Pandemic blog 32: writing

Taylor Swift surprised everyone by releasing a surprise new album, which she wrote and recorded entirely during the quarantine. My favorite song on it is the poignant “Invisible String”

which has an agreeable Penguin Cafe Orchestra vibe, see e.g.

(The one thing about “Invisible String” is that people seem to universally read it as a song about how great it is to finally have found true love, but people, if you say

And isn’t it just so pretty to think
All along there was some
Invisible string
Tying you to me?

you are (following Hemingway at the end of The Sun Also Rises) saying it would be lovely to think there was some kind of karmic force-bond tying you and your loved one together, but that, despite what’s pretty, there isn’t, and you fly apart.)

Anyway, I too, like my fellow writer Taylor Swift, have been working surprisingly fast during this period of enforced at-homeness. Even with the kids here all the time, not going anywhere is somehow good writing practice. And this book I’m writing, the one that’s coming out next spring, is now almost done. I’m somewhat tetchy about saying too much before the book really exist, but it’s called Shape, there is a lot of different stuff in it, and I hope you’ll like it.

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Pandemic blog 31: farmers’ market

First trip back to the Westside Community Market, which in ordinary times is an every Saturday morning trip for me. It feels like a model for people just sitting down and figuring out how to arrange for people to do the things they want to do in a way that minimizes transmission. We don’t have to eliminate every chance for someone to get COVID. If we cut transmissions to a third of what it would otherwise be, that doesn’t mean a third as many people get COVID — it means the pandemic dies out instead of exploding. Safe is impossible, safer is important!

They’ve reorganized everything so that the stalls are farther apart. Everybody’s wearing masks, both vendors and customers. There are several very visible hand-washing stations. Most of the vendors now take credit cards through Square, and at least one asked me to pay with Venmo. It’s easy for people to keep their distance (though the vendors told me it was more crowded earlier in the morning.)

And of course it’s summer, the fields are doing what the fields do, the Flyte Farm blueberries, best in Wisconsin, are ready — I bought five pounds, and four containers of Murphy Farms cottage cheese. All you need is those two things for the perfect Wisconsin summer meal.

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Pandemic blog 30: opening day

I have been generally feeling: it is OK to start relaxing restrictions on in-person contact, because there seems some decent chance that barring the most infectiogenic scenarios might be enough to keep outbreaks small and manageable. And that still might be true, in some contexts; in Dane County, we had a big spike of cases when the bars re-opened, and when the bars shut down again, the case spike went away, and hasn’t come back, though people are certainly out and about. But statewide, cases are growing and growing, and the situation is much worse in the South. I would fight back if you said this was a predictable consequence; nothing about this disease is predictable with any confidence. It could have worked. But I wouldn’t fight you if you said it was an expectable consequence, the consequence you thought most likely.

Similarly, if you rigorously jettison everyone with a demonstrated ability to play baseball from your team, and sign a collection of promising young players but keep them off the roster in order to avoid starting their service time, and then put that team on the field against major league competition, you might find that the nobodies and never-weres and used-to-bes find it within themselves to go on a scrappy “Why not?” run of success; or you might, as an expectable consequence, give up eight doubles and get beat 13-2.

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Pandemic blog 29: Neowise, custard

Comets are supposed to presage plague so I guess this one, Neowise, must have taken a wrong turn and showed up four months late? Anyway, AB and I saw it — bright enough that you don’t have to find a remote hillock with a northward view over lightless fields, you can just head over to Hoyt Park and look out over the city lights. With naked eye you can just barely see the comet, mostly in your peripheral vision; through binoculars, the tail is quite clear. AB was over the moon about it. She has seen a comet and a total eclipse in her first ten years of life; not bad!

This is my third comet, I think. Hale-Bopp, very visible, late 90s, everybody saw it. And the 1986 Halley’s, which was disappointingly dim; I don’t think I ever saw it with my eyes, but we were visiting my grandmother in Tucson and the UA observatory was letting people come in and look at Halley’s comet through their telescope, so I did that. But in the end seeing it through a big fixed scope at the observatory isn’t that different from seeing a picture of it.

After four months, I decided it was OK for me to eat purchased food again. Since then I have done it three times, and all three times it was Michael’s Frozen Custard. Eating only home-cooked food and Michael’s Frozen Custard is like the pescatarianism of quarantine. I had an idea in my mind that the second I ate a bite of restaurant food, all the weight I’d lost during the abstention would instantly reappear, like in the Simpsons episode where Barney stops drinking Duff beer but then has one. It didn’t happen. I might need more custard.

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Pandemic blog 28: Smart Restart

What’s going to happen to school in the fall? Madison schools are talking about having two days on, three days off, with half the kids going on Monday and Tuesday and half on Thursday and Friday.

I think if we open anything it has to be schools. And it seems pretty clear we are not not opening anything. If there’s no school, how are people with young kids supposed to work?

There’s decent evidence that young kids are less likely to get infected with COVID, less likely to spread it, and drastically less likely to become seriously ill from it — so I don’t think it’s crazy to hope that you can bring kids together in school without taking too much of a hit to public health.

What about college? UW-Madison is proposing a “Smart Restart” plan in which students come back to dorms, on-campus instruction starts in a limited fashion (big classes online, small classes taught in big rooms with students sitting far apart.) A lot of my colleagues are really unhappy with the fact that we’re proposing to bring students back to campus at all. I’m cautiously for it. I am not going to get into the details because more detail-oriented people than me have thought about them a lot, and I’m just sitting here blogging on Independence Morning.

But three non-details:

  1. Given the high case numbers among college students in Madison now, just from normal college student socializing, it’s not clear to me that asking them to come to class is going to make a notable difference in how much COVID spread the student population generates.
  2. Any plan that says “Protect the most vulnerable populations, like old people, but let young healthy people do what they want” that doesn’t include “vulnerable people who can’t safely do their jobs because their workplaces are full of young, healthy, teeming-with-COVID people get paid to stay home” is not a plan. We can’t make 65-year-old teachers teach in person and we can’t make diabetic teachers teach in person and we can’t make teachers with elderly relatives in the household teach in person.
  3. Any plan for re-opening schools has to have pretty clear guidelines for what triggers a reverse of course. We cannot figure out what’s safe, or “safe enough,” by pure thought; at some point we have to try things. But a re-opening plan that doesn’t include a re-closing plan is also not a plan.

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Pandemic blog 27: Impossible Stroganoff

We are down to once every three weeks at Trader Joe’s (I fill two whole carts with stuff, it’s an undertaking) which we supplement with other kinds of food purchases in between. I’m unhappy with the conditions industrial meatpackers are putting their workers in, so I’m picking up meat curbside at Conscious Carnivore, our local meat-from-nearby-farms-you’re-supposed-to-feel-vaguely-OK-about supplier. We get shipments from Imperfect Foods, which I’m a little concerned is some kind of hedge-fund-backed grocery store destruction scheme but helps fill in the gaps. And the really exciting food news is that Impossible Foods, the substitute meat company I learned about from my old math team buddy Mike Eisen, is now delivering!

This stuff is by far the most realistic fake ground beef in existence. We served Impossible cheeseburgers at CJ’s bar mitzvah and a member of the ritual committee was so convinced he was ready to pull the fire alarm and evacuate the shul for de-trayfing. Since I don’t cook milk and meat together in the house, there are a lot of dishes that just don’t happen at home. And one of them — which I’ve been waiting years to make — is my favorite dish from childhood, “hamburger stroganoff.”

This dish comes from Peg Bracken’s protofeminist masterpiece, the I Hate To Cook Book. Is that book forgotten by younger cooks? It’s decidedly out of style. Maybe it was even out of style then; my mom, I always felt, made hamburger stroganoff grudgingly. It involves canned soup. But it is one of the most delicious things imaginable and readers, the Impossible version is almost indistinguishable from the real thing.

Here’s Peg Bracken’s obituary, which leads with the famous lines from this famous recipe:

Start cooking those noodles, first dropping a bouillon cube into the noodle water. Brown the garlic, onion and crumbled beef in the oil. Add the flour, salt, paprika and mushrooms, stir, and let it cook five minutes while you light a cigarette and stare sullenly at the sink.

And here’s the recipe itself. If you’re vegetarianizing this, you can just use cream of mushroom soup for the cream of chicken and replace the bouillon with some salt (or veggie stock, if that’s your bag.)

8 ounces Noodles, uncooked
1 cube Beef Bouillon
1 clove Garlic,minced
1/3 cup Onion, chopped
2 tablespoons Cooking oil
1 pound Ground Beef
2 tablespoons Flour
2 teaspoons Salt
1/2 teaspoon Paprika
6 ounces Mushrooms
1 can Cream of Chicken Soup, undiluted
1 cup Sour Cream
1 handful Parsley, chopped

Start cooking those noodles, first dropping a boullion cube into the noodle water.
Brown the garlic, onion, and crumbled beef in the oil.
Add the flour, salt, paprika, and mushrooms, stir, and let it cook five minutes while you light a cigarette and stare sullenly at the sink.
Then add the soup and simmer it–in other words, cook on low flame under boiling point–ten minutes.
Now stir in the sour cream–keeping the heat low, so it won’t curdle–and let it all heat through.
To serve it, pile the noodles on a platter, pile the Stroganoff mix on top of the noodles, and sprinkle chopped parsley around with a lavish hand.

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Pandemic blog 26: writing

I was supposed to turn in a manuscript for my new (general-audience book) last week. It’s not finished. But I’ve written a lot of it during the pandemic. Of course it is very hard to be “productive” in the usual way, with the kids here all day. But being in the house all day is somehow the right setup for book-writing, maybe because it so clearly separates life now from my usual life where I am neither staying in the house nor writing a book.

I think the pages I’m putting out are good. As usual, the process of writing is causing me to learn new things faster than I can put them in the book and indeed there is now too much material to actually go in the book, but that means, at any rate, I can be selective and pick just the best.

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Pandemic blog 25: version

I’ve always thought of myself as an extravert, but am I? I read lots of articles about the almost unendurable pain of being cut off from physical contact with friends, relatives, and just random people out in the world. I’m lucky — I don’t experience that as pain. Partly, I guess, it’s because I haven’t really been contactless. I go for walks, I talk at a distance to friends I see; or I work on the porch and I talk to people I know who come by.

There are individual differences. I took CJ to the middle school to pick up the contents of his locker; it was the first time in two and a half months he’d been 100 feet from our house. He really doesn’t need variety. Me, I take my walks, and I go for bike rides with AB. I could really do things this way for a long time, forever if I had to.

I don’t have to. The restrictions on gatherings and business are starting to lift now; cases aren’t really declining, are maybe even going up a little, but there seems to be some sense that with testing protocols in place we can afford to experiment with a gradual, carefully monitored relaxation of restrictions.

It’s aggregates that matter. Not everybody has to be perfectly sealed off, which is good, because not everybody can be. But the easier it is for you to not see people, the less you should see people. From each according to, etc.

Pandemic blog 24: enter the gamma

I blogged last week about how to think about “R_0,” the constant governing epidemic growth, when different people in the network had different transmissibility rates.

Today, inspired by Kai Kupferschmidt’s article in Science, I look another look at what happens when the transmission rates vary a lot among people. And I learned something new! So let me set that down.

First of all, et me make one point which is silly but actually has mathematical content. Suppose 90% of the entire population is entirely immune to the disease, and the other 10% each encounter 20 people, sharing extensive air with each one . Since only 2 of those 20 are going to be susceptible, the dynamics of this epidemic are the same as that of an epidemic with an R_0 of 2. So if you look at the exponential growth at the beginning of the epidemic, you would say to yourself “oh, the growth factor is 2, so that’s R_0, we should hit herd immunity at about 50% and end up covering about 80% of the population,” but no, because the population that’s relevant to the epidemic is only 10% the total population! So, to your surprise, the epidemic would crest at 5% prevalence and die out with only 8% of people having been infected.

So extreme heterogeneity really matters — the final spread of the epidemic can be completely decoupled from R_0 (if what we mean by R_0 is the top eigenvalue like last time, which measures the early-epidemic exponential rate of spread.)

In my last post, I included a graph of how spread looked in non-heterogeneous populations generated by 6×6 random matrices I chose randomly, and the graph showed that the top eigenvalue and the eventual spread were strongly coupled to each other. But if you choose a random 6×6 matrix the entries are probably not actually going to be that far apart! So I think this was a little misleading. If the transmissibility has a really long tail, things may be different, as the silly example shows. What follows is a somewhat less silly example.

The model of heterogeneity used in this famous paper seems to be standard. You take transmissibility to be a random variable drawn from a gamma distribution with mean B and shape parameter k. (I had to look up what this was!) The variance is B^2/k. As k goes to infinity, this approaches a variable which is exactly B with probability 1, but for k close to 0, the variable is often near zero but occasionally much larger than B. Superspreaders!

Just like in the last post, we are going to completely jettison reality and make this into a static problem about components of random graphs. I am less confident once you start putting in rare high-transmission events that these two models stay coupled together, but since the back-of-the-envelope stuff I’m doing here seems to conform with what the epidemiologists are getting, let’s go with it. In case you don’t feel like reading all the way to the end, the punchline is that on these kinds of models, you can have early exponential growth that looks like R_0 is 2 or 3, but an epidemic that peters out with a very small portion of people infected; the “herd immunity is closer than we think” scenario, as seen in this preprint of Gomes et al.

Let’s also stick with the “rank 1” case because it’s what’s in the paper I linked and there are already interesting results there. Write X for our gamma-distributed random variable.

Then, sticking with the notation from the last post, the mean number of transmissions per person, the “average R_0”, is

(\mathbf{E} X)^2 = B^2

(I guess I wrote the last post in terms of matrices, where the average R_0 was just the sum of the entries of the matrix A, or \mathbf{1}^T A \mathbf{1}; here the “matrix” A should be thought of as a rank 1 thing w w^T where w is a vector with entries sampled from X.)

The top eigenvalue is just the trace of the matrix, since all the other eigenvalues are 0, and that is

{\mathbf E} X^2 = B^2(1+1/k).

Note already that this is a lot bigger than the average R_0 when k is small! In particular, there are lots of random graphs of this type which have a giant component but average degree < 2; that’s because they have a lot of isolated vertices, I suppose.

So what’s the size of the giant component in a graph like this? As always we are solving an integral equation

f = 1 - e^{-Af}

for a function f on measure space, where A is the “matrix” expressing the transmission. In fact, a function on measure space is just a random variable, and the rank-1 operator A sends Y to E(XY)X. The rank-1-ness means we can turn this into a problem about real numbers inteadd of random variables; we know Af = aX for some real number a; applying A to both sides of the above equation we then have

aX = \mathbf{E}(X(1-e^{-aX}))X

or

a =  \mathbf{E}(X(1-e^{-aX}))

But the latter expectation is something you can explicitly compute for a gamma-distributed variable! It just involves doing some integrals, which I rarely get to do! I’ll spare you the computation but it ends up being

a = B(1-aB/k)^{-(k+1)}

which you can just solve for a, and then compute E(1-e^{-aX}) if you want to know the total proportion of the population in the giant component. If k is really small — and Adam Kucharski, et al, back in April, wrote it could be as low as 0.1 — then you can get really small giant components even with a fast exponential rate. For instance, take B = 0.45 and k = 0.1; you get a top eigenvalue of 2.2, not inconsistent with the growth rates we saw for unimpeded COVID, but only 7.3% of the population touched by the infection! Another way to put it is that if you introduce the infection to a random individual, the chance of an outbreak is only 7%. As Lloyd-Smith says in the Nature paper, this is a story that makes “disease extinction more likely and outbreaks rarer but more explosive.” Big eigenvalue decouples from eventual spread.

(By the way, Kucharski’s book, The Rules of Contagion, is really good — already out in the UK, coming out here in the US soon — I blurbed it!)

What’s going on here, of course, is that with k this low, your model is that the large majority of people participate in zero interactions of the type likely to cause transmission. Effectively, it’s not so different from the silly example we started with, where 90% of the population enjoyed natural immunity but the other 10% were really close talkers. So having written all this, I’m not sure I needed to have done all those integrals to make this point. But I find it soothing to while away an hour doing integrals.

I don’t know whether I think k=0.1 is realistic, and of course, as Adam K. explained to me by email, who is a superspreader may change with time and context; so 7% is probably too low, since it’s not like once the infection “uses up” the superspreaders there can’t possibly be any more. Probably the variance of propensity to transmit over time should either actually be modeled dynamically or proxied by letting X be a less strongly skewed distribution representing “time average of propensity to transmit” or something like that.

In any event, this does make me feel much more favorable towards the idea that unmitigated spread would end up infecting less than half of the population, not 80% or more. (It does not make me favorable towards unmitigated spread.)

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Pandemic blog 23: why one published research finding is misleading

I really like John Ioannidis: his famous 2005 article “Why Most Published Research Findings are False” probably did more than any other paper to draw attention to the problems with blind use of p-value certification in medicine.

But he has a preprint up on medrxiv today that is really poorly done, so much so that it made me mad, and when I get mad, I blog.

Ioannidis has been saying for months that the COVID-19 pandemic, while bad, is not as bad as people think. Obviously this is true for some value of “people.” And I think he is right that the infection fatality rate, or IFR, is in most places not going to be as high as the 0.9% figure the March 16 Imperial College model used as an estimate. But Ioannidis has a much stronger claim; he thinks the IFR, in general, is going to be about 1 or 2 in a thousand, and in order to make that case, he has written a paper about twelve studies which show a high prevalence of antibodies in populations where not very many people have died. High prevalence of infection + few deaths = low IFR.

I think I am especially irritated with this paper because I agree that the IFR now looks lower than it looked two months ago, and I think it’s important to have good big-picture analysis to back that intuition up — and this isn’t it. There’s a lot wrong with this paper but I just want to focus on one thing that jumped out at me as especially wrong, and that is Ioannidis’s treatment of the Netherlands antibody study.

That study found that in blood donors, all ages 18-72 (Ioannidis says <70, not sure why), 2.7% showed immunity. Ioannidis reports this, then makes the following computation. About 15m of the 17m people in the Netherlands are under 70, so this suggests roughly 400,000 people in that age group had been infected, of whom only 344 had died at the time of the study, giving an IFR of a mere 0.09%. Some plague! Ioannidis puts this number in his table and counts it among those of which he writes “Seven of the 12 inferred IFRs are in the range 0.07 to 0.20 (corrected IFR of 0.06 to 0.16) which are similar to IFR values of seasonal influenza.”

But of course the one thing we really do know about COVID, in this sea of uncertainty, is that it’s much, much more deadly to old people. The IFR for people under 70 is not going to be a good estimate for the overall IFR.

I hashed out some numbers — it looks to me like, using the original March 16 Imperial College estimates, derived from Wuhan, you would derive an infection fatality rate of about 0.47% among people age 20-70. There are about 10.8m Dutch people in that range (I am taking all this from Wikipedia data on the age distribution of the Netherlands) so if 2.7% of those are infected, that’s about 300,000 infections, and 344 deaths in that group is about 0.11%. Lower than the Imperial estimate! But four times lower, not ten times lower.

What about the overall IFR? That, after all, is what Ioannidis’s paper is about. If you count the old people who died, the toll as of April 15 wasn’t 344, it was over 3100. If the 2.7% prevalence rate were accurate as a population-wide estimate, the total number of infected people would be about 460,000, for an IFR of 0.67%, more than seven times higher than the figure Ioannidis reports (though still a little lower than the 0.9% figure in the Imperial paper.) Now we definitely don’t know that the infection rate among old Dutch people is the same as it is in the overall population! But even if you suppose that every single person over 70 in the country is infected, that gets you to a little over 2 million infections, and an IFR of 0.15%. In other words, the number reported by Ioannidis is substantially lower than the theoretical minimum the IFR could actually be. And of course, it’s not the case that everybody over 70 already had COVID-19 in the middle of April. (For one thing, that would make the IFR for over-70s only slightly higher than the IFR overall, which contradicts the one thing about COVID we really know!)

There’s no fraud here, I hasten to say. Ioannidis tells you exactly what he’s doing. But he’s doing the wrong thing.

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