Wisconsin is slowly loosening its emergency health restrictions. Stores are allowed to open as long as they’re not in enclosed malls and no more than five customers are inside at once. People are moving around more than they were in April (though still quite a bit less than they were at the beginning of March):
The streets aren’t empty; last Sunday I walked over to Tim Yu‘s house to drop off a copy of an oral history of REM I knew he wanted to read, and everyone in the neighborhood was outside; I probably socalized more, sidewalk to porch, than I do on an ordinary Sunday. AB and I did a 25-mile ride, a new record for her, and there were plenty of people out on the bikepaths, unmasked. I played Frisbee with CJ at Wingra Park and a big group of teenagers was hanging out in close quarters, looking very much not like a family group.
On the other hand, at Trader Joe’s today, shoppers were making a visible effort to stay away from one another, and I counted only four people without masks. I overheard the Russian guy who works there say to one of this co-workers, “We all look like freedom fighters.”
I see this as a reasonable response to increased knowledge about the nature of the disease. Sustained indoor propinquity seems to be the dominant mechanism of transition.
Freedom fighters! The Wisconsin Supreme Court has struck down the state stay-at-home order issued by Governor Evers, except not exactly, because in order to find a reading of the statute that supported the outcome they asserted they had no beef with the governor’s order itself, only its implementation and enforcement by Andrea Palm, the State Health Secretary (or rather the State Health Secretary Designee because the Senate doesn’t feel like confirming anyone.) Anyway, as of now, nobody knows what the rules are. Some bars opened up and served crowds as normal. Seems like a bad idea. The smart political money in Wisconsin says this decision has nothing to do with COVID per se but is mostly an attempt to establish some precedent that the executive needs legislative approval to, well, execute things.
I don’t know what happens next. Maybe nothing. Stores were already open, people were already moving around. And large chunks of the state, including some of the places with the highest caseload like Green Bay, Kenosha, and Milwaukee, are still under county orders that the Supreme Court didn’t touch. Maybe people packing into newly open bars will create superspreading events and we’ll see a big wave of new cases and deaths in Waukesha and Platteville. And maybe they won’t! The main thing we know about COVID is we don’t know much about COVID. Why was there so much more spread in New York than there was in Chicago, and so much more in Chicago than in San Francisco? I don’t think there are any convincing answers. There’s graph theory in it, as in my last post, but it’s not just graph theory.
Wisconsin may very well not suffer any disastrous consequence from opening up with no real plan. But it’s hard to deny we’re taking a risk of a disastrous consequence. Let’s hope it doesn’t happen. That’s not a crazy hope. Most drunk drivers get home safe.
People are talking about R_0. It’s the number that you wish were below 1. Namely: it is the number of people, on average, that a carrier of SARS-CoV-2 infects during the course of their illness. All the interventions we’re undertaking are designed to shrink that number. Because if it’s bigger than 1, the epidemic grows exponentially until it infects a substantial chunk of the population, and if it’s smaller, the epidemic dies out.
But not everybody has the same R_0! According to some of the epdemiologists writing about COVID, this matters. It matters, for instance, to the question of how far into the population the infection gets before it starts to burn itself out for lack of new susceptible hosts (“herd immunity”) and to the question of how much of the population eventually falls ill.
Here’s an easy way to see that heterogeneity can make a difference. Suppose your population consists of two towns of the same size with no intermunicipal intercourse whatsoever. The first town has an R_0 of 1.5 and the second an R_0 of 0.3. Then the mean R_0 of the whole population is 0.9. But this epidemic doesn’t die out; it spreads to cover much of the population of Contagiousville.
I learned an interesting way to think about this heuristically from Tim Gowers on Twitter:
I've done a little bit of reading this morning about network epidemiology — that is, modelling disease spread on a graph and actually taking some account of the structure of that graph — and learned a fact that for the mathematically inclined is quite amusing. 1/
You can read Tim’s interesting thread yourself, but here’s the main idea. Say your population has size N. You make a graph out of the pandemic by placing an edge between vertices i and j if one of the corresponding people infects the other. (Probably better to set this up in a directed way, but I didn’t.) Or maybe slightly better to say: you place an edge if person i and person j interact in a manner such that, were either to enter the interaction infected, both would leave that way. If one person in this graph gets infected, the reach of the infection is the connected component of the corresponding vertex. So how big is that component?
The simplest way to set this up is to connect each pair of vertices with probability c/n, all such choices made independently. This is an Erdos-Renyi random graph. And the component structure of this graph has a beautiful well-known theory; if c > 1, there is a giant component which makes up a positive proportion of the vertices, and all other components are very small. The size of this component is nx, where x is the unique positive number such that
.
If c < 1, on the other hand, there is no big component, so the pandemic is unlikely to reach much of the population. (Correspondingly, the equation above has no nonzero solution.)
It is fair to be skeptical of this model, which is completely static and doesn’t do anything fancy, but let me just say this — the most basic dynamic model of epidemic spread, the SIR model, has an endstate where the proportion of the population that’s been infected is the unique positive x such that
.
Which looks pretty familiar!
Now what happens if you want to take into account that the population isn’t actually an undifferentiated mass? Let’s say, for instance, that your county has a red town and a blue town, each with population n/2. Two people in the red town have a probability of 2/n of being connected, while two people in the blue town have a probability of just 1/n of being connected, and a red-blue pair is connected with probability 1/n. (This kind of random graph is called a stochastic block model, if you want to go look at papers about it.) So the typical red-town person is going to infect 1 fellow red-towner and 0.5 blue-towners, for an R_0 of 1.5, while the blue-towner is going to have an R_0 of 1.
Here’s the heuristic for figuring out the size of the big component. Suppose x is the proportion of the red town in the big component of the graph, and y is the proportion of the blue town in the big component. Take a random red person; what’s the change they’re in the big component? Well, the chance they’re not connected to any of the xn/2 red-towners in the big component is
(oh yeah did I mention that n was infinity?) and the chance that they’re not connected to any of the blue-towners in the big component is
so all in all you get
and by the same token you would get
and now you have two equations that you can solve for x and y! In fact, you find x = 47% and y = 33%. So just as you might expect, the disease gets farther in the spreadier town.
And you might notice that what we’re doing is just matrix algebra! If you think of (x,y) as a vector v, we are solving
where “exponentiation” of a vector is interpreted coordinatewise. You can think of this as finding a fixed point of a nonlinear operator on vectors.
When does the outbreak spread to cover a positive proportion of the population? There’s a beautiful theorem of Bollobas, Janssen, and Riordan that tells you: you get a big component exactly when the largest eigenvalue λ of A, the so-called Perron-Frobenius eigenvalue, is larger than 1. In the case of the matrix studied above, the two eigenvalues are about 1.31 and 0.19. You might also note that in the early stages of the epidemic, when almost everyone in the network is susceptible, the spread in each town will be governed by repeated multiplication of a small vector by A, and the exponential rate of growth is thus also going to be given by λ.
It would be cool if the big eigenvalue also told you what proportion of the vertices are in the giant component, but that’s too much to ask for. For instance, we could just replace A with a diagonal matrix with 1.31 and 0.19 on the diagonal; then the first town gets 43% infected and the second town completely disconnected from the first, gets 0.
What is the relationship between the Perron-Frobenius eigenvalue and the usual “mean R_0” definition? The eigenvalue can be thought of as
while the average R_0 is exactly
where 1 is the all-ones vector. So we see immediately that λ is bounded below by the average R_0, but it really can be bigger; indeed, this is just what we see in the two-separated-towns example we started with, where R_0 is smaller than 1 but λ is larger.
I don’t see how to work out any concise description of the size of the giant component in terms of the symmetric matrix, even in the simplest cases. As we’ve seen, it’s not just a function of λ. The very simplest case might be that where A has rank 1; in other words, you have some division of the population into equal sized boxes, and each box has its own R_0, and then the graph is constructed in a way that is “Erdos-Renyi but with a constraint on degrees” — I think there are various ways to do this but the upshot is that the matrix A is rank 1 and its (i,j) entry is R_0(i) R_0(j) / C where C is the sum of the R_0 in each box. The eigenvalues of A are all zero except for the big one λ, which is equal to the trace, which is
or, if you like, mean(R_0) + variance(R_0)/mean(R_0); so if the average R_0 is held fixed, this gets bigger the more R_0 varies among the population.
And if you look back at that Wikipedia page about the giant component, you’ll see that this is the exact threshold they give for random graphs with specified degree distribution, citing a 2000 paper of Molloy and Reid. Or if you look at Lauren Meyers’s 2005 paper on epidemic spread in networks, you will find the same threshold for epidemic outbreak in section 2. (The math here is descended from work of Molloy-Reed and this much-cited paper of Newman, Strogatz, and Watts.) Are typical models of “random graphs with specified degree distribution” are built to have rank 1 in this sense? I think so — see e.g. this sentence in Newman-Strogatz-Watts: “Another quantity that will be important to us is the distribution of the degree of the vertices that we arrive at by following a randomly chosen edge. Such an edge arrives at a vertex with probability proportional to the degree of that vertex.”
At any rate, even in this rank 1 case, even for 2×2 matrices, it’s not clear to me how to express the size of the giant component except by saying it’s a nonzero solution of . Does the vector v have anything do do with the Perron-Frobenius eigenvector? Challenge for the readers: work this out!
I did try a bunch of randomly chosen 6×6 matrices and plot the overall size of the giant component against λ, and this is what I got:
The blue line shows the proportion of the vertices that get infected if the graph were homogeneous with parameter λ. Which makes me think that thinking of λ as a good proxy for R_0 is not a terrible idea; it seems like a summary statistic of A which is pretty informative about the giant component. (This graph suggests maybe that among graphs with a given λ, the homogeneous one actually has the biggest giant component? Worth checking.)
I should hasten to say that there’s a lot of interest in the superspreader phenomenon, where a very small (probability -> 0) set of vertices has very large (superlinear in n) number of contacts. Meyers works out a bunch of cases like this and I think they are not well modeled by what I’m talking about here.
A more technical note: the result of Bollobas et al is much more general; there’s no reason the vertices have to be drawn from finitely many towns of equal size; you can instead have the types of vertices drawn from whatever probability space M you like, and then have the probability of an edge between an vertex x and a vertex y be W(x,y) for some symmetric function on M^2; nowadays this is called the “graphon” point of view. Now the matrix is replaced by an operator on functions:
,
the probability g(x) that a vertex of type x is in the giant component is a solution of the integral equation
and a giant component exists just when the operator norm is greater than 1. This is the kind of analysis you’d want to use if you wanted to really start taking geography into account. For instance, take the vertices to be random points in a disc and let W(x,y) be a decreasing function of |x-y|, modeling a network where infection likelihood is a decreasing function of distance. What is the norm of the operator A_W in this case? I’ll bet my harmonic analyst friends know, if any of them are reading this. But I do not.
Update:Now I know, because my harmonic analyst friend Brian Street told me it’s just the integral over W(x,y) over y, which is the same for all y (well, at least it is if we’re on the whole of R^d.) Call that number V. He gave a very nice Fourier-theoretic argument but now that I know the answer I’m gonna explain it in terms of the only part of math I actually understand, 2×2 matrices. Here’s how it goes. In this model, each vertex has the same expected number of neighbors, namely that integral V above. But to say every vertex has the same expected number of neighbors is to say that 1 is an eigenvector for A. If 1 were any eigenvector other than Perron-Frobenius, it would be orthogonal to Perron-Frobenius, which it can’t be because both have positive entries, so it is Perron-Frobenius, so λ = V.
In fact I had noticed this fact in one of the papers I looked at while writing this (that if the matrix had all row-sums the same, the long-term behavior didn’t depend on the matrix) but didn’t understand why until just this minute. So this is kind of cool — if the kind of heterogeneity the network exhibits doesn’t cause different kinds of vertices to have different mean degree, you can just pretend the network is homogeneous with whatever mean R_0 it has. This is a generalization of the fact that two towns with no contact which have the same R_0 can be treated as one town with the same R_0 and you don’t get anything wrong.
A mall in Janeville struggles to stay alive. The stores can’t or won’t pay rent. Why is the landlord (a private equity company) demanding it? Surely they’d rather have a tenant three months from now then get one month of rent and then have no tenant because they bankrupted the tenant they had. Leila Schneps tells me that, in France, landlords who forgo the next three months’ rent get a property tax break that compensates them for the loss.
For $89, Frontier will leave the middle seat next to you empty. Presumably this offer only applies while Frontier can’t fill the planes anyway. So what happens if people don’t pay; they pack all the passengers into ten full rows and then don’t let anyone move to an empty row unless they cough up $89?
Oh yeah, and when I wasn’t reading the newspaper I was writing something in the newspaper! A piece for the New York Times about pooled testing, an old idea that’s come back to relevance as we try to figure out how to test huge numbers of Americans faster than we can produce test kits.
The Wisconsin Idea: the boundaries of the university are the boundaries of the state. They drill this into you right when you are arrive; that we are not just here to teach things to the 18-22-year-olds in the room with us, but to contribute to the advancement of the state as a whole. Then after a while you start to realize it’s not just a slogan; it’s the actual value system of the institution.
I’ve been seeing it this last month. UW-Madison faculty members are doing swift and amazing work, sometimes visibly, sometimes behind the scenes. Not me, really. It’s not a time for pure math. But my colleagues! Song Gao from geography made this dashboard showing changes in mobility by county based on cellphone tracking data. Colleagues in statistics and engineering worked with the state government to pin down exactly what was meant by “14 days of decline in cases,” one of our criteria to start opening businesses. Speaking of opening businesses, Noah Williams from economics and his team at the Center for Research on the Wisconsin Economy (caw!) wrote a report about the economic costs of the pandemic to the state and the ways we might go about opening more businesses, balancing production and safety. A friend at the med school is collecting blood from recovered patients so we can start to see how antibody levels relate to time since recovery. Thomas Friedrich from veterinary pathology is studying the genomes of viral samples around Wisconsin to understand how the disease is moving within the state. (It turns out that viruses, just like people, don’t actually move between Milwaukee and Madison that much.) My colleague/neighbor Mike Wagner from journalism just launched COVID-19 Wisconsin Connect, to foster discussion among the general public of what we’re going through. And the UW Library is working to document and archive the experience of the campus and the state during the pandemic, because we think we’ll remember exactly how this was, but we probably won’t. The Library will.
Beautiful real spring day today and AB and I got on the bike late in the afternoon and did an 11-mile ride, just to take in some air and some healing ultraviolet light. AB asked me “What if it snows again this year?” and it was my regrettable duty to concede that I couldn’t rule that out.
By contrast with the grocery store, almost nobody on the bike path was wearing a mask; maybe 10%? We didn’t either. It was pretty easy to stay far away from people, and everything I’ve been reading suggests that outdoor transmission is rare. Tell me if you think that’s antisocial.
You could see that people were working to comply, getting into single file and hugging the edge of the path when someone came the other way. All except the guy with a big red parrot perched on the back of his bike, who kept stopping and letting kids play with his parrot. When I mentioned this to a neighbor she said “You saw the parrot guy? I’ve heard about him but I’ve never seen him in real life!” Somebody on Reddit saw him a few months back, on the same path we were on. Last we saw him he was still heading southwest, deeper and deeper into Fitchburg.
We have mostly settled into a routine here. In the morning the kids have school. The 8th grade has a couple of on-the-screen meetings most days, but the 3rd grade has almost no real-time instruction, just assignments described in video clips by the teachers. That feels right to me; I’ve heard about kids in other cities asked to be in front of the laptop camera for six hours a day as if it made sense to hew to the usual schedule, and that sounds nuts to me. In the afternoon AB has “camp” for two hours where she does art projects with a counselor and a group of kids, mostly from here and there in the US, one from Costa Rica. CJ is still baking a lot — oatmeal raisin cookies yesterday.
It works, basically. I had a lot of ambitions for “things I’ve always wanted to do with the kids but we’re too overscheduled to get to them,” and most of those have been unrealized. I wanted us to play music together and record some tracks. I wanted CJ to do a coding project. I thought, being in the house all day, we could do some reorg and cleaning of the house. Those things didn’t happen (for the good reason that the kids didn’t actually want to do them.) On the other hand, both kids are doing AOPS courses, AB and I have learned to throw a Frisbee forehand, CJ and I watched The Mandalorian (much better overall than any of the last three movies). I always felt I should get into gaming with the kids and AB and I are now working our way through Pikuniku.
Of course, the fact that I can even think about opportunity as well as burden is because I am in the very lucky position of having a job that doesn’t go away during a pandemic, and I don’t have people in the house at high risk of serious illness, so I can safely accept the modest risk of infection that comes with shopping, taking walks, etc.
Nobody I know has died of this yet. Two people I know have lost parents, another an aunt, another a grandparent. I guess it depends what you mean by “know.” John Conway died of COVID last week. Is that somebody I know? He’s somebody I’d chat with when I was around the Princeton math department. His famous theorems are familiar, but in the round of admiration attending his death I learned one I didn’t know; given any six points in R^3, you can partition them into two groups of three in such a way that the resulting two triangles are linked. That’s cool, but the proof is even cooler — it turns out that the sum of the linking numbers over all 10 such partitions is always odd! My favorite kind of existence proof is “there are an odd number of these things so there aren’t zero of them.”
Our neighborhood bakery and our favorite breakfast place got PPP small business loans so they can continue to pay their employees for the next couple of months. That loan program ran out of money in 28 seconds or something but Congress is planning to fill the bucket with money again, they say. Campus is still closed but the undergraduates are rebuilding it in Minecraft. The governor has released a plan to start reopening schools and more businesses, once cases in Wisconsin start showing a consistent decline. That looked like it might happen soon, but now there’s a new outbreak in Green Bay, tied to spread in the meatpacking plants there. (Yes, non-Wisconsinites, that’s why the football team is called that.) The workers who keep the food supply going, just like the doctors and nurses treating patients, are unavoidably going to be exposed to a lot of risk, because, unlike me, they do work that can’t be done on a screen and can’t not be done.
Trader Joe’s on Friday: the first time I had to wait in line to get in the store. To maintain an appropriately low density, they don’t let a new shopper in until someone comes out. This week, about 90% of shoppers were masked. The people who weren’t were mostly college-age. The food supply still seems pretty normal; a few things, like butter, were out, but it’s Trader Joe’s — there’s always something they’re for some reason out of. I asked the store manager whether they were selling more beer than usual, and he said, beer, no, hard liquor, yes.
Large majorities in Wisconsin support the governor’s safer-at-home order, but there are always dissenters:
Protester from Menomonee Falls: “This isn’t common throughout the state. It’s common in Milwaukee. It’s common in Madison. Lock them down.”
You might be surprised to hear I have some sympathy for this point of view, though he needs to be more broad with his lockdown; Waukesha County, where Menominee Falls is, has just as high a case rate as Dane does.
But it’s not crazy to imagine that COVID spread might be slower in less dense regions; maybe so much slower that the pandemic could be kept in check with less stringent suppression measures. Let’s posit that, eventually, we open schools and some businesses in rural Wisconsin before we do the same in Milwaukee. So this guy gets his wish.
My concern is this: he is not going to then say “It’s just like I said, I want to work and be productive, I’m glad I’m able to do so and I support strong relief measures for my fellow Wisconsinites in Milwaukee who have to stay home for the sake of public health.” No, I think that guy is going to say “Why should my taxes be paying somebody in Milwaukee to sit at home when I have to work?”
I have occasionally worked to lose weight, never too seriously because my weight problem has never been too serious. I used to sometimes do the Scarsdale diet in sync with my dad and once, a few years back, I went six weeks without carbs.
Anyway, a month without restaurant food has gone by and I’m 13 pounds lighter. Even though I’m eating all the cakes and cookies the kids are baking, snacking at night, going through enormous amounts of eggs, doing everything wrong. I looked up the records from doctor’s appointments and this is the least I’ve weighed since 2011. Who knew all it took was an order from the Governor to stay at home and make my own food?
Already discussed: the fracas over the April 7 spring election, which should have been postponed, or held by mail if it was held at all. To my great surprise, Jill Karofsky, the liberal running to unseat Scott Walker appointee Daniel Kelly from the Supreme Court, did so, with a bang, winning by about 11 points. Incumbents usually don’t lose Supreme Court races here and I thought Democrats’ political attention was taken up by the Presidential primary, by now all but over. Since Trump’s election, conservative candidates have won only one out of seven statewide elections here, and that one (Brian Hagedorn for Supreme Court) was by half a percent.
Why did Karofsky win by so much? One natural theory is that the election being the same day as the Democratic primary helped bring Democrats to the polls. Boosting this: Bernie Sanders made the apparently strange decision to campaign in Wisconsin, stay in the race until election day, and then immediately drop out before the results were reported. It all makes sense if you understand his motive to be getting his voters to the polls to vote for Karofsky as well as him.
But did it work? This chart from Charles Franklin, who knows Wisconsin politics like nobody else, says otherwise:
Turnout was way up for a court election: 1,548,504 vs 1,207,569 in 2019. This is 2nd highest court turnout since 2000.
If it was the Democratic primary driving Democratic voters to the polls, there’d be a bigger turnout boost in more Democratic counties. There wasn’t. So either the primary didn’t really boost turnout at all, or Republicans were equally motivated to go to the polls and vote for Trump against — well, the state GOP didn’t allow Trump’s Republican primary challengers on the ballot, so against nobody.
Was turnout actually higher because of the pandemic? Maybe people are more likely to vote when they’ve actually got a ballot to mail than they are to find time on Election Day.
Our first Seder without family since 2006, when I broke my arm so badly a week before Pesach that I couldn’t travel: Dr. Mrs. Q, baby CJ and I did it alone. This year we had grandparents in by Zoom both nights. But I had to cook Seder dinner, which I’ve never done. We all have things we don’t do in the kitchen for no reason except it’s not our habit. For me it’s giant pieces of meat. Just not what I cook. Don’t know how to roast a chicken or a turkey, don’t ever make leg of lamb (butterfly? spatchcock?) and I have never, before this week, made a brisket. But it’s easy, it turns out!
I was extremely successful, to my surprise, in hiding the afikoman. Both nights I thought it was in too easy a place and both nights my kids required multiple hints and were very satisfied with the search. Either I’m more cunning than I thought or my kids are not born hunters.
We did a gefilte taste test this year; traditional vs. tilapia. Tilapia is better!
With two days left to go we have eaten just about all the eggs.
We have been playing 4-person Ultimate in the backyard, AB and I vs CJ and Dr. Mrs. Q. They always win. The team of AB and me is called “Violent Frisbee” and AB has made us a flag:
Election Day in Wisconsin today; every other state with an April primary moved theirs, but not us. The state legislature, seeing that a contagious illness centered in Milwaukee would be good for depressing Democratic turnout, decided the show must go on. I sent in my vote by mail. But a lot of other people tried to as well, upwards of a million, and the overburdened clerks couldn’t get all the absentee ballots mailed out in time. Some people still don’t have theirs, and that means they don’t get to vote.
I went to Metcalfe’s instead of Trader Joe’s because they have a better Passover selection, and because Trader Joe’s was closed after a worker there tested positive. Wore a mask again. This time the proportion of mask-wearers was close to half and included some of the employees. Metcalfe’s made its aisles one-way to avoid people passing each other, but the signage was confusing and compliance was weak. Shelves were pretty fully stocked but toilet paper/paper towels/hand sanitizers were one to a customer. I bought a giant brisket for seder, which is in the oven now. I’ve never cooked seder dinner before because we always have it at Dr. Mrs. Q’s mom’s house. This time we’re bringing her in via Zoom and hoping for the best.
The only store I saw open in the mall besides the grocery was the Sprint cellphone store. More restaurants than I’d have thought were open for takeout, but we haven’t gotten any takeout yet. The last meal I ate that I didn’t cook myself was a burrito at the Tucson airport on March 9.
“School” has begun. It seems they won’t be doing very much real-time instruction, which I think is for the best; mostly short meetings where teachers give and explain assignments for kids to do in their own time.
I now have two friends who’ve had COVID; both have recovered. I don’t know anyone who’s died.
AB and I have been playing frisbee in the backyard, trying to learn how to throw a forehand reliably. We’re getting sort of OK. Two houses down from us, two guys were working on the roof; one of them coughed at least three times, and wasn’t wearing a mask. Wear a mask! For the sake of your fellow roofer!
.
.
,
is greater than 1. This is the kind of analysis you’d want to use if you wanted to really start taking geography into account. For instance, take the vertices to be random points in a disc and let W(x,y) be a decreasing function of |x-y|, modeling a network where infection likelihood is a decreasing function of distance. What is the norm of the operator A_W in this case? I’ll bet my harmonic analyst friends know, if any of them are reading this. But I do not.
