Pandemics and persistent heterogeneity

It has become increasingly clear that the COVID-19 epidemic is characterized by overdispersion whereby the majority of the transmission is driven by a minority of infected individuals. Such a strong departure from the homogeneity assumptions of traditional well-mixed compartment model is usually hypothesized to be the result of short-term super-spreader events, such as individual’s extreme rate of virus shedding at the peak of infectivity while attending a large gathering without appropriate mitigation. However, heterogeneity can also arise through long-term, or persistent variations in individual susceptibility or infectivity. Here, we show how to incorporate persistent heterogeneity into a wide class of epidemiological models, and derive a non-linear dependence of the effective reproduction number R_e on the susceptible population fraction S. Persistent heterogeneity has three important consequences compared to the effects of overdispersion: (1) It results in a major modification of the early epidemic dynamics; (2) It significantly suppresses the herd immunity threshold; (3) It significantly reduces the final size of the epidemic. We estimate social and biological contributions to persistent heterogeneity using data on real-life face-to-face contact networks and age variation of the incidence rate during the COVID-19 epidemic, and show that empirical data from the COVID-19 epidemic in New York City (NYC) and Chicago and all 50 US states provide a consistent characterization of the level of persistent heterogeneity. Our estimates suggest that the hardest-hit areas, such as NYC, are close to the persistent heterogeneity herd immunity threshold following the first wave of the epidemic, thereby limiting the spread of infection to other regions during a potential second wave of the epidemic. Our work implies that general considerations of persistent heterogeneity in addition to overdispersion act to limit the scale of pandemics.

Here is the full paper by Alexei Tkachenko, et.al., via the excellent Alan Goldhammer.  These models are looking much better than the ones that were more popular in the earlier months of the pandemic (yes, yes I know epidemiologists have been studying heterogeneity for a long time, etc.).

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It has become increasingly clear that the COVID-19 epidemic is characterized by overdispersion whereby the majority of the transmission is driven by a minority of infected individuals.

It should have been obvious a long time ago as this is basically the HIV model. There is not much transmission but among the small number of people who do get it, a very small number of people spread it to a lot of people.

Which is a polite way of saying San Francisco bathhouses.

It is likely to be true of TB as well.

The political response has been, shall we say, very different in those cases.

Yes, you are right. It should have been obvious a long time ago, at least since the study of history became a discipline. I think there are two reasons for the rediscovery of such an obvious idea.

First, today we have much better ways to learn about heterogeneity both for social science and for social engineering (two different undertakings that are often confused). While scientists look for patterns in human behavior, engineers should pay special attention to differences. This is good news, and I think we are making progress despite the noise from those too old to accept new ideas (for an example, read about the progress in macroeconomics, a field in which today it's grotesque to rely only on the categories of national accounts; btw, I regret that Tyler has ignored the recent book by his GMU colleague Richard Wagner).

Second, today many politicians and their armies of anointed intellectuals want to rely on group differences unrelated to any serious knowledge of the natural environment in which we live. Yes, they want to fabricate their categories to grab power and money. For a grotesque example, read the details discussed by one of Tyler's main sources of wisdom https://www.washingtonpost.com/pr/2020/07/29/washington-post-announces-writing-style-changes-racial-ethnic-identifiers/

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@SMFS - good observations at an initial level. However, knowing TC, I think the Straussian reading of his post is this sentence: "However, heterogeneity can also arise through long-term, or persistent variations in individual susceptibility or infectivity. ". That's why TC posted this OP. It's the "however" sentence, on the theory that there's something special about C-19 superspreaders (consistent with his UFO observations and his sonic weapons Cuban embassy observations). TC sometimes posts speculative stuff.

As for the overall thrust of the article, I would file it under "speculative". It's not clear to me that superspreaders due to individual variability in infection (the 'however' clause above) even exist. More likely superspreaders are just close talkers, people who don't cover their mouth, and people who engage in risky behavior, not genetics.

What about farting? Can a superspreader just be a gassy person?

What about close farters?

wear a diaper.

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Actually there does seem to be a big difference among individuals in terms of viral shedding that’s biological, not just if they’re close-talkers etc.:

> Among the samples collected without a face mask, we found that the majority of participants with influenza virus and coronavirus infection did not shed detectable virus in respiratory droplets or aerosols…

> Our results also indicate that there could be considerable heterogeneity in contagiousness of individuals with coronavirus and influenza virus infections.

https://www.nature.com/articles/s41591-020-0843-2

Apologies if this has been brought up before. Here is a link to a 2008 article in Virology Journal by John Cannell et al., "On the Epidemiology of Influenza” (Hat tip to @HalifaxShadow on Twitter). It's short, non-mathematical, and very readable.

The premise is that influenza isn’t spread sick-to-well. Instead, transmission starts with a small nucleus of contagious-but-asymptomatic individuals. Most of the time, conditions aren’t right for many people to become infected. But when things become more permissive, an epidemic results. The 1980s originator of this hypothesis didn't know what that trigger was. Cannell et al identify the seasonal decline in serum Vitamin D levels as that key factor.

It seems "more likely than not" that similar dynamics are at play with SARS-CoV-2. And that Vitamin D is important. Everyone should consider taking at least 1,000 IU/day -- there is virtually no downside, and the cost runs about $15 per year.

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I love when people interpret something the way they want to interpret it and call it the “Straussian reading.”

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I think it's not exactly the same as super spreading. Super spreading is generally linked to behavior ( someone with many contacts) but this model works also if one assumes that infectivity and susceptibility are not constant across individuals.
So if some people are more susceptible and more infectious it works also.
The model really assumes that susceptibility and connectivity are gamma distributed ( Gomes model) and the rest more or less follows from SIR type models

seems to me superspreading events will also include a critical environmental component

three legs of the perfect storm: spreader, spreadee, and environment

+1, yes I think this is correct

Super spreader, elderly people, singing in a choir, makes for a super spreading incident.

Indeed.

Specifically, I am thinking of temperature, humidity, airflow turbulence, etc.

I suspect we will learn more about a "sweet spot"

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Coronavirus is spread mostly by Good Clean Fun (e.g., choir practice, dancing, going out to eat with family and friends, cruise ships, ski vacations, etc.), so we quickly shut down the world to stop it.

HIV was spread mostly by promiscuous anal sex, so San Francisco took years to shut down its gay bathhouses.

That makes it sound like the Promiscuous Gay Sex lobby had more power in the 1980s than the Good Clean Fun lobby has in 2020. Interesting ...

Jeez, who knew nursing home residents, sick to the point of being near death, were so active?

Dunno about the US but in the UK there are signs that hospital staff have been a major vector of the disease. The NHS's reputation of being hopeless at infection control is many decades old.

I'm guessing that this is a major reason why people with other medical problems shunned the hospitals. I suspect this would have happened whether or not there had been a lockdown.

More generally I am suspicious of claims that lockdown demonstrably caused some phenomenon or other. How do you distinguish the lockdown and simple fear as causes of changes in behaviour?

There were far fewer cases of healthcare worker infection in the UK than in Spain or Italy. I don't think what you posit is correct.

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What was interesting about the bath house thing is that the Gay lobby was willing for lots of Gay men to die as long as no one suggested that their actions were in any way unusual or to be condemned. Until the latest lot of drugs, the only way to reduce the HIV infection rates was to change behavior. But that they were not willing to do.

And I notice San Francisco has just allowed the bath houses to re-open again despite Covid-19. Single issue lobbies are often very effective even if they are rather single minded.

But the paranoid view is that Good Clean Fun types vote Republican. So Democrat mayors and governors are willing to take a popularity hit - it will be a while before they are up for election anyway - as long as they can make Trump voters suffer. Ideally as long as they can make Trump voters riot. Thus distracting from the degree to which the Democrats and the media, but I repeat myself, have incited and encouraged rioting and looting. They can avoid some of the bad publicity if they can say *both* sides are doing it - and hence the country is spiraling out of control because of Trump.

As opposed to what we have which is the Left saying to people who might vote for Trump - Don't vote Trump, or else. The Don't Make Me Hurt You election.

"Good Clean Fun types vote Republican". Lol, just about every person out there would like these decidedly non anal sex venues to open back up, but many are not enthusiastic about risking lives to do it, nor do they want this thing to go on for many, many months because we can't get it under control once and for all.

+1. Other developed countries have this under control and are reopening. The difference between us and other developed countries is not that we have Democratic governors and mayors (for one thing, most governors are Republican, and mayors of large cities are generally from liberal or left parties in most countries), but that we have Trump.

The “paranoid view” expressed above is really projection because Republicans actually are using the pandemic as pretext to shut down immigration and international travel. Nobody wants to shut down “good clean fun” venues; doing so is viewed as a temporary necessity to fight the pandemic and I confidently predict that those venues will be reopened as soon as the virus is under control whereas I can’t say the same for immigration and international travel.

The Boogeyman theory of epidemiology. Confining oneself instead to empirical facts, the difference between us and other countries is several weeks of large-scale, close-packed protests and rioting. And look -- cases are on the rise in Oregon and Washington!

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Sorry but what other countries do you think have this under control? Everyone seems to be struggling with it still

It is absurd to blame Trump. There is little anyone can do to control this and America is a Federal system. Cuomo chose to put the sick into nursing homes. Trump did not make him. Trump probably could not stop him.

There is plenty of evidence that the Left wants to shut down as much good clean fun as they can. They have done so even if there is no actual health risk. Fishing was banned early for instance.

The Left will do their best to make sure there is no re-opening before the election because they need to crush the economy to have any chance of winning. So looting and rioting will remain permissible but Church services will not.

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Every time Germany or France or China get five to ten cases in area they shut everything down.

What large, non island nation has this thing under control?

You know who probably has this under control? NYC? Why because they’re near herd immunity and aren’t letting outsiders in.

Wrong, clearly NY has it under control because Cuomo banned Cuomo Chips

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Worst comment of the day

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what is the right wing counterpart to trump delusion syndrome? delusion delusion? TDDS

it must be sad and scary to live in a mental model where governors and health professionals shut down entire states just to tweak trump’s base

Trump being an ineffective leader is a factor, but it's absurd to say he's the critical factor.

The UK, Italy , France, Belgium and Spain have all had more deaths per capita. Trump doesn't explain that.

The US just passed France, but Peru and Chile passed the US.

+1, good catch. It's also likely that Brazil will pass the US soon also.

Though, looking at the big picture, I am somewhat apprehensive about a true second wave (different strain) rolling across the world as the Northern hemisphere starts it typical flu season in the next 4-5 months. If this happens, none of the current statistics will matter much.

Yeah maybe. The relationship between cases and deaths is dubious cuz ramping up testing, but it does seem that mortality rates are a lot lower in the states currently seeing lots of infections. Maybe it's better to get COVID during the summer. It is possible that by stumbling into Sweden, the US ends up getting beyond the virus sooner than others and with a less bad winter ahead. Who knows?

Those are thoughts I have had. Is the strain in the US predominantly more infectious and less lethal? Or is the US worse at quarantine but with a better health care system? Does NY/NJ/MA/CO have an inferior health care system to the rest of the country? Or did they make worse choices? (sending patients to nursing homes, intubating patients to a greater degree) Or did they just have a higher density thus overwhelming the health care system? Or some combination of all of the above?

"The relationship between cases and deaths is dubious cuz ramping up testing"

I think the Confirmed cases is a pretty arbitrary number, Deaths is probably a far better metric. Specifically excess mortality numbers.

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Coronavirus seems very different from HIV because HIV (at least in developed countries with controls on blood transfusions and the like) is only spread through risky unprotected sex, so someone can individually reduce their HIV risk to zero with relatively minimal precautions. By contrast, the coronavirus is spread very easily and people can get infected from it just from normal daily activities and not know where they got it. Based on that, it is more justified to close sites that could super-spread COVID while adopting a more libertarian attitude towards HIV that the individual who chooses to engage in risky unprotected sex voluntarily assumes the risk of contracting HIV.

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But the question is, wny shut down the activity rather than make sure it is in fact good "clean" fun and making it clear that the purpose of the regulation was to prevent an unaware asymptomatic person from infecting a bunch of other people.

And the larger question is why did we not do massive screening testing using pooled testing to reduce the cost in order to isolate asymptomatic infected people before they could infect (more?) people and people likely to have opportunities to infect many others -- care givers especially in nursing homes, retail clerks, waiters, transit workers, etc? This would have been a better strategy in any case, but especially valuable with greater heterogeneity.

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> (2) It significantly suppresses the herd immunity threshold;
> (3) It significantly reduces the final size of the epidemic.

Not necessarily a robust takeaway. It depends, among other things, on habitation density as well.

Mumbai seroprevalence survey estimates 40% prevalence, comprising 57% within slums and 16% without - however not everyone in the latter group participated,
so likely an underestimate. Also, due to the test kit characteristics, the actual prevalence in the former will be a few % higher.

https://twitter.com/mybmc/status/1288156896029896704

@daksya - no, you misread these bullets (1) to (3). They logically follow if indeed superspreaders are due to genetic variability. Another way of putting it (at the extreme): suppose superspreaders are the only people who ever get Covid-19 and the rest of us are immune (this is not the case but I'm using this to illustrate my point), and superspreaders are 10% of the population. Suppose herd immunity is when R0 is such that 30% of the relevant population is infected. Thus, we get herd immunity in this scenario when 0.10*0.30 of the overall population is infected, or, 3% rather than 30% as in an ordinary population where such special superspreaders don't exist. That's what this paper is arguing, see the "however" clause in my OP above.

Listen to me and you will never go wrong. Never. I'm in the 1%. You don't get into the 1% unless you are smart (or lucky).

Your illustration does not affect my point above. Even if superspreaders are solely a function of innate traits and not behaviour, in very high density habitations, each superspreader will lead to many more afflicted before they no longer remain infectious. So there will be considerable overshoot and hence not constrain "final size of the epidemic" as much as expected.

No, not really, but let's leave it at that.

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we show how to incorporate persistent heterogeneity into a wide class of epidemiological models,

So we get to pick whatever model looks good, on the basis of incomplete or imaginative data.

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The heterogeneity argument is appealing because it (a) is opaque and (b) avoids blame and responsibility. I'm reminded of the heterogeneity argument for racial and ethnic differences. In this study, the authors "estimate social and biological contributions to persistent heterogeneity . . . ." Social and biological contributions? Could it be they are connected? For example, racial and ethnic groups exhibit different social behaviors. Are different social behaviors learned or biological? Other researchers have found biological differences between races and ethnic groups. We are familiar with the smart gene and the not so smart gene with regard to intelligence. Could there be a smart gene and a not so smart gene with regard to contagions? I'm just thankful that I was born with the smart gene. I'd hate to go through life with a not so smart gene when I'm applying for college or a job or when a contagion is spreading in the neighborhood. If it's all heterogeneity, what's the point of all those protests? The protesters are barking up the wrong tree.

Ahhh argument by subterfuge.....

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-1 for content
-1 for format (block of text)
-1 per non sequitur

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"whereby the majority of the transmission is driven by a minority of infected individuals"

Though a first patient in a church choir may be identifiable, it is choir practice that that resulted in the subsequent infections. It is not that one singer that caused 80 infections, it is simply the first infected singer is the reason that practice became an example of a documented super spreading event.

Whether on a cruise ship or aircraft carrier, or in a hospital ward or nursing home, the first case is simply what started the infections spreading to large numbers of people within.

This is not an argument that certain individuals - doctors for example - may be in a position where they are more infectious than other patients, or that one can trace a chain through a particular politician's or socialite's public interactions.

It is simply that such individuals are likely a small fraction of the cases compared to recurring situations/events where conditions are ideal for the virus to spread.

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This is very, very similar to the model posted last Sunday by Tyler (Gabriela M. Gomes et al). They use a gamma-distributed susceptibility and with the coefficient of variation CV for the susceptibility α , the force of infection becomes λ = / = 1 + CV^2

So R and the susceptible are now related by = R0 S^ λ =1 for herd immunity and Herd immunity is reached at HIT = 1 - (1/Ro) ^(1/(1+CV^2)) instead of at 1 - 1/Ro

For example if R0 =3 and CV =2, R0 becomes 1.7, a much lower value and goes even lower at higher values of CV.

This was the Gabriela Gomes et al paper
https://www.medrxiv.org/content/10.1101/2020.07.23.20160762v1

The blog changes the formatting unfortunately , in the second line should read λ = / i.e second moment of alpha in the numerator, first moment in the denominator

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Correct. Having read almost all of these modeling papers, I'm increasingly of the belief that herd immunity may be obtained at the lower range of estimates. I posted the following yesterday over on Econbrowser:

While cross region comparisons are interesting, I know of no region other than maybe New Zealand that has not seen a major outbreak after initially getting the virus under control. There are obvious reasons for this but the main one is nonchalance among the public regarding physical distancing and the wearing of masks. The other key factor is the lack of knowledge about what percentage of the public that will need to get infected in order to achieve herd immunity. For some communicable diseases that number can be very high (measles and chicken pox). We do not know the answer for SARS-CoV-2. there have been a lot of very fine papers published using different modeling approaches to come up with an answer. Most standard epidemiological models early on were looking at an infection rate of 40-70%. I now feel comfortable with a 30-60% rate and it may tend towards the smaller side. Lest anyone take comfort in a low rate, simple mathematics results in 297K deaths with an infection rate of 30% and mortality rate of 0.3%. The sarcastic optimist would gleefully say that with 150K deaths we are half way home.

One quibble with your last point about 300k deaths. The death rate is also highly heterogeneous. The IFR for people under about 40 or so is something like 1/10,000.

So if you can get most of the 30% infected from that group, you would only have about 10,000 deaths after the 100m infections needed for herd immunity. Of course that’s an extreme case and you’ll never isolate the elderly so well in practice. But it does show that there was potentially a very small lower bound on the total fatalities, both in the past and going forward.

What you write is a possible outcome. There were a number of highly educated people who thought there would be fewer than 20K deaths back in February based on the data from the Diamond Princess outbreak. The problem is we still don't know enough to make a good guess and mortality is inching up to what was observed in New York City. I would certainly like to see things on the low side and have felt things could have been opened in a far safer manner.

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So the opening statement claims that overdispersion is becoming the clear main cause of transmission. But then goes on to study the effects of persistent heterogeneity of individual susceptibility, an alternate hypothesis. Was this just a math exercise for the heck of it?

No, it's not an alternative hypothesis. Persistent heterogeneity may be the cause for the overdispersion.

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They just consider there is variation (the heterogeneity) in susceptibility and infectivity and these are long term characteristics of each individual. The variation can be social ( many contacts type) or biological ( more/less susceptible). They make this susceptibility follow a gamma distribution with shape parameter and rate inverse of each other. Alpha = 1/ beta = 1/ CV^2 with CV the coefficient of variation.
This changes Ro and consequently lowers the herd immunity threshold significantly. Similar to Gomes et al but a little bit more elaborated and better presented

Thanks. Makes more sense now

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Whenever I read a post about heterogeneity I am reminded of heterodoxy. To me, it's up in the air. Here is an update on the spread of coronavirus in the air and how that affects both the frequency of transmission and the severity of symptoms (including a link to a recent study on the protection surgical masks provide for the wearer): https://www.nytimes.com/2020/07/30/opinion/coronavirus-aerosols.html [This article is in the opinion section of the NYT, but don't be mislead.]

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Fauci to teachers: 'In many respects, unfortunately, though this may sound a little scary and harsh—I don't mean it to be that way—is that you're going to actually be part of the experiment of the learning curve of what we need to know."

Liberals idolized him as the embodiment of Dr. Technocrat vs anti science Orange Man until they have skin in the game, literally.

http://blogs.edweek.org/teachers/teaching_now/2020/07/anthony_fauci_to_teachers_youll_be_part_of_the_experiment_in_reopening_schools.html

Not so. He has been criticized severely for not messaging cleary form day one that everyone needs to wear masks to protect other people. Even when he changed his recommendation, he did not explain the reason to wear masks and to distance socially. This has induced behavior change based on self interest and self assessment of risk to oneself, rather than concern for others.

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Models. How do you model masks becoming a political issue? How do you model *removing* safe distancing signs as a political imperative? Or an indoor rally in a hot zone?

Does the tragic death of (good guy) Herman Cain come out of your model in the end? Like on the paper tape in a Tracy-Hepburn movie?

I'm very saddened at the moment, more so because I don't see Cain's story as unique. I've got a feeling that a lot of our MR friends would have made the same choices for the same reasons, and exposed themselves to the same unnecessary risks.

And no, obviously you can't model this. It's way too far into a very bizarre realm of political complexity.

Supplemental:

GOP Staffers Detail Ridicule for Wearing Masks at Capitol

All I can say is that you gotta have a pretty deep faith in "models" if you think you can average across that.

Let alone predict it months ago, from starting conditions.

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Yes, ultimately models do poorly because they must predict human behavior, not just virus behavior.

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My trending of town to town trends here in Connecticut show this clearly. The original infections traveled from NYC by Commuter trains and the hot spots were located there. Subsequent trends showed orders of magnitude differences in towns of similar population located next to each other. None of the strong trends showed any significant impact from lockdowns but did show post peak reductions similar to the patterns in NYC. I'd call that effective herd immunity

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I know some of these authors very well. They are a cut above the usual epidemic modelers. They may not be right, but I care what they say 1000x more than traditional teams like the Imperial College team.

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