COVID-19 April 02, 2020

Today's total infection rate graph, created from the tallies at the end of the day yesterday, continues to show a consistent trend with the new, slower rate of exponential growth that was established about 5 days ago.  I want to emphasize that these data are very likely influenced by poor reporting and testing.  For example, this recent article from CNN reports that there are up to 160,000 tests backlogged at a single lab.  I think it is likely that at least part of the flattening is real--the rate of new infections really has gone down.  Nearly 80% of the country is now under some kind of quarantine.  That is going to have an effect, and most of those policies have now been in effect long enough to show, even if there is a lag in testing and reporting.

What is troubling is that the daily rate of deaths, and therefore the total number of deaths, appears to be rising without much of a break in slope.  Note that the 10-day daily death rate in orange is just barely below the rate for all the data (blue).  In principle, deaths should be correlated with the number of cases, but with a time lag.  If the number of cases has slowed, why haven't deaths?  We'll have to keep an eye on total deaths and especially the death rate to see if it starts to flatten more.  If not, that suggests that the testing data is significantly biased and problematic.

Another concern is that with so much of the country already sequestered, why hasn't the rate of new infections begun to stabilize more quickly?  Or, stated another way, if we wish to reduce the rate of new infections further, it seems like it's going to take even more draconian measures to do so. Either that, or we are going to have to live with this semi-flattened curve for a loooonnnnnggggg time.  The curve is not going to flatten naturally until a significant portion of the population has been infected and become immune, or, possibly, until warmer weather begins to affect the stability of virions.  The latter idea is only a possibility.   No one is sure how the virus will respond under warmer conditions.

Finally, I'll point out that the population is not uniformly distributed across the country.  Much of the rapid growth we saw was due to New York.  Other cities could be next to experience this growth, and that could cause the curve to steepen again.  The social distancing policies that are in place across much of the country should help mitigate this risk, but it is still there.  Florida only recently put in place relatively weak orders that are going to take at least 10 days to have an effect.  The major cities in Florida could spike dramatically within that period of time and set the US curve as a whole along a steeper path in the same way that New York did.

COVID-19 April 01, 2020

Assuming the states' health department data are accurate, a new, slower exponential growth rate of new cases has been established over the last four or five days.  The doubling time is now approximately 5 days with a decadaling time of over 16 days.  Whereas we would have hit 1,000,000 total cases by the end of this week at the prior rate, that has now been pushed off well into next week.  But, it is still coming.  The curve has flattened. It has not rolled over.

There was an upward blip in new cases in yesterday's numbers.  We'll have to keep a watch on that.  It could be just a statistical aberration associated with poor testing.  It was clear that several states had a bunch of new test results dumped on them yesterday.  Georgia went from 126 to 1028 cases.  Louisianna jumped from 485 to 1212.  At the same time, the State of Washington has failed to report anything for the last two days.  Their data server has crashed due to the large volume. Since they are a hot spot, those numbers could be relatively important.

The number of deaths continues to grow exponentially with little change.  Note that the 10 day best fit matches the long-term almost exactly.  The daily death rate is close 1,000 and will hit that mark in a day or two.

COVID-19 March 31, 2020

If you take the numbers at face value, there has been a definite and unarguable flattening of the curve for the number of cases.  The total number of deaths does not reflect this flattening, but it looks to be starting even in the death rate data. I would caution taking all the numbers at face value, however.  There are still obvious issues with testing and reporting, and it's quite possible these are affecting the U.S. aggregated data.  It is certainly affecting some states, as the data are contrary to the behavior of a natural system.  The slowing death rate is a good thing, since the dead are almost always reported, and those data are likely to be more robust than the data for the number of infections.

The usual plots of total cases, daily change in cases, death, and death rate are below.  After that, I provide a quick analysis of the ongoing "New York Effect" and introduce a plot generated by a colleague for Colorado.

In a previous post, I introduced the "New York Effect", which was the result of the New York cases being such a large fraction of the U.S. total.  As went New York so went the U.S.   The fraction of total cases coming from New York has dropped with time.  In the graph below, I plot the whole U.S., New York, and the U.S. after removing New York cases.

The data (diamonds) do not trace out straight lines; the data is flattening over time.  The dashed lines show a 4th order polynomial fit to the data.  Note that New York (orange) is falling away faster than the rest of the data.  The rest of the U.S. (gray) is flattening, but not as quickly.  As the fraction of New York cases continues to decrease, all of the U.S. (blue) will be less and less influenced by the numbers in New York.  

The figure below for Colorado was provided courtesy of my colleague Dr. Scott Anderson and used with his permission.  It shows hints that the testing in Colorado is falling short.  Over the last few days, note that the total number of tests conducted to date is falling away from the best fit curve. If that trend continues, it suggests that the testing cannot keep up with the demand.  The number of positives is also falling away, which is exactly what you'd expect if there are insufficient numbers of tests being given.  It is also possible that the demand has simply decreased, but that doesn't seem like a good explanation.  All indications are that there is far more demand than available tests.  The number of hospitalized continues to rise with a doubling time of about 2.7 days.  This is a lagging indicator of initial infection since an individual must go through the incubation period and then become sick enough to be hospitalized.  In very rough numbers, it is also clear that each of the lines is separated by about a factor of 10.  So, of all those tested, about 10% end up positive.  Of those, about 10% are hospitalized.  And of those, about 10% die.  Again, these are very rough numbers.  Those rough numbers suggest a mortality rate on the order of a few percent for those known to be positive, which is consistent with global calculations. 

COVID-19 March 30, 2020

The numbers and graphs for today look promising, but I am very skeptical that the numbers accurately reflect reality.  The number of cases shows a distinct flattening and even the death rate, which is a lagging indicator, is falling below the trend line.

The reason for my skepticism is that there are an increasing number of states with numbers that, if correct, indicate a very non-standard behavior of infection growth.  I'll discuss a few examples.

Missouri
The new case numbers over the last number of days were 72, 101, 146, 168, 168, 65.  For the first four days, the numbers were increasing as would be expected.  The day before yesterday, they level off with no increase in the rate of new cases.  And then yesterday, the fall to 65.  The leveling off is perhaps plausible, although unlikely to happen so quickly.  The fall to 65 is bizarre.

Georgia
The numbers here are 172, 254, 221, 278, 476, 445, 237.  The large ups and downs and especially the drop to 237 yesterday are not consistent with any model of infection growth that I'm aware of.  Maybe everyone decided to go to church yesterday instead of the emergency room.

Louisianna
The numbers are similarly odd to Georgia: 335, 216, 407, 510, 441, 569, 225.

Washington
248, 111, 627, 493, 610, 586.  What's with the drop to 111 and then the sudden increase to the largest value in the series of 627 followed by a large drop the next day?

Massachusetts
679, 579, 823, 1017, 698.  Is the trend up, down, or sideways?

There are also some states where the number of cases has jumped out at an extreme rate in addition to odd bouncing around.  Here are just a couple of examples.

Connecticut
203, 257, 137, 279, 233, 469.  This state bounces around in odd ways similar to others but ends with a large increase instead of a decrease.

California
369, 433, 471, 795, 842, 1657.  At least all these numbers are going in the same direction, but did the rate of new cases suddenly go to a doubling every day as of yesterday?

What all the examples above indicate, and the many more I didn't highlight, is that there are highly inconsistent and problematic reporting issues.  This makes it very hard to accept the validity of the numbers and trends.  Even if the day to day noise in individual states is removed by aggregation, it's not at all clear that the total numbers across the U.S. are valid, or if they are artificially low (or, less likely, high) due to inadequate testing protocols.

I'll continue to track the numbers, but I have less and less confidence in their quantitative value.  The total number of new cases each day are still increasing while testing capabilities continue to lag.  This should mean that a larger fraction of cases go undetected or untested each day.  That would lead to a systematic decrease in reported cases that does not reflect the reality of the situation.

COVID-19 March 29, 2020 and a Deeper Analysis of the "New York Effect"

I'm posting this early.  The numbers are mostly in for the day, and I may sleep in a bit tomorrow.  I'll update figures and data as needed tomorrow when I get around to it, but it should be pretty minor.

The pandemic marches on, largely as expected.  The graphs of total cumulative cases, daily case rate, cumulative deaths, and daily death rate are below.  Be aware that individual state numbers are very suspect from day to day.  They go up as expected, then suddenly drop one day, and then increase again the next day.  That is not how nature behaves.  That's how a dysfunctional testing and reporting system behaves.  I'm hoping that the aggregated numbers filter out the noise and that looking at states over the period of several days or more smooths out the idiosyncrasies.  I don't know that this can continue as the number of cases increases exponentially and the testing cannot keep up. 

While stuck at home over this weekend with all the time in the world, I thought I would take a deeper look into the impact that NewYork has had on the total numbers for the U.S.  It's a lengthy post.  Dive in if you have time.

New York had accounted for 20% or less of the total cases prior to about March 16 (day 75), as shown in the figure below.  Then, over about a 5 day period it surged to nearly 50% of the total cases, before starting to decrease again.

There are at least a few possible reasons for the rapid rise in the percentage of total U.S. cases:
(1) There was a particularly contagious strain of the virus that appeared or evolved in New York.
(2) Areas outside of New York were controlling the infection better than within New York.
(3) There were social patterns in New York that resulted in a greater infection rate.

The data provided at NextStrain would seem to rule out the first hypothesis.  Assuming the NextStrain data is accurate, the U.S. is dominated by a single strain of virus.  Furthermore, the virus is mutating slowly with very few nucleotide differences between the different strains.

Map courtesy of NextStrain.org.  Each color represents a different strain of the COVID-19 virus. Variant ORF1b in red dominates the U.S. Map downloaded March 28, 2020 (17:46 MDT).

The second hypothesis is also unlikely.  New York was one of the first locations to implement increasingly strict containment policies.  Or, conversely, it's safe to say that most of the rest of the country was doing little to nothing.

That leaves hypothesis #3 (or other possibilities not considered).  This hypothesis seems the likely culprit, and it's a lesson in why social distancing is critical to stemming the spread.

It turns out that the patient zero for the outbreak is likely a lawyer in New Rochelle, NY.  That person was hospitalized in early March (around the 5th, as best as I can tell). Here's where it gets interesting.  The attorney practices in a law firm in New York City and also happens to practice orthodox Judaism.  New York City is, of course, an incredibly dense city where each person is likely to come into contact with far more people than a typical American.  Equally important, however, is the Jewish orthodox culture.  Having grown up in the Eastern suburbs of Cleveland where there is a large population of orthodox and ultra-orthodox (Hasidic) Jews, I am reasonably familiar with the religious sect.

Both religious sects go temple at least daily, often several times per day.  It is a community practice and requires that a quorom of at least 10 men are present.   This is in addition to other community gatherings.  The culture is fairly insular, although still open to society, and children are often schooled in private religious schools (yeshivas) where there's a focus on the study of the torah and talmud.  The attorney's son attended a yeshiva in Manhatten.  The school had a sizeable boarding population.

Given the above, it should be no surprise that there was an explosion of cases in New Rochelle within two weeks after patient zero showed symptoms.  By March 10, the Governor ordered a 1 mile containment zone centered around the synagogue of which the attorney was a member.  Services were canceled and schools were closed.  But, by the time that happened the fate of New York was sealed.  In less than a week, those infected prior to the closures became symptomatic, and the orthodox culture and the density of New York City meant that the number of people infected by any given person was higher than on average across the U.S.  On March 17, the fraction of total cases from New York began to rise, reaching almost 50% in a matter of days.  The peak fraction was reached on March 23, about two weeks after the lockdown.  This is consistent with the 5 day average incubation period and additional testing and reporting delays.  The fraction has since started to decline as the rate of new infections has started to decline and the number of cases outside New York continued to increase.

The explosion of cases was likely the result of two things: (1) The high number of interactions of people in a dense city like New York, and (2) The high interconnectivity of the orthodox community.  Together, it was a time bomb for a highly infectious disease like COVID-19.  1 person was the source of a large fraction of the 45,000+ infections in the state, all within a little over two weeks. One person! Social distancing and the eventual lockdown partially mitigated both of the key factors in the rapid spread.

New Rochelle is not unique.  There have been numerous reports of groups of people ignoring quarantine, often claiming a religious exemption.   The prior rapid spread in S. Korea was traced to a religious group.  I'll refrain from digressing to theodicy and the reconciliation of evil and suffering with a god(s), and instead refer you to Epicurus' trilemma, who pretty much had it figured out millennia ago.  Hume, killed it even more deader.  Philosophy, reason, logic, and mathematics are not highly valued in the U.S.  Anti-intellectualism is instead the prime virtue.  Science, facts, data, and the use of such is for elites. And so we are almost certainly doomed to see New Rochelle repeated throughout the U.S. in the coming weeks, months, and perhaps years.

Below is a graph of the cases in NY and the U.S. plotted in semi-log format, as I have been doing since my first post.  Recall that the slope of the line is indicative of the exponential growth rate.  There are three distinct regions in the graph that I've highlighted with shading and lines that parallel the data.

The first yellow region represents a "normal" exponential growth rate.  During this early time, the rate of growth in New York (orange) data was roughly the same as the U.S. as a whole; the orange and blue lines are parallel (they have nearly the same slope).  In the middle region, the slope of the New York data increases dramatically.  The increase mirrors the rapid growth in the fraction of cases in New York previously presented.  As the fraction of cases from New York became a larger fraction of U.S. cases, the slope of the whole U.S. curve began to change.  Beginning around day 80 (March 21), the growth rate for New York and the rest of the U.S. returned to a more "normal" growth rate.  This reflects a decrease in the fraction of cases from New York and the overall decrease in exponential growth rate in New York.  Note, however, that the slope is still positive.  New York and the whole U.S. is still in an exponential growth phase.

I've created one last plot to illustrate what happened in New York and its effect on the U.S. statistics.  Below is the same data plotted above, but without the use of a log axis.  In linear axis coordinates the exponential growth is obvious.

If you follow the best-fit orange curve for New York you'll find that around day 75 (March 16) the data (orange dots) began to rise above the best-fit curve.  This means that the number of cases was increasing faster than the historical exponential rate.  The deviation of the data from the curve reach a peak around day 82 (March 23).  By day 84, the data actually fell below the best fit curve.  It's still growing exponentially, as shown in the standard semi-log plot.  It is in fact, growing at a rate similar to the U.S. as a whole, which is to say that the orange and blue curves are more or less going up in parallel (they have the same slope after all).  The curve in New York has flattened a bit compared to where it was a week ago, but only back to a more "normal" growth rate.  At the same time, the U.S. data was pulled slightly above the best fit curve starting around day 75, because the numbers were becoming increasingly dominated by the large number of cases in New York.  The U.S. has now returned to more "normal" growth.   If we remove the effect of New York, we basically find that the infections in the U.S. have been growing at a nearly constant exponential rate with little to no evidence of flattening.  The flattening seen in the total data is really just a reflection of the "New York Effect."

In a previous post, I noted that the predicted day that the U.S. would reach 100,000 cases was, for at least two weeks, March 26.  But that prediction was increasingly influenced by the data in New York, and once the quarantine policies began to take hold, the exponential growth rate flattened enough to push the actual day to March 27.

Yet another lesson of this lengthy post is that it is not sufficient to just look at graphs of aggregated U.S. data.  You do have to understand, at least a little bit, what's behind the numbers.  For a good period of time, the collective U.S. data we've been looking at has been strongly and disproportionately influenced by what's been going on in New York.  That means that the data was not necessarily accurately reflecting what was going on in the country as a whole.  New York still has a substantial fraction of total cases, so that effect has not completely diminished.

And a final lesson of this lengthy post is to realize that with respect to this pandemic, the U.S. is not one country.  We are in many ways a collection of 50 states, plus territories, that act and behave semi-autonomously, particularly in the complete and utter absence of federal coordination.  The trends in the U.S. will almost certainly be traced to just a handful of locations and regions that will unduly dominate the statistics at any given time.


EDIT: An earlier version of this post indicated that patient zero was affiliated with the Hasidic sect of Judaism. That was incorrect, and I regret the error. The post has been corrected to reflect his affiliation with the orthodox sect. Thank you to those who noted the error.

COVID-19 March 28, 2020

Yesterday, March 27th, we did pass the 100,000 cumulative case mark.  Although the exponential models had been indicating for nearly two weeks that it should have happened a day earlier, changes in the trend just a couple of days prior shifted the event by one day.  It doesn't seem like much of a difference, but it is a big deal.  See the divergence of the best-fit curves in the figure below?  The lower, yellow curve is the trend from the last three days. The orange is 10 days.  The blue is all the data.  Follow each of those curves into the future and note when they predict we will hit 1,000,000.  The 3-day curve indicates April 8.  The 10-day is April 6. The full data predicts April 5.  That's a four to five day difference, and that's a lot considering April is just days away.  Let's hope the short-term trend holds and slows even more.  The curve is flattening, albeit slowly.  The daily rate of change curve shows the same.

Unlike the cumulative cases, the death rate is accelerating. The short-term curve is slightly above the long-term trend.  This is because it is a lagging indicator, as discussed yesterday.  In principle, we won't see a flattening of this curve for at least 10 days (and likely more) beyond where the cumulative cases began to flatten.  Thus, if the exponential model with the expected lag holds, we should reach 10,000 deaths unimpeded on or around April 5, a little over one week from now.

COVID-19 March 27, 2020

The trends for today are interesting and complicated.  Yesterday, I noted that the short-term trend had flattened a bit and that the 100,000 case mark would be pushed off by a day.  For almost two weeks, the exponential model predicted 100,000 by the end of the day on March 26.  That changed a few days ago, and I suggested that was due to a decrease in new cases in New York, which accounted for nearly half the total cases.  New infections in New York have flattened out a bit.  They are still growing, but not at the same rate as before.  Besides the flatter short-term best-fit line (first figure below), the influence of New York is most obvious in the daily rate of change (second figure).   Note that last three days in the daily rate of change curve clearly fall below the long-term trend. 

I also noted yesterday that other states were exploding, and that they could cause a secondary increase in infections.  This appears to be happening.  New Jersey, which I thought had also stabilized based on data from prior days went from 727 cases to 2,474.  This almost certainly has to be testing issues, but it may also indicate continued uncontrolled growth.  Perhaps more concerning is the rapid increase in mostly southern states where the first line of defense appears to be prayer rather than science.  AL went from 68 to 197.  MS went from 57 to 108.  MO went from 101 to 146.  And so on.

In the daily rate of change graph, the rate of change is once again on a fairly steep upward climb.  There was an offset three days prior, which brought a temporary one day reprieve.  States outside of NY are now undergoing explosive growth.

Unfortunately, the total number of dead is now large enough to begin tracking.  Today, I introduce the total death charge and the daily death rate.  Unlike infections which have testing issues, the dead are always counted.  We can't be entirely sure that we still aren't missing some deaths that were not properly associated with COVID-19, but the data should be overall more reliable. 

The downside of tracking deaths is that it is a lagging indicator.  People that die from COVID-19 were infected a week or more ago.  The average incubation period prior to symptom onset is about 5 days.  Then it takes three or five (or more) days until the victim succumbs to the disease.  So, the death numbers on a given day are a measure of reality something like 10 days ago.  The total deaths as of yesterday, March 26, was 1147.  That's more representative of the population that was infected on March 16 and that began to show symptoms around March 21.   If we go back to March 16, we find there were only 4,672 total cases.  On March 21 there were 23,710. We are now at 100,000 total cases.  The reality of today's situation will not be reflected in the deaths for many days to come. I'll also note that the 10-day exponential fit for total deaths is exceeding the long-term fit.