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Bad Horse


Sufficiently advanced friendship is indistinguishable from magic.

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Mar
29th
2020

Problems Interpreting the Covid-19 Data · 2:09am March 29th

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We've seen a lot of graphs like this, showing the total number of Covid-19 cases worldwide, like this graph from worldometers.info:

But then I saw these graphs on the New York Times website:

This shows both the number of cases found (the red bars) and the number of tests done (orange bars).  Eyeballing the two right-most graphs, it looks like the number of cases found was determined mainly by the number of tests done.  If you're testing an exponentially-increasing number of people, you're going to find an exponentially-increasing number of cases even if the virus isn't spreading.

So I downloaded the data from the Italian Department of Civil Protection (took seconds--thank you, Italy, for providing the data file!), copy-pasted tables for the US from the Centers for Disease Control and Prevention (CDC) and The COVID Tracking Project (CTP), and typed in the data from the Korea Centers for Disease Control and Prevention (took 3 hours--you could've at least put it all in one table, South Korea!).  Then I used Google Spreadsheet to plot the fraction of people tested on each day who tested positive.

Here's the log graph of the number of cases tested each day in the US from 3/3 to 3/26:

In fact, the number of tests administered on each day accounts for nearly all of the variance in the number of positives and in the number of covid-19 deaths.  All of these data sets fit a curve yday = y0 × crday very well, with an R2 of .989 to .999.  The values of r for deaths, positives, and tests given are 1.32, 1.32, and 1.29.

The data looks as if the increase in positives and deaths attributed to covid-19 has been determined mostly by the number of tests given.  So graphing the number of total positive results may be completely misleading. The data we should be looking for is what fraction of the population has the virus on each day, and the best proxy for that which I can think of is the fraction of the test population that has the virus on each day.

But graphing that gives us quite a surprise.

Remembering that the disease has an incubation period of 5 days, and Bi et al. reported a serial interval (SI, the mean time between when someone is infected and when they infect someone else) of 6.3 days (that's an estimate of a lower bound or the SI, because of how the data was collected), it appears transmission of covid-19 was already declining by Feb. 17 or 18, before South Korea banned large gatherings in Seoul on Feb. 21 and declared a "highest alert" on Feb. 23. The steps that had been taken before that--banning flights from Hubei, China, and large-scale voluntary self-isolation and wearing of face masks--may have been sufficient.

The fraction of people with symptoms who tested positive had stabilized in Italy by March 5.

The US data is wonky for at least three reasons:

  • The CDC did relatively few tests in January and February--on most days, less than a hundred.
  • On Feb. 27, testing moved abruptly from being done on a small scale by the CDC, to being done on a large scale, locally, and the number of tests being given immediately exploded exponentially (and has continued to do so).
  • Since the CDC no longer knew how many people were being tested, I transitioned from using the CDC's data to using the Covid Tracking Project's data starting March 3.

    • Judging from where they overlap, the CTP data appears to be delayed by one more day, so I used their March 4 data for March 3, etc.
    • These data sets conflict with each other at times on the number of cases reported, but by less than 10%.
    • Both report the number of new people who've tested positive, but the CDC reports number of specimens tested, while the CTP reports number of people tested.  This accounts for some unknown amount of the difference between the left and right halves.

So we shouldn't think that the sudden change in the graph from 2/27 to 3/3 reflects a sudden change in the infection rate.  It's the result of a sudden change in the testing rate.

If we do a linear regression on the section from the CTP's data (3/3 to 3/26), we see the fraction of tests that were positive is now declining (with the caveat that the left half of the graph below is noisier than the right half):

How can we reconcile this lack of an increase in the fraction of positives with the explosive increases predicted from models? Liu et al. (see references below) reported that "The doubling time of NCP [covid-19] was 2.4, 2.8, and 3.6 days nationwide, in Wuhan, and in Guangdong Province, respectively,"  Read et al. said "we find doubling times of 2.1, 1.8, and 2.0 days." The US detected its first case 68 days ago, on 1/22. If these parameters are right, we should

I can think of some reasons we might see this data even if the virus is spreading exponentially:

  1. Suppose that (A) the number of tests given is driven by the number of people who want tests, rather than by the number of tests available; and (B) a gradually-increasing panic leads people to get tested based on less-and-less stringent criteria.  In that case, the fraction of people who don't have Covid-19 and get tested will increase faster than the fraction of people who do have Covid-19 and get tested.
  2. Some influenza virus with the same symptoms as Covid-19 could be spreading just as fast as Covid-19, but in much larger numbers, causing more and more people who don't have Covid-19 to get tested.
  3. The covid-19 test used in the US now has a false-negative rate (the probability that a test of someone with the virus is negative) greater than 3/4, and most of the people tested have the virus. This article recounts 3 estimates of the false negative rate: 0 (obviously optimistic), 0.15, and 0.25.

I don't consider any of these very probable, though a perfect storm in which each of them was a little bit true might suffice.

"But, wait!" you might say.  "This virus has an original transmission rate R0 > 2.  It must spread exponentially!"

Unless… you don't suppose the greatest epidemiologists around the world collectively forgot to divide the number of positives per day by the number of tests given that day?

Yes, they did. (In fairness, that data doesn't seem to be available.)

Here are the scientific papers cited by the Epidemic Calculator, a nifty website that models the spread of a disease in some detail, and which Tomas Pueyo used in his recent Medium article calling for panic panic panic NOW:

Publication
Location studied
Reproduction number R0
Adjusted for # of tests?

Kucharski et. al, Wuhan, 3.0 (1.5-4.5), no
Li, Leung & Leung, Wuhan, 2.2 (1.4-3.9), no
Wu et. al, Greater Wuhan, 2.68 (2.47-2.86), no
WHO Initial Estimate, Hubei Province, 1.95 (1.4-2.5), not mentioned
WHO-China Joint Mission, Hubei Province, 2.25 (2.0-2.5), no
Liu et. al, Guangdong, 4.5 (4.4-4.6), no
Rocklöv, Sjödin & Wilder-Smith, Princess Diamond, 14.8, implicit--see below
Read et al., Wuhan, 3.11 (2.39-4.13), no--see below
Bi et. al, Shenzhen, 0.41, implicit--see below
Tang et. al, China, 6.47 (5.71-7.23), no

1. This R0 estimate was omitted from the Epidemic Calculator's table.

Not one of those studies accounted for the number of people tested on each day, probably because they weren't given those numbers.  Two of the studies didn't need to, because of how they sampled.

  • Rocklöv et al. studied passengers all confined together on the cruise ship Diamond Princess.  All passengers were tested, so there's no need to adjust for testing rate.  But an R0 measured within a crowded cruise ship is hardly relevant to R on-shore.
  • Read et al. estimated an ascertainment ratio, the fraction of existing cases which have been detected.  But they assumed this was constant over time, which implicitly assumes that the fraction of people tested remains constant over time.
  • Bi et al. used a dataset which measured R directly via contact-based surveillance (testing all the contacts that a carrier has come into contact with).  Their measurement of R (not R0) = 0.4 is the only estimate of R in this entire set of references that we can consider reliable.

We still have a problem, though: R = 0.4, the only estimate we have that was measured using good methodology, seems improbably low.

The problem with that problem is that R is actually a poor statistic to estimate the spread of disease.  It's the rate of transmissions per infected person. That's not a real thing that a virus can have.  The real thing is the odds of transmission per contact. With a few neurologically interesting exceptions like rabies, a virus can't control how many people you contact.  Bi et al. report that 80% of infections were caused by 8.9% of the traced cases--the superinfectors, people who had thousands of contacts while infected. So computing an "R for a virus" is literally meaningless unless you assume that the entire population--in this case, the entire world--has the same distribution of number of contacts per person.

Which it doesn't.  Even within a single nation, this distribution will be quite different in areas of high vs. low population density.  The "R" model is, like most of our governments, city-centric; it assumes that's what's best for big cities is best for everybody.  Right now, rural stores, businesses, and even hospitals in America are closing because they can't afford to pay the rent, the payroll, or the health insurance for workers who aren't working, even though the R in rural areas is, so far as I know, approximately zero.  Unfortunately no agencies report cases with a precise location, or even the local population density.

IIRC, each of the major outbreaks was linked to a large meeting, or to the market in Wuhan (itself a city of 11 million people).  So studying outbreaks to learn R0, as every one of the studies above did, is meaningless--outbreaks only happen when there's such a huge gathering of people that R is blown sky-high.  Your sample has been self-selected for extremely high R.

So even if those early studies of the outbreaks in China were done correctly--which they might have been, if the field researchers tested or surveyed the same number of people every day--it doesn't matter.  The transmission characteristics of an outbreak don't represent the transmission characteristics of the virus most of the time.  An outbreak is noticed and studied only because it's an anomaly. Estimating the transmission rate of a virus by studying outbreaks is like estimating your chances of winning the lottery by interviewing lottery winners and asking them how many times they had played the lottery.  At least 25 other cruise ships besides the Diamond Princess carried Covid-19 onboard, but nobody studied them to estimate transmission rates, because they weren't as big outbreaks, so they couldn't provide as much data.

So, if what is meaningful and useful is knowing the probability or rate of transmission per contact, what do we know about that?

Tang et al. used a Markov Chain Monte Carlo (MCMC) method to estimate the probability of transmission per contact as 2.1x10-8.  That's one in 210 million.  Carlo Bi et al. estimated the secondary attack rate (the chance that a random contact of a carrier will contract the disease) within a household as .112-.158.  That's your chance of contracting Covid-19 if you're living with someone who has it.  So Pueyo's assumption that literally every person in America will eventually contract Covid-19 is absurd.

To sum up:

  • The virus is probably not spreading exponentially in South Korea, Italy, or the US as a whole anymore, although it may still spread exponentially in some times and places (particularly big cities and popular tourist spots).
  • All but one of the studies that were used to build the models of the spread of Covid-19 looked at the beginnings of sudden outbreaks instead of ongoing transmission, and/or failed to look at the fraction of people who test positive over time rather than the number of people who test positive over time.
  • That one good study measured a transmission rate R = 0.4, far below what's needed to sustain an epidemic.
  • The R-based model of epidemics is unsuitable for studying a worldwide pandemic, because a disease will have a different R in every region.
  • A better model would be based on the transmission per contact, and extensively model or measure the distribution of contacts per case.
  • Government response should take into account the different transmission rates that occur in urban and rural areas.

I don't want to discourage anyone from taking precautions. As for the US, I think the CDC and the states both should immediately begin gathering and reporting more data. How much testing is being done, for a start--neither the CDC nor the states report that at present. To figure out what's going on, we also need data on how consistently post-mortem tests are performed, how consistently guidelines for testing criteria are followed, and what the time lags are at every step in the process from gathering specimens, to testing them, to reporting the results.

And all governments should re-evaluate the data before taking any more drastic measures. The correct analysis should become clearer in another couple of weeks, because we'll know something big is coming if the number of cases continues to rise exponentially. Also, sometime around then, there should be enough tests to test everyone with symptoms.

Comments ( 72 )

If you've been following me for a while, you may have become cynical enough to guess the answer

I assure you, my cynicism comes from independent research.

Either way, I've been temporarily laid off, have had to file for unemployment, am shut in my apartment, and irritated that the Chinese government could have stopped all of this several months ago.

If the virus is not spreading exponentially, then why are deaths increasing exponentially? By eye the dynamics of the cases curve seems to mirror the dynamics of the deaths curve but shifted by a few days (looking at the US data for the worldometers site) . That the two curves are largely in agreement would seem to suggest that the increase in testing is not contributing unduely to the increase in cases.

Generally, testing (at least in the US) is mainly done on suspected cases, so the increase in testing partially reflects an increase in suspected cases. The relatively flat positive rate just reflects the guidelines for choosing suspected cases.

Your point about R_0 being different in rural vs urban environments is well taken, however. The type of control measures taken in NYC are probably not the same as those necessary in Montana, for example.

Bless you for giving us facts, not fears.

5230825 A good question. I'm sure it spreads exponentially in some times and places; otherwise it wouldn't still be around. But the exponentially-increasing worldwide cases and deaths could be accounted for by nations all around the world giving an exponentially-increasing number of tests.

AFAIK, only deaths of people who had been diagnosed with Covid-19 have been counted as Covid-19 deaths. Since we're rationing tests, I doubt anybody is testing corpses for Covid-19. Those deaths will get blamed on influenza, pneumonia, old age, diabetes, or some other pre-existing condition.

Also, given that the deaths attributed to covid-19 are concentrated among people over 70 with heart disease, diabetes, chronic respiratory disease, and/or cancer, we should ask how many of these people really died of covid-19. One could do either of these tests:

  • Look up the number of people in the world who die of these diseases every day. See if it's going down during the covid-19 epidemic.
  • Find an actuarial table and compute the expected number of sick or elderly people with covid-19 that we would expect to have died anyway without covid-19.

Bad Horse, I know you've read Yudkowski.

If you're testing an exponentially-increasing number of people, you're going to find an exponentially-increasing number of cases even if the virus isn't spreading.

We know the virus is spreading, so this is meaningless. If it wasn't there's nothing happening at all. At best this is an absence of evidence is evidence of absence arguement.

Exponential growth means growth proportional to the number of people who have it. Everyone who is infected has a chance to spread it, so this is more or less true. Yes, R is a garbage metric, or at least it shouldn't be used alone to determine response. But on average, the more people have it the faster it spreads. Thats exponential growth. If it's not growing exponentially, how do you think it is growing?

You can change the co-efficient in the exponential via measures such as social distancing and mask wearing , which is what South Korea and other Asian nations have done, which has indeed slowed the spread a lot.

Remembering that the disease has an incubation period of 5 days, it appears transmission of covid-19 was already declining by Feb. 17 or 18, before South Korea banned large gatherings in Seoul on Feb. 21 and declared a "highest alert" on Feb. 23.

Almost everyone in South Korea was wearing masks before official government intervention, which would cause the slow down in transmittion.

As you say, every region of the US is different. Do you really want to bet that no parts of the US will have their medical facilities overwhelmed by this, causing massive deaths? Not to mention the third world, which can't get a lot of help right now as the West needs all the doctors and medical resources it can use.

I don't see why you think the fraction of people who tested positive is such a great metic. Without looking at data, we can guess that in the beginning when there is only testing at outbreaks, it will give high fractions, then if testing is rolled out to other places faster than the virus spreads, it will give low fractions until the virus catches up. The nature of exponential growth is slow then fast. If you test an area that has just recently been infected, few people will have it, then that will change.
Taking fraction of people testing postive only really works as a metic if you are contantly taking random samples of the population though time, which you can't do.

This is not the case if your test threshold is relatively consistent. The reason there are more tests is there are more people with symptoms.

I'd trust the doctors on this one.

5230833
The life expectancy for an 80-89 year olds in China is 6.58 years. Median time to death for fatal COVID-19 patients was observed to be about 20 days from the onset of symptoms, and for people over 80, the mortality rate was 14.8% in studies of the Chinese outbreak. I don't think 14.8% of all people >80 years of age are going to drop dead in the next three weeks. (There are 23 million 80+ year olds in China, so this would correspond to 3.4 million deaths over three weeks, and you would only expect to see 0.56 million total deaths of all ages in China over three weeks.) Modeling survival as an exponential decay with a half-life of 6.58 years, we would only expect to see about 0.61% mortality among octogenerians over a three week span (even if you double the period to six weeks as 20 days is only the median time to death, you still only expect 1.2% of octogenerians to die of natural causes in that time span). Thus, COVID-19 may increase mortality among octogenerians by over an order of magnitude.

Does the 14.8% represent the proportion of the >80 year-olds that are worse off in health? The study of life expectancy among the elderly in China found that ~ 11.8% of octogenerians were disabled (they require assistance in the essential activities of daily living like bathing, dressing, eating, or going to the bathroom). Among disabled octogenerians, life expectancy is only 1.1 year. Still, you would only expect to see 3.6% of that population die over the course of three weeks, nowhere near the 14.8% mortality seem among all infected 80+ year olds (regardless of whether they have additional health problems). Even the 70-79 age group (mortality rate of 8%) has twice the expected death rate of disabled octogenerians.

The health conditions that pre-dispose COVID-19 patients to greater mortality do not come close to explaining the observed mortality of the disease.

5230834
As I just mentioned to equestrian_sen Catalysts Cradle, it must spread exponentially some of the time, in some places, or else it wouldn't still be here. I'm not saying that isn't happening anywhere. I'm saying that the doomsday predictions, like Tomas Pueyo's claim that 5 million Americans will die from it in the next 2-3 months if we don't do something, are based on models with unrealistic assumptions. I'm saying those exponential graphs misrepresent the data, and that more-accurate graphs aren't as scary.

As you say, every region of the US is different. Do you really want to bet that no parts of the US will have their medical facilities overwhelmed by this, causing massive deaths? Not to mention the third world, which can't get a lot of help right now as the West needs all the doctors and medical resources it can use.

I didn't say anything like that. I said that the data people are looking at is the wrong data. We should fit our models to the right data.

Well lets also consider that this virus preys on the very old and the very young quite hard and is a hella upper respiratory infection, while coming from one of the larger population density areas in the world. Knowing that they are a huge import export hub and the likely hood of virus travel I do believe the travel bans and such were a decent idea. I know its totally impossible to keep them from entering the country because vital trade and such but it does put a limiter on how much of the virus can spread.

So I can see putting a lockdown in place in our country can actually be beneficial idea even with the lack of doing division. Also consider this China is a huge air travel hub and since I live in one of the 10 states that have our own travel hubs, and live in a town that links to that travel hub fairly directly. A international flight is basically 6 to 10 hours of breathing re filtered air of 50 to 100 people and if even 2% have that virus the likelihood of that plane becoming carriers is pretty damn high. That also doesn't count the spread since as a travel hub you don't always live in the same area your traveling through. So while current rates of infection are low, thats also after they basically stopped incoming people from coming from said countries that had infection rates.

So in all do I feel that people are overreacting a bit, yes. Do i feel that measures taken probably kept the infection rate down in the USA, yes. Do I feel that keeping a lockdown over a month is probably going a bit far, yes. Do I feel that people have given into Paranoia somewhat, yes. Do I feel bad about people loosing their jobs because in the lockdown a lot of jobs deemed non essential were shuttered, yes. Do I also feel that healthcare in america is going to get a big overhaul since all those benefits are getting negated during a health crises cause johnny business man basically made it only affordable if your employed, yes.

Will small and medium businesses suffer because our economy isnt made to let people take non paid weeks off at a time and that society in its instant gratification is suffering from bills and debt being due during a time of non profit and that banks will foreclose on them because said people are always living paycheck to paycheck with no padding to keep them afloat, yes.

5230849
This is a fair response, but if you allow for the possiblility of massive spread causing many deaths if governments do nothing, I think you are most of the way to agreeing qualitatively with Pueyo and others. It's an X-risk arguement, whatever our models, there is uncertainty and the possiblility of huge consequences. So I just mean to say focusing on those arguing for action and preparation ( I find 'panic' to be disingenuous almost every time it is used) is probably less useful than looking at places like Vox and WHO which have been constantly downplaying the possibility that this could get very bad. And I think my arguement about fraction of positive tests still stands, your improved models don't exclude the possibility this gets very bad, so it's reasonable to take action ("panic")

5230848

The health conditions that pre-dispose COVID-19 patients to greater mortality do not come close to explaining the observed mortality of the disease.

You made a good case for that. I concede that, between the known recent origin of the virus, and the number of deaths among people diagnosed with it, there must have been considerable exponential growth, leaving the question of why that would suddenly stop.

So we have conflicting data under any interpretation. But that doesn't mean we should ignore and throw out my observations that infection rate probably isn't going up, and that the estimations of R were flawed. We need to try to reconcile all the data.

5230844
I thought that for a while, too, earlier today. But then why does the number of people without the virus who have symptoms increase just as rapidly as the number of people with the virus who have symptoms?

By strange coincidence, I woke up this morning wondering what a text-based XKCD would look like.

Seriously, it strikes me that you feel about math the way I feel about language: you can't stand sloppy usage because that does violence to something you hold dear.

5230865
Again, I don't think there is evidence that spread had stopped. Think of testing this way:

1) Testing is done on people whom you suspect may have the disease. (This is done for Bayesian reasons; the positive predictive power of the test will be greater if the prior likelihood of a positive is higher).

2) Therefore, an increased number of tests reflects an increases number of suspected cases, which should reflect an increase in the number of actual cases.

3) When might this not hold? When the fraction of suspected cases that are positive changes. For example, if you make the criteria for suspected cases more lenient, you may be picking up additional mild cases that would have been missed before, inflating the number of detected cases without actually changing the prevalence of the disease.

4) However, your data for the US shows that the rate of positive tests is relatively stable, with only a slight decrease. This would only require a small decrease in the number of cases at later timepoints to correct for the increased testing (essentially, you would try to normalize the data such that the positive test rate is constant over time).

5) I'm pretty sure of you normalize the data in this way, you will still see an exponential growth in cases in the US (I'm not an epidemiologist, so I don't know if this would meaningfully affect r_0 or other parameters of the models being used).

"A virus can't control how many people you contact"

That isn't true – and is actually one of the reasons why Coronavirus, Influenza, and others are so dangerous. Since their symptoms are very mild for a significant percentage of the infected, they are able to travel and interact with a lot of people — unlike a more deadly disease.

Additionally, "contact" in this case would be better phrased as "can pass the virus to". In this case, "successful" diseases are the ones whose symptoms actually help them spread – like spreading viruses over a large area when you cough, or how a disease like Cholera causes diarrhea, which will spread the disease to more bodies of water.

5230874
I meant what I literally said--a virus can't control how many people you contact. Whether you know you have the virus or not. I wasn't relying on people reducing contact; I was trying to point out that the number of contacts per person is highly variable.

5230866
This would happen if you test close contacts of infected individuals.

5230861
Pueyo is trying to get people to sign a petition for immediate, drastic enforcement of stricter measures across the entire US, so I don't agree qualitatively with him. I don't claim to know what should be done in other countries, but it seems that efforts in South Korea, Italy, and the US, taken voluntarily before any nationwide or statewide bans had gone into effect, were enough to slow the virus down to the threshold of R = 1.

Re. what you said about fraction of positive tests, I think that what you're saying requires that the threshold of symptoms needed to get tested has been steadily declining. I don't think that's happened in the US. On the contrary, (1) the positive rate was extremely low for the first month, and (2) we've recently gotten guidelines for doing stricter testing. Trump made his policy declaration about stricter testing just last week.

And it couldn't very well have happened in Korea, where they kept the number of people tested per day nearly constant for over a month, and the number who tested positive kept declining.

I feel confident that fraction of tests that are positive is a better metric than total number of tests. But I'll admit I'd have to know a lot more about who has been chosen for testing, and why, to be confident that R ~ 1 in Italy and the US today. There could still be an exponential rise hidden in that data. But not one as steep as we've been led to believe.

5230872
I (hopefully) never said the virus had stopped spreading. It must still be spreading, or else the fraction who test positive would be declining.

3) When might this not hold? When the fraction of suspected cases that are positive changes. For example, if you make the criteria for suspected cases more lenient, you may be picking up additional mild cases that would have been missed before, inflating the number of detected cases without actually changing the prevalence of the disease.

I mentioned this point in my post, though I added it after posting, and it might not have been there when you loaded the page.

I'm not sure I understand points 4 and 5. I have a notion that you mean that there's an exponential in the data that I've hidden by normalizing by number of tests given on each day. I don't think that can be the case.

5230856
I think our views are similar. I think travel bans and bans on large meetings are reasonable ideas. But I think shutting down rural businesses is not, and admitting only people who need care urgently to hospitals will carry a heavy penalty for a long time even after that rule is lifted, because getting an appointment to see a doctor took 1 or 2 months even before covid-19.

Another thing worth pointing out: remember that your data shows people who tested positive over specimens tested (where multiple specimens can come from a single individual). The mean duration of a hospital stay is about 11 days for someone diagnosed with COVID-19. Assuming they are treated daily to check levels of the virus, the number of tests should always be at least 10x the number of cases, and one reason why the number of tests grows with the number of cases.

5230849

As I just mentioned to equestrian_sen, it must spread exponentially some of the time, in some places, or else it wouldn't still be here.

What? I don't think I've ever talked to you about covid-19. Unless you're talking about the equestrian_sen you talk to in your head.

Yeah, the rural midwest is remarkably separated even on the best of times. When Kansas finally got over a hundred cases, that works out to roughly 1,000 square miles of state per infected person. (although technically most of the cases are in Kansas City). We (in my area) have little to no mass transit, our bus system is these little puddle-jumpers around town, and to see more than one person in a car means it's a family.

The Diamond Princess had all the conventional signs of being the catastrophe the media thought C-19 was going to mean for the rest of the world: 4,000 passengers and crew in close confinement for a long period, mostly elderly, damp and chill environment, etc... If it had played out the way the media thought, all 4,000 people would be infected (because exponential growth, of course), with at least ten to twenty percent fatalities or worse since they're old and.... Testing showed 712 contracted the disease (18% or so) of which about half did not have symptoms, leaving 378 who actually got sick, and 7 died. That's still too many, but there's literally no way to bring that down to zero on a global or countrywide scale without cranking up other factors that can prove to be far more dangerous.

5230844 The reason there has been a spike in tests in the US is that the FDA and the CDC fell flat on their face during the early phases of this outbreak. It is not until recently that enough tests have become available to test the "Well, they may have it but we're not sure and they're not in serious trouble, so we'll save it for the ones who really need it." Even Politifact agrees. That's why looking at only numbers of positive tests in isolation is not an accurate measure of the actual spread of a disease. (Also, inaccurate tests can totally screw up any attempt at tracking disease progression. Even as little as a three percent false positive/negative rate.)

5230906 Our church shut down services, which I can see, because their median age is about 65-70, and one case could go through the congregation like the grim reaper. That little virus hits like a sledgehammer.
5230879 But *when* do you test close contacts? Too early, and you won't pick it up. Too late and it's too late. Even testing somebody today does not mean they won't pick up a case tomorrow.

Really appreciate the write-up, Horse. Just came back from traveling out west to help the family move into a new rural home in Nowhere, USA, and have pretty much been out of the loop on what's been going on. I'll definitely take a look at these linked and referenced sources.

Unless… you don't suppose the greatest epidemiologists around the world collectively forgot to divide the number of positives by the number of tests?

Is dividing by the number of tests a standard practice in estimating R0?

Seems very on-point to me, and I've been trained to be highly skeptical of statistical arguments in general. I was double-checking similar claims about COVID-19 last week, and I had similar issues with the numbers being put out. Thanks for doing the legwork on this one, Bad Horse.

I think the most notable takeaways for me here are:

That there's no rural-centric model for any of this (because there's no rural-centric model for anything of importance, generally speaking), which makes the USA very poorly-equipped to learn from highly urbanized countries in how to deal with such pandemics. Our urban centers do need to control the curve as much as possible, and granted that the lack of medical infrastructure in rural areas means it isn't hard to overload them, either - but we need to keep a realistic interpretation of facts in mind regardless.

The rate of infectivity of this virus has clearly been grossly over-exaggerated in some instances. That doesn't mean it's not infectious, or that we shouldn't be taking extreme care to curb it, but again, truth matters. We need a realistic idea of how much this will spread, why, and what steps can have the biggest impact - especially if we're going to engage in discussions of what can remain operational and what can't under lockdown.

Honestly, all of this comes off more as a damning indictment of our medical infrastructure than anything else. If we had enough tests, we'd have a much clearer idea of the infection rate; if we had enough medical capacity, we wouldn't need to take such drastic actions to control the curve. It's a clearly-identifiable weakpoint in our social infrastructure, made many times worse by things like uninsured folks who can't afford to get tested or treated (or even physically get to the hospital for such treatment) continuing to spread the disease when sick. A huge combination of ugly factors that make even rural neighborhoods terrified to do anything but lock down, lest their disgraceful shortage of supplies and infrastructure be overtaxed.

The virus is probably not spreading exponentially in South Korea, Italy, or the US as a whole anymore,

Y'know, maybe you're right. I haven't so much as run a t-test in eight years, I won't make an even bigger ass of myself by trying to look at the data.

But the US is interesting because unlike Italy or South Korea, it's a failed state which lacks the medical capacity to do either of the strategies put in place by those two countries. "Forcing people to come to work sick" has been a method of labor discipline imposed by US bosses for decades. Italian car factory workers went on strike because they're not essential workers and they weren't being provided with PPE, US unions are atrophied and their strikes are illegal wildcat ones out of necessity. The disease is already overwhelming the New York medical system, and New York has taken it kinda seriously. Mississippi has not only refused to lockdown, but the state governor has overrode all local and city lockdowns in place.

You're running a comparison based on two states that put control measures - not necessarily lockdowns, but measures to track and control the spread - in place and are determined to enforce them. I don't think you can say the same about America (or Russia or Brazil, who are both on track to have insane outbreaks).

If you're right, you'll be standing on a big pile of "COVID-19 recovered cases" in six week's time, and I'll be over here with my dick hanging out looking like an idiot.

I don't want to think about what either of us will be standing on if you're wrong.

You're weighting the initial spikes of percentage 'tested positive' done in early March too heavily in order to get the result you want. (edit: I fully understand how harsh this accusation is relative to your value system, but I'll stand by it. You are doing this to get the result you want to argue for)

That data is much more noisy because the sample set is much smaller. (there's also a tendency, in conditions of limited tests, to avoid doing them unless the outcome seems important: wasting tests on possibly-not-infected people is the later part of the trend line, not the early limited-tests part) The actual trend line is more accurately represented by 3/11 onward where the sample set is increasingly larger, and shows that not only is testing increasing exponentially, but that (for now) the percentage testing positive is also increasing every day. This represents the conditions of a pandemic as we understand it.

The data I see on this is invariably plotted log, and the angle of the straightish line is the thing of interest. Some of these plots, such as out of New York, still appear to hint at an exponential curve even when plotted log, which is literally the opposite of what we'd like to see.

I prefer when you 'Bad Horse' about Greek literature. That behavior is almost certainly less likely to kill people, depending on how mad people get about Greek literature.

5230908
No; all of the data was reported as number of new people tested per day, except that from the CDC, which, as I explained in the notes under the graph for the US, reported number of specimens tested. The CDC only did enough testing to confirm each case. They're monitoring the spread, not managing daily care.

I'm almost certain sick patients aren't tested daily; they're kept until symptoms subside.

  • You have to keep them until the symptoms subside, regardless of whether the virus is still active.
  • You can release them when symptoms subside, because they've acquired immunity and aren't contagious.
  • They'll probably still test positive on a PCR text, and certainly will on an immunological test, even after recovering.

5230979 AJ, I downvoted you for assuming that you know what I want, and for assuming that I would argue for the outcome I wished for, making a personal attack rather than just arguing with the data. Not sorry.

That said: You should specify which graph you're talking about. I think you're talking about the US graph. You're correct that the points on the left end, before ~Mar. 10, are more noisy. That's a sample of 17,227 tests, quite large enough that we don't have to worry about noise on a per-case basis. But we still might want to worry about the sample randomness; the earlier points were probably gathered from a smaller number of test centers, And we might worry about replicability; people might not have gotten their routines down yet, and made more errors. And they might have changed a lot of other things in that first week.

All that said, I don't care much if the right-hand side of the graph is more correct, and shows a gradual increase of covid-19+ people. Obviously the virus hasn't stopped spreading, or that line would be curving down. I NEVER SAID THAT THE VIRUS HAS STOPPED SPREADING. I SAID THAT THE MODELS PEOPLE ARE USING, AND THE FORECASTS THEY ARE MAKING, ARE BASED ON BAD DATA AND GIVE CONCLUSIONS MUCH WORSE THAN WE SHOULD REALLY EXPECT. That is what I said. That is all. I'm not pursuing a political agenda or seeking a desired outcome. I observed that the whole world is panicked based on a shitty analysis of data.

Sorry for shouting, but the comments seem full of people who think I'm saying we should shut down the CDC or something, and people who think that charging forward with shitty data analysis as our guide is somehow being "cautious".

Liu et al. reported that " The doubling time of NCP [covid-19] was 2.4, 2.8, and 3.6 days nationwide, in Wuhan, and in Guangdong Province, respectively," Read et al. said "we find doubling times of 2.1, 1.8, and 2.0 days." This is OBVIOUSLY not happening in the US. If there's a "doubling time" (which is not a great way to model the spread of a virus, because the doubling time must continually increase due to acquired immunity), it's measured in months, not days.

The data I see on this is invariably plotted log, and the angle of the straightish line is the thing of interest.

That's what I'm talking about! That data is counting number of cases, not fraction of tests that were positive. (You could probably find true rapid exponential increase in fraction of positive tests at the start of an outbreak--because that's what defines an outbreak. We're dealing with a statistical distribution of virus transmission rates, and at any given time, some fraction of samples will have a high-enough transmission rate to be sharply exponential.)

5230948
Dividing the number of positive samples by the sample size is just basic statistics and common sense. We aren't interested in how rapidly the number of tests is increasing.

5231016
Dividing positives by the sample size is common sense when the sample is representative of the population. But only people who have a high prior of being positive are even tested. So I don't see what this fraction of positives is supposed to measure. It probably has little to do with the fraction of infected people in the general population, which would be what we're interested in, no?

5231024
You're correct that this fraction of positives will be different than the fraction of infected people in the general population. But if the prior odds of a test subject testing positive remain the same--an assumption which would be flawed if the criteria needed for being tested changes over time--then this fraction will be proportional to the fraction of infected people in the general population.

I should have mentioned that in the post; thanks for prodding me. The assumption that test criteria remained roughly constant, or at least didn't gradually loosen, is probably the weakest point in my argument. Test centers have constant guidelines, but whether people on the ground followed those guidelines consistently, I don't know. You'd probably have to interview people at test centers to learn what's going on. re. test criteria.

5230922 Derp. I meant Catalysts Cradle. You've both got Apple Bloom avatars and talk about math.

5231012
Can we agree that the number of deaths is a good measure to track the spread of the disease? The one disadvantage of deaths is that, because it takes a while (2-3 weeks) for an infection to result in a death, the dynamics of the increase in deaths reflects the dynamics of infection 2-3 weeks ago. However, these numbers should not be affected by changes in the number of tests.

Here's what it looks like if I take the cumulative number of deaths in the US (data from https://covidtracking.com/us-daily/) and fit it to an exponential curve:
i.imgur.com/asBcA1l.png
Where Date is the date in the month of March, N_0 is the expected number of cumulative deaths on Date = 0 (i.e. Feb 29), and t is the time constant for the exponential growth. The data are fit well by an exponential growth function with a time constant of 3.6 days (corresponding to a doubling rate of 2.6 days). If anything the residuals from the fit suggest that the exponential growth model is underestimating the number of deaths in recent days (expect for data from 3/28 which may reflect delays in reporting). The doubling time found from this analysis is consistent with the numbers found elsewhere around the globe that you cited in your previous posts.

Because the dynamics in the growth of the number of deaths reflects the dynamics of infections 2-3 weeks ago, these dynamics largely reflect the spread of the disease before many major social distancing measures had been put into place. Data over the next several weeks should tell us whether these measures have been able to slow the spread of the disease. Of course, I do agree with you that these numbers are not reflective of the entire country; likely, they mostly reflect transmission dynamics in the large urban areas that are driving most cases in the US. As I said in previous posts, I agree that it is probably appropriate to take different control measures in a dense urban city vs a sparse rural area (but that said, we don't have much data on the transmission dynamics in rural areas to guide policy making).

However, the analysis of the number of deaths largely confirms that, in the absence of control measures, the disease has been spreading exponentially in the US with a doubling time ~ 2-3 days.

I look at number of deaths. Nobody cares about number of tests: more tests is better. And you are at perfect liberty to downvote anything you like, and to be mad at me for slighting you personally. I see you backpedaling or redefining what you meant, and I'm tired of your behavior particularly as it touches on this subject.

You said this: "If we do a linear regression on the section from the CTP's data (3/3 to 3/26), we see the fraction of tests that were positive is declining" and this: "All that said, I don't care much if the right-hand side of the graph is more correct, and shows a gradual increase of covid-19+ people. Obviously the virus hasn't stopped spreading, or that line would be curving down."

You know perfectly well that the graph YOU made to emphasize your point is the percentage of TESTED people testing positive over time, on a population (of tested individuals) that is expanding exponentially, attempting to track a population of infected people that is said to be also expanding exponentially which is the whole point of all these desperate measures to 'flatten the curve'.

I'm angry with you because you are misinterpreting your own figures for some reason, possibly because the results of the situation offend your notions of how the world ought to be conducted. You are again at liberty to do whatever you like around that, but not in a vacuum and not without push-back, including push-back that questions your motivations for doing this. It's dishonest.

If the rate of increase stays the way it is, whether that's exponential or some other function, your line will be flat, not curving down. Repeat, you're graphing percentage of infected over tests and infected people spread the disease unless they are isolated. If spread stops happening and testing increases exponentially, negatives begin to outnumber positives. If the disease is also increasing exponentially, you see a flat line and your line stays the same, documenting the trajectory of all this, which also gives us information on how quickly hospitals will be overwhelmed: we already see death spikes in places like Italy, some of which may be coronavirus-related, some of which may be attributed to the collapse of the health care system.

You yourself say the right-hand side of your own graph (as tests proliferate so widely that they are no longer reserved for the very sick or very wealthy) shows a gradual increase, but it is not 'a gradual increase of covid-19+ people' and that is still dishonest.

It is a gradual increase in a metric that ought to be a flat line plotted on a logarithmic curve, documenting the worsening of a situation that is already expanding in exponential fashion. If we are looking at a normal pandemic of unprecedented destructiveness your line WOULD be flat. If the pandemic only reverts to exponential expansion, which is already overwhelming our health system and we ain't even BEGINNING to get started with the big numbers here, your line would not be 'gradually increasing'. If the testing is also expanding and the virus 'stopped spreading', your line would fall off a cliff, it would just drop. It's doing the opposite.

You made a whole blog post to state wrong conclusions that endanger people. Stop it.

5231041
I wasn't gonna say anything, but fuck it.

To add on to what Applejinx is saying, Bad Horse, it doesn't really matter what you were trying to say. What you have accomplished, with a clickbait title and some data that I really don't have the background or desire to double-check, is "This isn't as bad as people say it is and also everyone else in the world has done their math incorrectly."

Applejinx is correct. That is dangerous.

If you're wrong, and someone gets hurt as a result of what you've said, you've hurt that person. Think about what you're doing before you do it.

https://m.youtube.com/watch?v=S2ME0KXYUd4

I’m also concerned that large (~50%) chunk of identified patients are asymptomatic

5231016

You haven't really answered the question: you're making some extremely bold claims about these epidemiologists, and it isn't sufficient to imply that this is 'basic statistics and common sense'. Unless it's standard to do this, when calculating R0, I'm inclined to think the epidemiologists know what they're doing.

Furthermore you make this claim:

Bi et al. used a dataset which measured R directly via contact-based surveillance (testing all the contacts that a carrier has come into contact with). Their measurement of R (not R0) = 0.4 is the only estimate of R in this entire set of references that we can consider reliable.

But the authors go on to say:

In Shenzhen, SARS-CoV-2 transmission is most likely between very close contacts, such as individuals sharing a household. However, even in this group less than 1 in 6 contacts were infected; and, overall, we observed far less than one (0.4) onward transmission per primary case. As noted above, low transmission levels may in part be due to the impact of isolation and surveillance; but it is equally likely unobserved transmission is playing some rule(sic). We also estimate reasonably high rates of overdispersion in the number of cases each individual causes, leaving open the possibility that large COVID-19 clusters occur even if surveillance and isolation are forcing R below one; events that could potentially overwhelm the surveillance system.

This work has numerous limitations. As in any active outbreak response, the data were collected by multiple teams under protocols that, by necessity, changed as the situation developed. Hence, there may be noise and inconsistency in definitions. Of note, the definition of a confirmed case changed to require symptoms near the end of our analysis period (Feb. 7); but sensitivity analyses show that truncating the data at this point does not qualitatively impact results. It is, likewise, impossible to identify every potential contact an individual has, so contact tracing focuses on those close contacts most likely to be infected; hence our observed R is assuredly less than the true R in the population. Asymptomatic travellers will be missed by symptom-based surveillance; and, even if tested, some asymptomatic contacts may be missed due to the imperfect sensitivity of the PCR test.

In other words, the reported R0 in this paper is considered by the authors to be less than the actual R for COVID-19, and there's a lot of reasons for that, including tests not being perfectly accurate But it's also possible that testing and isolating everyone as much as possible, could be forcing the R down-- which incidentally is exactly what you want the management efforts to do. Which would seem to contradict the argument you appear to be trying to make.

Neece #42 · March 29th · · 1 ·

Italy has 1/6 of the US population and it's already at 10,000 deaths, with no signs of stopping soon. And they've been in lockdown for less than three weeks.

Sure, US has less worker rights than third world countries and an economic massively slanted to fuck the little guy and therefore said little guy is suffering horribly right now, but considering said little guy can't afford to get their parents/grandparents to the hospital for two weeks either any idea propagation AT ALL that tries to take this epidemic ANY less seriously it's downright criminal. Thousands of deaths that CAN be mitigated by an economic crisis that'll have profound effects on how we see the healthcare and wealth distribution around the world, and let's not forget that those same deaths have a bigger impact on the economy than their prevention too.

If you're testing massively more people you might get less positives simply because you reached the "saturation" point of you sample. If it's an urban center that's in lockdown, good, it shows the measure is working! If it's spread around the entire country and therefore including places that are naturally more isolated then your samples are completely schewed and you need more information than given to have any idea of how it's behaving.

No offense horse but straight out saying that epidemiologists around the entire world don't know how to do their jobs and that your analysis shows they're taking shit - and that's the message that your post passes, even if unintentional - is rather conceited and DANGEROUS. A single reader that reads this and decides that you're right, that they need not to worry, represents deaths in potential that could be avoided. This kills, and will kill even more when every single precisely free ICU is busy taking care of Covid patients. Don't try to mitigate what's happening, that's beyond irresponsible and may genuinely lead to deaths.

Maybe not of a granny either, but of a five year old girl that got in an accident but couldn't find a bed to recover because some asshole decided he was above the worries and got their parents into the beds too soon.

5231056 So, you don't have time to check what I've written, but you know it's dangerous. :trixieshiftright:

If there were no cost to shutting down the world's economy, then, sure, why not? Hell, we should shut it down permanently; nobody would have to work anymore.

But there is a cost to shutting down the economy. It might even cause more lives than it saves. Running around saying the sky is falling is not virtuous unless the sky is in fact falling. Telling somebody who's got strong evidence that the sky is not falling, that they should shut up about it, is also dangerous.

5231029

Can we agree that the number of deaths is a good measure to track the spread of the disease?

No. Nobody is testing dead people for the virus. People who die from the virus without being tested are not counted as deaths from the virus. Increasing the number of people tested exponentially will increase the number of deaths attributed to covid-19 exponentially even if the number of new infections per day stays constant.

5230950

Seems very on-point to me, and I've been trained to be highly skeptical of statistical arguments in general.

When I was in college I was given a choice between a course in statistics, which everybody took, or a course in elementary linear algebra, which everybody avoided because it was neither elementary nor linear nor algebra.

And I? I took the road less travelled by. Which I've always regretted because nobody's ever tried to lie to me with an eigenvector.

5231068
Asymptomatic patients may be a good thing. They're not coughing, so they're probably not very infectious (although there have been cases of transmission by asymptomatics, I've seen no data on their transmission rate). They slow the spread of the virus, by having immunity to it.

Bi et al. wrote,

Focusing on cases detected through contact-based surveillance adds nuance to previous characterizations of COVID-19. Since PCR testing of contacts is near universal, we can assume these cases are more reflective of the average SARS-CoV-2 infection than cases detected through symptomatic surveillance. In the contact-based surveillance group, any tendency for cases to be male or older (beyond the underlying population distribution, see Table S3) disappears. Further, in this group, 20% were asymptomatic at the time of first clinical assessment and nearly 30% did not have fever. This is consistent with a reasonably high rate of asymptomatic carriage, but less than suggested by some modeling studies, though PCR has imperfect sensitivity

TL;DR: Of the contacts of infected people, 20% of those who tested positive had no symptoms.

Li. et al wrote,

A total
1,075 confirmed cases were reported in Guangdong Province, in which 220 (20.5%) cases were secondary cases, 51 cases were identified with positive of 2019-nCoV but did not report any symptoms

That's 4.7% asymptomatic; a lower bound, because it includes mostly cases detected due to symptoms. If all 51 asymptomatic cases were secondary cases (found through contact-tracing, not by looking for symptoms), that would be 23% asymptomatic--an upper bound, because they didn't say how many of the asymptomatic cases were secondary cases.

Tang et al. estimated (via numeric optimization) the chance that an infected person will have no symptoms as 13%.

5231041

spread stops happening and testing increases exponentially, negatives begin to outnumber positives.

Yes, which I've admitted before. I NEVER SAID SPREAD HAD STOPPED HAPPENING.

If the disease is also increasing exponentially, you see a flat line and your line stays the same, documenting the trajectory of all this

No you don't, not if you're increasing your testing exponentially! Look at the graph I've now put at the top of the post. The number of tests given has a doubling time of 2.67 days, and the number of positives has a doubling time of 2.68 days. You can't get a flat line unless the virus is spreading at least that quickly, AND you manage to always test the same fraction of the people who actually have covid-19. That would be hard enough, but then you have to explain why the number of people presenting with covid-19 symptoms who do not in fact have covid-19 increases exponentially at the same rate.

It comes down to Bayesian reasoning. It is possible to see this data if the infection is spreading with a doubling time of 2.67 days, but that is possible only if the number of people tested is, coincidentally, increasing at exactly the same rate, and you've consistently been able to test the same fraction of all those who have the virus, and you get an exponentially-increasing number of volunteers coming in to be tested who have symptoms of covid-19 but don't have covid-19.

If we have actually succeeded in detecting all the initial cases, and in tracking and testing all of their contacts, and scaled up testing at exactly the same rate as the virus spread, and a mass hysteria of imagined covid-19 symptoms spreads at the same rate as the virus, then, yes, we could see the data I found. But each of those things is highly improbable. Whereas the hypothesis that we see this flat line in fraction testing positive is because the fraction of people in the population with the virus is roughly constant, requires no improbable assumptions.

The main difficulty with my analysis, I think, is that we know of only 2 people the US who were infected by 1/14, and that increased to 107,329 on 3/26. That's a doubling every 4.5 days. But even if that rate is still the same, that's a much more-manageable situation than people get using the parameters given in the studies of outbreaks.

I'm suggesting that the virus began spreading more rapidly, with a doubling time below 3.4 as in previous outbreaks, but slowed down dramatically after people began taking precautions, reaching a near-equilibrium level by March 5. Not implausible; that would still be much less-impressive than the speed with which the virus was halted in South Korea.

5231096

No. Nobody is testing dead people for the virus. People who die from the virus without being tested are not counted as deaths from the virus. Increasing the number of people tested exponentially will increase the number of deaths attributed to covid-19 exponentially even if the number of new cases per day stays constant.

Longtime Biden adviser posthumously tests positive for coronavirus: https://thehill.com/homenews/news/489907-longtime-biden-adviser-posthumously-tests-positive-for-coronavirus
Woman posthumously tests positive for COVID-19, records first virus death in Sequoyah County: https://www.swtimes.com/news/20200328/woman-posthumously-tests-positive-for-covid-19-records-first-virus-death-in-sequoyah-county
A Georgia healthcare worker was found dead in her home, and a posthumous test found she was infected with the new coronavirus: https://www.businessinsider.com/coronavirus-georgia-healthcare-worker-found-dead-in-her-home-2020-3

Posthumous testing has been going on since the beginning of the outbreak, for example, see this NPR article from Mar 3 (note that my deaths data go from Mar 11-Mar 28):

The newly revised tally includes two people who died in the Seattle area last week and have since been found to have been infected with the novel coronavirus, the health department in King County, Wash., announced Tuesday.

The posthumous diagnoses means the pair — a woman in her 80s and a man in his 50s — were likely the first in the U.S. to die from the disease. They both died on Feb. 26.

https://www.npr.org/sections/health-shots/2020/03/03/811690163/9-coronavirus-deaths-now-reported-in-washington-state

CDC recommendations on collection of postmortem specimens:

Collection of Postmortem Swab Specimens for COVID-19 Testing

For suspected COVID-19 cases, CDC recommends collecting and testing postmortem nasopharyngeal swabs (NP swabs) and if an autopsy is performed, lower respiratory specimens (lung swabs). If the diagnosis of COVID-19 was established before death, collection of these specimens for COVID-19 testing may not be necessary. Medical examiners, coroners, and pathologists should work with their local or state health department to determine capacity for testing postmortem swab specimens.

https://www.cdc.gov/coronavirus/2019-ncov/hcp/guidance-postmortem-specimens.html#SpecimenCollection

I don't know why you're trying so hard to encourage people to get themselves and their loved ones killed through negligence, but it's becoming clear to me that you will let nothing stop you.

Shame on you, and I am sorry I ever knew you, 'Bad Horse'.

edit, and your own words: "I'm suggesting that the virus began spreading more rapidly, with a doubling time below 3.4 as in previous outbreaks, but slowed down dramatically after people began taking precautions, reaching a near-equilibrium level by March 5". You are suggesting the pandemic, in the United States, has completely stopped doubling by March 5. How dare you? And then with the downthumb because you're so mad at how hurtful I am being!

Shame on you.

edit again: SOME of what caused me such fury, Horse has dialed back. That at least is good. I'm not retracting a word: people went to the trouble of downthumbing and all, I will leave up what they objected to. If it seems strident, trust me that Bad Horse was being the provocateur and gadfly with lives at stake, which I will not forget.

5230872 I'm guessing that your point 4 meant that the testing centers have been adjusting the number of people that they test to keep the fraction of positives constant. But if the fraction of the population with the virus is increasing, that would mean they keep decreasing the number of tests. See the chart I put up today near the top of the post, showing the number of tests per day in the US as reported by CTP.

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