Pandemic blog 11: Why do curves bend?

When you plot the number of reported deaths from COVID on a log scale you get pictures that look like this one, by John Burn-Murdoch at the Financial Times:

A straight line represents exponential growth, which is what one might expect to see in the early days of a pandemic according to baby models. You’ll note that the straight line doesn’t last very long, thank goodness; in just about every country the line starts to bend. Why are COVID deaths concave? There are quite a few possible reasons.

  1. Suppression is working. When pandemic breaks out, countries take measures to suppress transmission, and people take their own measures over and above what their governments do. (An analysis by Song Gao of our geography department of cellphone location data shows that Wisconsinites median distance traveled from home decreased by 50% even before the governor issued a stay-at-home order.) That should slow the rate of exponential growth — hopefully, flip it to exponential decay.
  2. Change in reporting. Maybe we’re getting better at detecting COVID deaths; if on day 1, only half of COVID deaths were reported as same, while now we’re accurately reporting them all, we’d see a spuriously high slope at the beginning of the outbreak. (The same reasoning applies to the curve for number of confirmed cases; at the beginning, the curve grows faster than the true number of infections as testing ramps up.)
  3. COVID is getting less lethal. This is the whole point of “flattening the curve” — with each week that passes, hospitals are more ready, we have more treatment options and fuller knowledge of which of the existing treatments best suits which cases.
  4. Infection has saturated the population. This is the most controversial one. The baby model (where by baby I mean SIR) tells you that the curve bends as the number of still-susceptible people starts to really drop. The consensus seems to be we’re nowhere near that yet, and almost everyone (in the United States, at least) is still susceptible. But I guess one should be open to the possibility that there are way more asymptomatic people than we think and half the population is already infected; or that for some reason a large proportion of the population carries natural immunity so 1% of population infected is half the susceptible population.
  5. Heterogeneous growth rate. I came across this idea in a post by a physicist (yeah, I know, but it was a good post!) which I can’t find now — sorry, anonymous physicist! There’s not one true exponential growth rate; different places may have different slopes. Just for the sake of argument, suppose a bunch of different locales all start with the same number of deaths, and suppose the rate of exponential growth is uniformly distributed between 0 and 1; then the total deaths at time t is \int^1_0 e^{\alpha t} d \alpha which is (1/t)(e^t - 1). The log of that function has positive second derivative; that is, it tends to make the curve bend up rather than down! That makes sense; with heterogeneous rates of exponential growth, you’ll start with some sort of average of the rates but before long the highest rate will dominate.

I’m sure I’ve skipped some curve-bending factors; propose more in comments!

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Pandemic blog 10: I’m masked in the supermarket

After thinking it over in previous posts I realized I had no rejoinder to the argument that we should all be wearing masks to go shopping, so I wore a mask to go shopping. Nothing fancy or ultra-filtering, just an elastic paper mask from a box. I worried I would feel awkward, but instead I felt cool, like a bandit. When I last went shopping, 9 days ago, almost no one was wearing a mask; now it’s up to 20 or 25 percent of the customers. Maybe people are reading my blog! I didn’t ask. None of the Trader Joe’s employees wear masks and I wonder whether they’re allowed to.

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Pandemic blog 9: The Class of 1895

I was wondering about what the last major pandemic, the Spanish flu of 1918, looked like in real time, so I looked at the 25th anniversary report of the Harvard Class of 1895, published in June 1920 and written in 1919. To my surprise, the flu is barely mentioned. Henry Adsit Bull lost his oldest daughter to it. A couple of classmates worked in influenza hospitals. Morton Aldrich used it as an excuse for being late with his report. Paul Washburn reported being quite ill with it, and emphasizing that it might be his last report, demanded that the editors print his curriculum vitae with no editorial changes. (Nope — he was still alive and well and banking in the 1935 report.) I thought 1894, whose report was written more in the thick of the epidemic, might have more to say, but not really. Two men died of it, including one who made it through hideous battles of the Great War only to succumb to flu in November 1918. Another lost daughter.

But no one weighs in on it; I have read a lot of old Harvard class reports, and if there’s one thing I can tell you about an early 20th century Harvard man, it’s that he likes to weigh in. Not sure what to make of this. Maybe the pandemic didn’t much touch the lives of the elite. Or maybe people just died of stuff more and the Spanish flu didn’t make much of an impression. Or maybe it was just too rough to talk about (but I don’t think so — people recount pretty grisly material about the war.)

Back to the present. The Wisconsin Supreme Court ordered all jury trials halted for two months for the safety of jurors, witnesses, and officers of the court; an extremely overwrought dissent from Justice Rebecca Bradley insists that if a right is in the constitution it can’t be put on pause, even for a couple of months, even in a pandemic, which will be news to the people in every state whose governors have suspended their right to assemble.

CJ made a blueberry bundt cake, the best thing he’s made so far, aided by the fact that at the Regent Market Co-op I found a box of pectin, an ingredient I didn’t even know existed. Powdered sugar there was not, but it turns out that powdered sugar is literally nothing but regular sugar ground fine and mixed with a little cornstarch! You can make it yourself if you have a good blender. And we do have a good blender. We love to blend.

Walked around the neighborhood a bit. Ran into the owner of a popular local restaurant and talked to him from across the street. He’s been spending days and days working to renegotiate his loan with the bank. He thinks we ought to be on the “Denmark plan” where the government straight up pays worker’s salaries rather than make businesses apply to loans so they can eventually get reimbursed for the money they’re losing right now. (I did not check whether this is actually the Denmark plan.) Also saw my kids’ pediatrician, who told me that regular pediatrics has been suspended except for babies and they’ve closed the regular clinic, everything is consolidated in 20 S. Park.

I’ve been spending a lot of time thinking about different groups’ COVID projections, claims and counterclaims. I’ll write about it a little in the next entry to show how little I know. But I think nobody knows anything.

Tomorrow it’ll be two weeks since the last time I was more than a quarter-mile from my house. We are told to be ready for another month. It won’t be that hard for us, but it’ll be hard for a lot of people.

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Pandemic blog 8: enter the hermit

It’s family blogging time! Since school is out we need some kind of writing activity so we’re all blogging, not just me. I did not require any particular subject. CJ is blogging about the movies he’s watching in his friend groups’ “movie club ” — he has the Marvel bug now and is plowing through the whole collection on Disney+. AB’s blog is called “The Nasty Times: Foods that Were Never Meant To Be Eaten” and each entry is about a food she considers nasty. The first entry was about mushrooms and she is currently composing “Why Onions Do Not Belong in Sloppy Joes.” I know, I know, who doesn’t like mushrooms and onions? Well, me at AB’s age — I made my mom take them out of everything, much to her annoyance. Now I’m getting my comeuppance.

I have two big longboxes of comics in the basement, almost all from 1982-1986, and AB and I spent part of the morning starting to sort and organize them. Perfect example of a task that feels like productivity and is not important in any way and yet — satisfying. Also nice to see old friends again, covers I haven’t seen in years but are familiar to me in every detail. This one seemed fairly on point:

I am still thinking about the masks. Why so unpopular in the US? Maybe it works like this. You are told (correctly) that wearing a mask doesn’t provide strong protection. Let’s say (making up a number) it only reduces your chance of transmitting or contracting the virus by a half. To many people that is going to feel like nothing: “I’m not really protected, what’s the point?” But in the aggregate, an easy, cheap measure that reduces number of transmissions by 50% would be extremely socially valuable.

talk about class of 1895

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Pandemic blog 7

Made a big, creamy, cheesy casserole with rotini and a million artichokes and peas, the vegetables out of the freezer of course. Times like this bring out the 60s housewife in me. Everyone is saying it’s good to get out of the house and see the sun from time to time, even just on your porch, but there hasn’t really been any sun here; it’s Wisconsin-technically-spring, in the 40s and kind of dreary. I go play basketball with the kids in the driveway each day in the chill. CJ can beat me almost all the time now.

AB and I listened to all the songs on Spotify called “Coronavirus.” There are already a ton; we didn’t actually listen to all of them, there were too many. A lot of them are in Spanish.

Daniel Litt organized a number theory conference, all held on Zoom with more than 130 people watching. To my surprise, this worked really well. People are starting to organize lists of online seminars and at this point there are more seminars I could be “going” to each day than there are when life is normal.

I’ve heard talk about starting baseball with the All-Star Game and having the World Series at Christmas.

Some people are hoping that maybe we’re drastically underestimating the prevalence of infection; maybe the reason curves are starting to bend isn’t the effect of our social isolation measures but the fact that a substantial population has already been affected and acquired temporary immunity, without ever knowing they were sick, and so maybe we’re vastly overestimating the proportion of cases which turn into serious illnesses. Wouldn’t that be great?

At the moment I don’t know anyone who’s died but I know people who know people who’ve died. At this point, do most people in the United States know people who know people who’ve died?

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Pandemic blog 6: safer at home

I went to Trader Joe’s this morning. It was an extremely pleasant oasis of normality. Everything was as it always is, except for the guy standing out front apparently doing nothing but who I guessed was there to control inflow in case the store got too crowded. (Verified by a friend who was at the store early this afternoon, by which point the guy was only letting someone in when someone else came out.) When I was there, the shoppers were somewhat sparse, but even so there was a kind of awkward impromptu ballet of people trying to imitate repelling particles as best they could. My friends in New York are saying the grocery stores are out of flour, eggs, milk, meat, and pasta, but here everything is stocked as normal. I filled my cart really high, not because I’m hoarding (we have enough shelf-stable starch and cans and root vegetables to last us a while, we’re fine) but because I now know that when all four of us, one of them a hungry teenager who’s now taller than I am, are eating three meals a day in the house, we actually consume a lot more food than I usually buy.

I didn’t wear a mask to the store — but why didn’t I? Everyone is saying that you are probably not going to get COVID from touching contaminated surfaces, as long as you are good about handwashing. They think the spread is really person to person — he coughs on you, you cough on me. Wrapping a scarf around the lower part of your face isn’t an N95 mask (remember when I didn’t know what an N95 mask was?) but any form of barrier has to block some reasonable portion of whatever droplet cloud a person coughs out, right? And that’s the game, to block a reasonable proportion of transmissions, to get that exponential constant down below 1. A few people in the store were wearing masks, maybe 1 in 20.

All the talk in the store was about the rumor that Governor Evers was signing a statewide shelter-in-place order, and when I got home I found out it was true. (Despite reassuring information about surfaces, I am trying not to take my phone out when I’m out in the world, to avoid potentially contaminating it.) Ours isn’t called “shelter in place,” it’s called “safer at home,” which I guess is meant to sound softer. What this is going to mean, I think, is that a lot of workplaces which are currently operating are going to stop. And that maybe I should have planned more state park walks with the kids last week because now it’s forbidden.

CJ’s middle school friends have a film club; they watch a movie and then discuss it for two hours the next day on FaceTime. He’s watching Guardians of the Galaxy right now. Last night we made Cincinnati chili, which I’ve never done before. Boiling the meat has always sounded gross to me but it really does make for a meaty-but-not-greasy chili. One small upside: I am making things you have to simmer for an hour, something I rarely do when I have to start dinner after I get home from work.

All in all, starting from the baseline that the news is very bad, the news is not bad. In Italy, which has been in hard lockdown for what, a week? the rate of new cases is starting to decline. (The mathematician Luca Trevisan is in northern Italy and his blog is a very good snapshot of what it’s like to be in the middle of the outbreak there.) China, after two months of lockdown and quite a long spell without major new infections, is starting to loosen up; what happens next seems pretty important. A big new wave of infection or have they really beaten it?

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Pandemic blog 5: exponential

When do you go the grocery store? If you’re concerned about your own risk of infection, the logic of exponential growth insists that today is always better than tomorrow. But the community is better served by each person waiting as long as they can, so as to slow the overall exponential constant.

What is the exponential constant? People are constantly graphing the number of confirmed cases in each country, state, locality on a log-linear scale and watching the slope, but I don’t see how, in a principled way, to untangle the effects of increased testing from actual increases in infection. I guess if one hypothesizes that there’s something like a true mean rate you could plot state-by-state nominal cases against tests done and see if you can fit exp(ct)*(tests per capita) to it. But there are state-to-state differences in testing criteria, state-to-state differences in mitigation strategy, etc.

AB and I made chocolate chip cookies today. Dr. Mrs. Q and CJ watched Inside Out. Weather’s warmer and I think we’ll get some driveway basketball in. We listened to “The Gambler” in honor of Kenny Rogers, deceased today. I had forgotten, or didn’t know, what an ice-cold love letter to death it is. “Every hand’s a winner, and every hand’s a loser, and the best that you can hope for is to die in your sleep.” Damn.

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Pandemic blog 4: food

CJ is eating a sandwich. Strawberry preserves, capers, lettuce, and spicy dijon mustard on whole wheat toast. He says it’s excellent. They say COVID can temporarily disrupt the sense of taste and smell. (He doesn’t have COVID.)

We go through the fresh food pretty fast when all four of us are eating three meals a day at home. Also, that makes more dishes than our dishwasher can really hold. Somebody mentioned spaetzle on the Internet yesterday and it touched off a primal urge for spaetzle in me, so I made spaetzle. It’s good to remember that you can have noodles even if you don’t have noodles. (But we have plenty of noodles.) I used this Serious Eats recipe and fried the spaetzle in a lot of butter and onions. Very good and I’ll do it again.

AB had trouble sleeping the other night and I tried to bore her to sleep by listing all the breakfast cereals I could think of, which didn’t work, because talking about breakfast cereal is pretty interesting. But now I’m craving Raisin Bran which is weird because I don’t even like Raisin Bran.

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Pandemic blog 3: usefulness, rules

One wants to feel useful. Of course, to an extent we are being useful by staying at home and not coming close to anyone outside the family. It’s incremental but somehow all this shared solitude generates a community spirit. We’re all in this, alone, together!

Of course when math is what you do you wish math were more useful, not in the true-but-abstract way we talk about math being useful, but useful today, in the face of what faces us. My colleagues Maria Chikina and Wes Pegden wrote up a model suggesting that we might do better to isolate younger people less than we are but older people much, much more. (Though whenever I leave the house I see older people are visibly out and about. Maybe it’s not up to “us.” Also, I am not really very much younger than older people anymore.) I was talking with Lior Silberman and Rachel Ward about pooling samples to mitigate what seems to be a shortage of COVID tests. Except nobody seems to really agree on whether there’s a shortage of tests or a shortage of administrative wherewithal to deploy the tests. Or maybe it’s swabs. There may not be enough swabs. And some people think we’re past the point in the United States where testing is useful, and maybe everybody should just treat themselves as if they’re infected. Anyway, they say you can test negative when you’re already infected but the viral load hasn’t built up enough for your sputum to set off the RNA detector.

Some rules everyone agrees on. If you must go outside, is to keep six feet of distance between you and anyone else. (Someone on Twitter asked whether they could still have sex, and got the answer: “If you can do it from six feet away, great!”) Wash hands for twenty full seconds, interlace the fingers to get the in-betweens, dig nails of one hand under the nails of the other to get any virus lurking under there. (Other sources recommend twisting the nails against the opposite palm in a circular motion for the same effect.) Some people are getting takeout, other people consider that risky. They say the virus can live 24 hours on cardboard, a few days on plastic and metal. Our neighborhood bookstore is closed for browsing but Joanne the proprietor is taking special orders. Two of AB’s came in so we stopped by to pick them up. I seemed more concerned about staying six feet from her than she was about staying six feet from me. I felt like a hero for having a no-touch credit card. She said the hard part was not really understanding when it was going to end. On the way home a guy with a stroller crossed the street so as not to pass us on the sidewalk.

The May conference in Germany on arithmetic statistics I was organizing for the Simons Foundation was postponed to an unnamed later date. Our moduli spaces conference here in Madison, which my student Soumya Sankar put a ton of work into, planned for next week, was cancelled too, of course. Just eight days ago we thought we were deciding whether or not to cancel it.

It’s so small in the big picture but I find myself moved by the small things I’m used to that are cancelled. United has shut down the direct flights from Madison to San Francisco and Los Angeles. I have to doubt they’ll come back soon, in the austere travel environment to come. The Isthmus, our alternative paper that’s been in press for 44 years, is shutting down. Maybe it would have gone out of business anyway. Most of those papers have.

Games played with kids: Mastermind 1. Monopoly 1. Big Boggle 4. Set 4. I am planning for Scrabble tomorrow.

I wrote to a friend on Monday, “I think we are going to be, like, playing 30 mins of Scrabble and then sniping at each other and I’ll relent and let AB watch Worst Cooks on the iPad. And I’ll look at Twitter and fume at all the academic parents who are like “it was so good for the kids and I to finally get the chance to act out Socrates’s dialogues, thanks coronavirus!!” So with that in mind I am somewhat embarrassed to say that I actually did a lot of math with AB today. Things I had been meaning to sit down and work through with her but didn’t find time when I didn’t have hours of time I needed to somehow fill. There are moments, lots of moments, where she rears back from instruction but today, for whatever reason, she was into it, and we just kept going. Area of isosceles triangles via Pythagorean theorem, then Heron’s formula — she liked very much the idea that they don’t teach it in regular geometry class. (They don’t, do they?) She is pleased with the word “semiperimeter.” In the context of areas, some approximation of square roots by decimals. Positive and negative exponents. And finally the Euclidean algorithm. In school they’re reducing fractions to simplest form and this was satisfyingly magical, to show how you can learn what to divide by without dividing.

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Pandemic blog 2: arm metaphor

Years ago I broke my arm, broke it really badly. Head of the radius broke right off and was floating around inside the skin. And the next 48 hours involved things like EMTs taking me to the hospital, setting up surgery, getting surgery, being on a lot of painkillers, etc. At no moment did I think about what was going to happen over the next six months. Everything was “what has to happen right now to get me from this thing to the very next thing.”

But then of course I was past the surgery and past the crisis and had to come face to face with the fact that I had months and months of tedious and kind of painful physical therapy ahead of me in order to turn my screwed-back-together arm into a machine that, while different in certain permament ways, approximated the function it used to have. Readers, it sucked. Probably it’s a good thing I wasn’t thinking about it too much while still waiting, opiated, for them to screw the titanium plate into my bone.

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