New Paper on Herd Immunity Thresholds

Link post

Pre­vi­ously: On R0

This new pa­per sug­gests that herd im­mu­nity could be achieved with only about 10% in­fected rather than the typ­i­cally sug­gested 60%-70%.

They claim this is due to differ­ences in con­nec­tivity and thus ex­po­sure, and in sus­cep­ti­bil­ity to in­fec­tion. They claim that the best model fit for four Euro­pean epi­demics at 16%-26% for England, 9.4%-11% for Belgium, 7.1%-9.9% for Por­tu­gal, and 7.5%-21% for Spain.

This be­ing ac­cu­rate would be ex­cel­lent news.

The 60%-70% thresh­old com­monly thrown around is, of course, ut­ter non­sense. I’ve been over this sev­eral times, but will sum­ma­rize.

The 60%-70% re­sult is based on a fully naive SIR (sus­cep­ti­ble, in­fected, re­cov­ered) model in which all of the fol­low­ing are as­sumed to be true:

  1. Peo­ple are iden­ti­cal, and have iden­ti­cal sus­cep­ti­bil­ity to the virus.

  2. Peo­ple are iden­ti­cal, and have iden­ti­cal abil­ity to spread the virus.

  3. Peo­ple are iden­ti­cal, and have iden­ti­cal ex­po­sure to the virus.

  4. Peo­ple are iden­ti­cal, and have con­tacts com­pletely at ran­dom.

  5. The only in­ter­ven­tion con­sid­ered is im­mu­nity. No help from be­hav­ior ad­just­ments.

All five of these mis­takes are large, and all point in the same di­rec­tion. Im­mu­nity mat­ters much more than the ‘naive SIR’ model thinks. What­ever the thresh­old for im­mu­nity might be for any given ini­tial re­pro­duc­tion rate, it’s nowhere near what the naive SIR out­puts.

Often they even take the num­ber of cases with pos­i­tive tests to be the num­ber of in­fec­tions, and use that to pre­dict for­ward or train their model.

This naive model is not a straw man! Such ob­vi­ous non­sense mod­els are the most com­mon mod­els quoted by the press, the most com­mon mod­els quoted by so-called ‘sci­en­tific ex­perts’ and the most com­mon mod­els used to de­ter­mine policy.

The effec­tive re­sponse is some com­bi­na­tion of these two very poor ar­gu­ments:

  1. “Un­til you can get a good mea­sure­ment of this effect we will con­tinue to use zero.”

  2. “Tel­ling peo­ple the thresh­old is lower will cause them to take less pre­cau­tions.”

Nei­ther of these is how knowl­edge or sci­ence works. It’s mo­ti­vated cog­ni­tion. Pe­riod.

See On R0 for more de­tails on that.

So when I saw this pa­per, I was hop­ing it would provide a bet­ter per­spec­tive that could be con­vinc­ing, and a rea­son­able es­ti­mate of the mag­ni­tude of the effect.

I think the mag­ni­tude they are sug­gest­ing is very rea­son­able. Alas, I do not think the pa­per is con­vinc­ing.

The Model

The model in­volves the use of calcu­lus and many un­ex­plained Greek let­ters. Thus it is im­pres­sive and valid.

If that’s not how sci­ence works, I fail to un­der­stand why they don’t ex­plain what the hell they are ac­tu­ally do­ing.

Take the model de­scrip­tion on page four. It’s all where this let­ter is this and that let­ter is that, with non-ex­plicit as­sump­tion upon non-ex­plicit as­sump­tion. Why do peo­ple write like this?

I tried to read their de­scrip­tion on page 4 and their model made zero sense. None. The good news is it made so lit­tle sense that it was ob­vi­ous that I couldn’t pos­si­bly be suc­cess­fully mak­ing heads or tails of the situ­a­tion, so I deleted my at­tempt to write up what I thought it meant (again, to­tal gib­ber­ish) and in­stead I went in search of an ac­tual ex­pla­na­tion later in the pa­per.

All right, let’s get to their ac­tual as­sump­tions on page 19, where they’re writ­ten in English, and as­sume that the model cor­rectly trans­lates from the as­sump­tions to the re­sult be­cause they have other peo­ple to check for that.

They be­lieve the in­fec­tivity of ‘ex­posed in­di­vi­d­u­als’ is half that of in­fec­tious ones, and that this pe­riod of be­ing an ‘ex­posed in­di­vi­d­ual’ takes four days to de­velop into be­ing in­fec­tious. Then they are in­fec­tious for an av­er­age four days, then stop.

That’s not my model. I don’t think some­one who caught the virus yes­ter­day is half as in­fec­tious as they will be later. I think they’re es­sen­tially not in­fec­tious at all. This mat­ters a lot! If my model is right, then if you go to a risky event on Sun­day, some­one see­ing you on Mon­day is still safe. Un­der this pa­per’s model, that Mon­day meet­ing is dan­ger­ous. In fact, given the per­son has no symp­toms yet and the per­son they caught it from still doesn’t, it’s very dan­ger­ous. That’s a big deal for prac­ti­cal plan­ning. It makes it much harder to be rel­a­tively safe. It makes it much harder to use­fully con­tact trace. Prob­a­bly other im­pli­ca­tions as well.

What it doesn’t change much is the ac­tual re­sult. Th­ese are not the maths we are look­ing for, and their an­swers don’t much mat­ter.

That’s be­cause they’re con­trol­led for by as­sum­ing the origi­nal R0, slash what­ever as­sump­tion you make about the mean level of in­fec­tivity and sus­cep­ti­bil­ity.

Tech­ni­cally, yes, there’s a differ­ence. Every­thing is con­tin­u­ous, and the ex­act timing of when peo­ple are how in­fec­tious changes the pro­gres­sion of things a bit. A bit, but only a bit. If what we are do­ing is calcu­lat­ing the herd im­mu­nity thresh­old, you can pick any curve slope you want for ex­actly when peo­ple in­fect oth­ers. It will effect how long it takes to get to herd im­mu­nity. Big changes in av­er­age de­lay times would mat­ter some (but again, over rea­son­able guesses, I’m think­ing not enough to worry about) for how far we can ex­pect to over­shoot herd im­mu­nity be­fore bring­ing in­fec­tions down.

But the num­ber of in­fected re­quired will barely change. The core equa­tion doesn’t care. Why are we jump­ing through all these hoops? Who is this go­ing to con­vince, ex­actly?

This is ac­tu­ally good news. If all the as­sump­tions in that sec­tion don’t mat­ter, then none of them be­ing wrong can makes the model wrong.

Se­cond as­sump­tion is that ac­quired im­mu­nity is ab­solute. Once you catch Covid-19 and re­cover, you can’t catch it again. This pre­sum­ably isn’t strictly true, but as I keep re­peat­ing, our con­tinued lack of large scale re­in­fec­tion makes it more ap­prox­i­mately true ev­ery day.

Third as­sump­tion they sug­gest is that peo­ple with similar lev­els of con­nec­tivity are more likely to con­nect, rel­a­tive to ran­dom con­nec­tions be­tween in­di­vi­d­u­als. This seems ob­vi­ously true on re­flec­tion. It’s not a good full pic­ture of how peo­ple con­nect, but it’s a move in the right di­rec­tion, un­less it goes too far. It’s hard to get a good feel for how big this effect is in their model, but I think it’s very rea­son­able.

Fourth as­sump­tion is that there is var­i­ance in the de­gree of con­nec­tivity, and so­cial dis­tanc­ing low­ers the mean and var­i­ance pro­por­tion­ally (so the var­i­ance as a pro­por­tion of the mean is un­changed). They then note that it is pos­si­ble that so­cial dis­tanc­ing de­creases differ­en­ti­a­tion in con­nec­tivity, which would effect their re­sults. I don’t know why they think about this as a one-way is­sue. Per­haps be­cause as sci­en­tists they have to be con­cerned with things that if true would make their find­ing weaker, but ig­nore if they would make them stronger and are spec­u­la­tive. They sug­gest a vari­a­tion where so­cial dis­tanc­ing re­duces con­nec­tivity var­i­ance.

I would ask which di­rec­tional effect is more likely here. It seems to me more likely that so­cial dis­tanc­ing in­creases var­i­ance. If R0 be­fore dis­tanc­ing was some­where be­tween 2.6 and 4, and it cuts it to some­thing close to 1, that means the av­er­age per­son is cut­ting out 60% to 75% of their effec­tive con­nec­tivity. By con­trast, at least half the peo­ple I know are cut­ting more than 90% of their con­nec­tivity, and also cut­ting their phys­i­cal ex­po­sure lev­els when con­nect­ing, on top of that. In many cases, it’s more than 95%, and in some it’s 99%+. If any­thing, the ex­ist­ing in­tro­verts are do­ing larger per­centage cuts while also feel­ing bet­ter about the lifestyle effects. Whereas es­sen­tial work­ers and kids who don’t care and those who don’t be­lieve this is a real thing likely are not cut­ting much con­nec­tivity at all.

I’ve talked about it enough I don’t want to get into it be­yond that again here, but I’d ex­pect higher var­i­ance dis­tri­bu­tions than be­fore. The real con­cern is whether the con­nec­tivity lev­els dur­ing dis­tanc­ing are no longer that cor­re­lated to those with­out dis­tanc­ing, be­cause that would mean we weren’t get­ting the full se­lec­tion effects. The other hid­den vari­able is if peo­ple who are im­mune then seek out higher con­nec­tivity. That effec­tively greatly am­plifies so­cial dis­tanc­ing. Im­mu­nity pass­ports two months ago.

Fifth, they mod­eled ‘non-phar­ma­ceu­ti­cal in­ter­ven­tions’ as a grad­ual low­er­ing of the in­fec­tion rate. This is sup­posed to cover masks, dis­tanc­ing, hand wash­ing and such. They said 21 days to im­ple­ment dis­tanc­ing, then 30 days at max effec­tive­ness, then a grad­ual lift­ing whose speed does not im­pact the model’s re­sults much.

They then take the ob­served data and use Bayesian in­fer­ence to find the most likely pa­ram­e­ters for their model.

To do that, they made two ad­di­tional sim­plify­ing as­sump­tions.

The first was that the frac­tion of cases that were iden­ti­fied is a con­stant through­out the pe­riod of data re­ported. This is false, of course. As time went on, test­ing ev­ery­where im­proved, and at higher in­fec­tion rates test­ing gets over­whelmed more eas­ily and peo­ple are less will­ing to be tested. They are us­ing Euro­pean data, which means there might be less im­pact than this would have in Amer­ica, but it’s still pretty bad to as­sume this is a con­stant and I’m sad they didn’t choose some­thing bet­ter. I don’t know if a differ­ent as­sump­tion changes their an­swers much.

The sec­ond was that lo­cal trans­mis­sion starts when coun­tries/​re­gions re­port 1 case per 5 mil­lion pop­u­la­tion in one day. An as­sump­tion like this seems deeply silly, like flip­ping a switch, but I pre­sume the model needed it and choos­ing the wrong date to start with would be mostly harm­less. If it would be a sub­stan­tial im­pact, then shall we say I have con­cerns.

They then use the serolog­i­cal test in Spain and used it to calcu­late that the re­port­ing rate of in­fec­tions in Spain was around 6%. That seems to me to be on the low end of re­al­is­tic. If any­thing, my guess would be that the serolog­i­cal sur­vey was an un­der­count, be­cause it seems likely some peo­ple don’t show im­mu­nity on those tests but are in­deed im­mune, but the re­sult­ing num­ber seems rel­a­tively low so I’ll ac­cept it.

They then use the rate of PCR test­ing rel­a­tive to Spain in the other con­tries to get re­port­ing rates of 9% for Por­tu­gal, 6% for Belgium and 2.4% for England. That 2.4% num­ber is dra­mat­i­cally low given what we know and I’m sus­pi­cious of it. I’m cu­ri­ous what their guess would be for the United States.

Then they took the best fit of the data, and pro­duced their model.

Any­one Con­vinced?

Don’t all yell at once. My model Doesn’t think any­one was con­vinced. Why?

The pa­per doesn’t add up to more than its key in­sight, nor does it prove that in­sight.

Either you’re go­ing to buy the core in­sight of the pa­per the mo­ment you hear it and think about it (which I do), in which case you don’t need the pa­per. Or you don’t buy the core in­sight of the pa­per when you hear it, in which case noth­ing in the pa­per is go­ing to change that.

The core in­sight of the pa­per is that if differ­ent peo­ple are differ­ently vuln­er­a­ble to in­fec­tion, and differ­ent peo­ple have differ­ent amounts of con­nec­tivity and ex­po­sure, and those differ­ences per­sist over time, then the peo­ple who are more vuln­er­a­ble and more con­nected get in­fected faster, and thus herd im­mu­nity’s thresh­old is much lower.

Well, no shirt, Sher­lock.

If the above para­graph isn’t enough to make that point, will the pa­per help? That seems highly un­likely to me. Any­one will­ing to think about the phys­i­cal world will re­al­ize that differ­ent peo­ple have rad­i­cally differ­ent amounts of con­nec­tivity. Most who think about the phys­i­cal world will con­clude that they also have im­por­tantly differ­ent lev­els of vuln­er­a­bil­ity to in­fec­tion and abil­ity to in­fect, and that those two will be cor­re­lated.

Most don’t buy the in­sight.

Why are so few peo­ple buy­ing this seem­ingly triv­ial and ob­vi­ous in­sight?

I gave my best guess in the first sec­tion. It is seen as an ar­gu­ment, and there­fore a solider, for not deal­ing with the virus. And it is seen as not le­gi­t­i­mate to count some­thing that can’t be quan­tified – who are you to al­ter the hard num­bers and ba­sic math with­out a bet­ter an­swer you can defend? Thus, mod­esty, and the choice of an es­ti­mate well out­side the realm of the plau­si­ble.

Add in that most peo­ple don’t think about or be­lieve in the phys­i­cal world in this way, as some­thing made up of gears and cause and effect that one can figure out with logic. They hear an ex­pert say ‘70%’ and think noth­ing more about it.

Then there are those who do buy the in­sight. If any­thing, I am guess­ing the pa­per dis­cour­ages this, be­cause its most promi­nent effect is to point out that ac­cept­ing the in­sight im­plies a su­per low im­mu­nity thresh­old, thus caus­ing peo­ple to want to re­coil.

Once you buy the in­sight, we’re talk­ing price. The pa­per sug­gests one out­come, but the pro­cess they use is suffi­ciently opaque and ar­bi­trary and de­pen­dent on its as­sump­tions that it’s more proof of con­cept than any­thing else.

It’s mostly per­mis­sion to say num­bers like ‘10% im­mu­nity thresh­old’ out loud and have a pa­per one can point to so one doesn’t sound crazy. Which is use­ful, I sup­pose. I’m happy the pa­per ex­ists. I just wish it was bet­ter.

There’s noth­ing es­pe­cially ob­vi­ously wrong with the model or their fi­nal es­ti­mate. But that does not mean there’s noth­ing wrong with their model. Hell if I know. It would take many hours pour­ing over de­tails and likely im­ple­ment­ing the model your­self and tin­ker­ing with it be­fore one can have con­fi­dence in the out­puts. Only then should it provide much ev­i­dence for what that fi­nal price should look like.

And it should only have an im­pact then if the model is in prac­tice do­ing more than stat­ing the ob­vi­ous im­pli­ca­tions of its as­sump­tions.

If this did pa­per did con­vince you, or failed to con­vince you for rea­sons other than the ones I give here, I’m cu­ri­ous to hear about it in the com­ments.

How Var­i­ant is Con­nec­tivity Any­way?

I think very, very var­i­ant. I hope to not re­peat my prior ar­gu­ments too much, here.

Out of cu­ri­os­ity, I did a Twit­ter poll on the dis­tri­bu­tion of con­nec­tivity, and got this re­sult with 207 votes:

Divide USA into 50% of non-im­mune in­di­vi­d­u­als tak­ing rel­a­tively less Covid-19 risk and 50% tak­ing rel­a­tively more. What % of to­tal risk is be­ing taken by the safer 50%?

Less than 10%: 27.5%

10%-15%: 22.2%

15%-25%: 19.3%

25-50%: 30.9%

I would have voted for un­der 10%.

This is an al­most ex­actly even split be­tween more or less than 15%, so let’s say that the bot­tom 50% ac­count for 15% of the risk, and the other 50% ac­count for 85% of the risk.

If we as­sumed the na­tion was only these two pools, and peo­ple got in­fected pro­por­tion­ally to risk taken, what does this make the herd im­mu­nity thresh­old?

Let’s con­tinue to be con­ser­va­tive and as­sume ini­tial R0 = 4, on the high end of es­ti­mates.

For pure SIR, im­mu­nity thresh­old is 75%.

With two classes of peo­ple, im­mu­nity thresh­old is around 35%.

Ad­ding even one ex­tra cat­e­gory of peo­ple cuts the herd im­mu­nity thresh­old by more than half, all on its own.

If this 8515 rule is even some­what frac­tal, we are go­ing to get to herd im­mu­nity very quickly.

Hope­fully this was a good ba­sic in­tu­ition pump for how effec­tive such fac­tors can be – and it seems more con­vinc­ing to me than the pa­per was.

Can We Quan­tify This Effect?

Yes. Yes, we can.

We haven’t. And we won’t. But we could!

It would be easy. All you have to do is find a sur­vey method that gen­er­ates a ran­dom sam­ple, and find a dis­tri­bu­tion of con­nec­tivity. Then give ev­ery­one an­ti­body tests, then ex­am­ine the re­sult­ing data. For best re­sults on small sam­ples, also give the sur­vey to peo­ple who have already tested pos­i­tive.

This is not a hard prob­lem. It re­quires no con­trol­led ex­per­i­ments and en­dan­gers no test sub­jects. It has huge im­pli­ca­tions for policy. Along the way, you’d also be able to quan­tify risk from differ­ent sources.

Then you can use that data to cre­ate the model, and see what thresh­old we’re deal­ing with.

That’s the study that needs to hap­pen here. It prob­a­bly won’t hap­pen.

Un­til then, this is what we have. It’s not con­vinc­ing. It’s not mak­ing me up­date. But it is a study one can point to that sup­ports an ob­vi­ously cor­rect di­rec­tional up­date, and comes up with a plau­si­ble es­ti­mate.

So for that, I want to say: Thank you.