over, lastly in this blog post of mine (esp. second bullet point): madore.org/~david/weblog/… — it's in French but Google Translate generally does a decent job for these things.) •11/24

So anyway, the idea is that while these phenomena are hard to model so it's impossible to compute the collective immunity threshold accurately, the simplistic formula 1 − 1/R will lead to a possibly gross overestimation. {END SUMMARY} •12/24

But there are a number of caveats here: one is that collective immunity being reached does not mean that the epidemic disappears, it merely means that it starts to wane. There can be an “overshoot” between herd immunity and the final attack rate. •13/24

(In general, I don't think this overshoot is a problem, in fact epidemics tend to undershoot rather than overshoot the herd immunity threshold, which is why they come in waves. But if the epidemic is fast enough, it can happen. More about this: madore.org/~david/weblog/…‌) •14/24

Another caveat is that R fluctuates in time for many reasons other than the accruement of immunity: this might be for sociological or environmental reasons. Immunity gained for one value of R no longer fully works for a higher value, so a rebound can happen. •15/24

Another caveat is that we need to distinguish “unconditional” herd immunity, which applies to a population which neither knows nor cares about the infection, and “conditional” one, where (and while!) the population takes various precautions which lower the effective R. •16/24

I've written about this distinction, and explained why it makes the whole concept rather slippery to define, in the following thread: threadreaderapp.com/thread/1324405… •17/24

And of course, more important caveats are that both the value of R (for a given time and place) and the attack rate (number of people having been infected so far) are known with considerable uncertainty (especially the latter). •18/24

The value of 2.5, or sometimes 3, which is sometimes stated for covid, is a debatable conventional value, which stems from early estimates of the Chinese phase of the pandemic. But people in other countries may behave differently, especially now they have heard of covid. •19/24

So 60% is an essentially meaningless value, for a loosely defined concept (herd immunity conditional to what?), pulled out of a simplistic formula 1 − 1/R by plugging in an equally simplistic value of R₀ = 2.5. I don't think it has any relevance whatsoever. •20/24

And in general, I expect collective immunity to kick in sooner, perhaps much sooner. BUT, and here's an important but, it can also kick in later! In fact, that's the whole deal about heterogeneity: the immunity threshold of the whole is lowered by variations in the parts. •21/24

So if you've found a place where more than 60% of people have been infected and the epidemic is still raging, well, it proves that there's a lot of heterogeneity between places on Earth because, you know, most places haven't reached nearly 60%. •22/24

And if you've cherry-picked that place to be one with a high attack rate, it will also probably be a place with a high reproduction number, so it's not surprising that its collective immunity threshold is way above 60%. •23/24

To summarize, the fact that some places on Earth have reached phenomenal attack rates and perhaps not even gained collective immunity in no way proves that elsewhere it can't be gained at a lower threshold. In fact, it might even suggest the contrary! •24/24