## Pandemic blog 36: undecided

Oh yeah, on top of everything else, there’s an election. Ordinarily, this time of year, I’d be spending some of a nice weekend day knocking on doors around Madison and making sure everyone’s registered to vote. This year, no door-knocking, not on the Democratic side, at any rate. So I did some phone calling, though it’s not something I love doing. And among other people I talked to one actual honest-to-God undecided voter.

Talking to strangers about politics makes you realize lots and lots of people don’t fit into the political boxes you understand from Online or TV or the Paper of Record. This voter was an African-American Iraq war vet in Georgia. He didn’t like what Trump had said about veterans. He didn’t like Harris’s record as California AG. He doesn’t like that Biden is in favor of a mask mandate, which he sees as “dictatorship” like what he saw in Iraq. He thinks the cost-of-living adjustment for Social Security is too low and the elderly can’t live on what they get. He thinks a one-time \$1200 payment is too little stimulus for ordinary people and most of the COVID relief went to big companies instead of into people’s pockets. He wants to know why we couldn’t have had monthly COVID relief checks like other countries. He thinks there should be term limits for Congress and judges. He worries that Biden is old and that Harris will become President and that based on his experience in combat women tend to “falter in the heat.” He thinks both candidates are “playing the race card.” He thinks Congress bickers too much and doesn’t do anything. He thinks we should throw them all out and start fresh.

I did not convert this guy to Joe Biden — how could I? I told him he had thought about the issues a lot more than anybody else I’d talked to, which is true. I tried to give him some sense that the reason I’d spend my afternoon calling strangers on the phone is that there’s a real difference between Joe Biden and Donald Trump and that difference directly affects things he cares about. He’s still undecided.

What’s going on with some of the topics previously covered?

Slimming: The initial weight loss reported slowed down, but hasn’t stopped, even though I started eating take-out from restaurants in July and have been doing so pretty regularly. Now at about 18 pounds below pre-pandemic weight. Why, I wonder? Is it really just the lunch out at work and the snack at the coffeeshop forgone?

Pandemic elections: 100,000 people in Dane County have already returned their absentee ballots for November. The city is setting up “Democracy in the Park” events where voters can turn in their ballots to city pollworkers; Republicans are trying to have those events declared illegal, because (this is me editorializing) they make it easy and convenient for people to vote whose votes they’d rather not see cast. There is a lot of noise about slowness of the mail, but it’s been fast here, and I mailed my ballot in; received by the clerk in just two days. The underlying worry here is that political actors will try to retroactively have legally cast ballots invalidated after Election Day, leaving voters with no recourse. The fact that mailed-in absentees are expected to be predominantly Democratic (only 44,000 ballots returned so far in Crucial Waukesha County) creates an obvious means of attack. I don’t really think that’ll happen but people are thinking about it under their mental breath.

Writing: The book is almost done! A draft is written, I’m going through and revising and putting in more endnotes now. To me it seems completely different from How Not To Be Wrong, while Dr. Mrs. Q says it seems exactly the same, which seems a kind of sweet spot: I can hope the people who liked the other book will like this one, while feeling for myself that I’m not putting out the same product again and again like a hack.

Impossible Meat: We’re still eating a lot of it! I have absolutely learned to read it as meat and no longer think of it as a substitute. But we’ve converged on using it exclusively in sauces; as a burger, it still doesn’t totally satisfy.

Smart Restart: After the big surge with the opening of classes, UW-Madison shut down in-person instruction for two weeks and put the two first-year dorms where cases were concentrated into isolation. The positivity rate on campus has dropped back down to around 1% and the campus outbreak doesn’t seem to have created sustained exponential growth in Madison’s general population; but it does seem to have brought our daily case load back up to where it was months ago, from which it is, again, only very slowly dropping. When R_0 is a little less than 1, even a brief bump up in prevalence can be very expensive in terms of long-term cumulative case numbers. Now we are starting football again. Is that smart? There won’t be any fans in Camp Randall (which means the economic catastrophe for local businesses of a year without a football season is going to happen unblunted.) Then again, there’s something hypocritical about me saying “Hell no, why take the risk” since I’ve been watching and enjoying baseball. The enjoyment of millions of fans actually does have value. MLB, because lots and lots of money is riding on this, has mostly kept its players and employees from suffering outbreaks. The Big Ten can probably do the same — if it cares to. What I worry about is this. By all accounts, in-person teaching hasn’t been spreading COVID either. But when we had in-person teaching, everyone felt things were more normal, and thinking things were more normal, they relaxed their social distancing, and that generated thousands of cases. There was indirect spread. Will football generate the same?

## Pandemic blog 34: teaching on the screen

A small proportion of UW-Madison courses were being given in person, until last week, that is, but not mine. I’m teaching two graduate courses, introduction to algebra (which I’ve taught several times before) and introduction to algebraic number theory, which I’ve taught before but not for quite a few years. And I’m teaching them sitting in my chair at home. So I thought I’d write down a bit about what that’s like, since depending on who you ask, we’ll never do it again (in which case it’s good to record the memory) or this is the way we’ll all teach in the future (in which case it’s good to record my first impression.)

First of all, it’s tiring. Just as tiring as teaching in the classroom, even though I don’t have to leave my chair. This surprised me! But, introspecting, I think I actually draw energy from the state of being in a room with people, talking at the board, walking around, interacting. I usually leave class feeling less tired than when I walked in.

On the screen, no. I teach lectures at 10 and 11 and at noon when both are done I’m wiped out.

My rig, settled on after other setups kept glitching out: Notability open on iPad, I write notes as if on blackboard with the Apple Pencil, iPad connected by physical cable to laptop, screensharing to a window on the laptop which window I am sharing in Microsoft Teams to the class while the laptop camera and mic capture my face and voice.

What I have not done:

• Gotten a pro-quality microphone
• Set up a curated “lecture space” from which to broadcast
• Recorded lecture videos in advance so I can use the lecture hour for discussion
• Used breakout rooms in Teams to let the students discuss among themselves

All of these seem like good ideas.

So far (but I am still in the part of both courses where the material isn’t too hard) the students and I seem to find this… OK. My handwriting is somewhat worse on the tablet than it is on the blackboard and it’s not great on the blackboard. The only student who has told me they prefer online is one who reports being too shy to speak in class, sometimes too shy even to attend, and who feels more able to participate by typing in the chat window with the camera turned off. That makes sense!

I have it easy — these courses have only thirty students each, so the logistical work of handling student questions, homework, etc. isn’t overwhelming. Teaching big undergraduate courses presents its own problems. What happens with calculus quizzes? In the spring it was reported that cheating was universal (there are lots of websites that will compute integrals for you in another window!) So we now have a system called Honorlock which inhabits the student’s browser, watches IP traffic for visits to cheating sites, and commandeers the student’s webcam (!) to check whether their eye motions indicate cheating (!!) This sounds awful and frankly kind of creepy and not worth it. And the students, unsurprisingly, hate it. But then how does assessment work? The obvious answer is to give exams which are open book and which measure something more contentful about the material than can be tested by a usual quiz. I can think of two problems:

• Fluency with the basic manipulations (of both algebra and calculus) is actually one of the skills the class is meant to impart: yes, there are things a computer can do it’s good to be able to do mentally. (I don’t think I place a complicated trig substitution in this category, but knowing that the integral of x^n is on order x^{n+1}, yes.
• Tests that measured understanding would be different from and a lot harder than what students are used to! And this is a crappy time to be an undergraduate. I don’t think it’s a great idea for their calculus course to become, without warning, much more difficult than the one they signed up for.
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## Pandemic blog 33: Smart Restart

I thought it was gonna work.

Really! I thought we could sort-of-open college again and not cause a big outbreak. Most of our students live here year-round. By all accounts, there have been fraternity parties all summer. We had a spike of cases in the campus area when bars opened back up at the end of June, which subsided when the county put back those restrictions (though never back down to the levels we’d seen in March, April, and May.) At the end of July I wrote “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.”

And maybe it could have; but it didn’t. As soon as school started last Wednesday, the percentage of student tests coming back positive, started growing, about 20% higher every day. On Saturday, nine Greek houses were quarantined. A week into school, with about 8% of tests positive, the University halted in-person classes and completely quarantined two first-year dormitories with two hours notice. Food is being brought in three times a day. Hope you like your roommate.

A lot of people, unlike me, saw this coming.

Maybe we can beat this back. Who knows? We did in July. But this outbreak is bigger.

Public schools in Madison are fully online right now. With a summer to prepare it’s working better than it did last spring. But it’s not great, and I would guess that for poor kids it’s a lot worse than “not great.” Private schools are allowed to be open in grades K-2, and a court decision that came down today has, at least for now, allowed them to open to all grades. More outbreaks? To be a broken record, who knows? The argument for opening K-2 sounds pretty good to me; while it’s not definite, most people seem to think younger children are less likely to spread and contract the disease, and that age range is where having kids at home limits parents most. Schools in Georgia have been open, and there have been lots of school outbreaks, and those schools get closed for a while and then reopen, but it doesn’t seem to have created big wave of cases statewide.

This article is good. Beating COVID isn’t all-or-nothing, but people seem to see it that way. If the bar’s open, that means it’s safe, and you can drink with whoever you want, as close as you want. No! Nothing is safe, if you mean safe safe. But also nothing is a guarantee of disaster. If everybody would do 50% of what they felt like doing, we could beat it. Or maybe 75%, who knows. But it feels like if we don’t insist on 0%, people will understand us to mean that 100% is OK. I don’t have any good ideas about how to fix this.

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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.

## 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 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.

## 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 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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