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.

COVID-19 March 26, 2020

The leveling off of new infections in New York and New Jersey appears robust, although it is still not possible to rule out testing and reporting issues.  Since these two states together account for approximately half of all cases in the U.S., a change in their behavior impacts the U.S. totals.  In the total case figure below I have added a third best-fit line that uses only the last three days.  It shows a shallower slope than both the lines for all the data and the 10-day fit.  It's a flattening of the curve, and if it holds, it's good news.  The daily rate of change figure also shows a departure from the 10-day and total data best-fit lines.  Still, the daily rate is large--over 10,000 new cases every day.

Statistics can be deceiving.  Because NY and NJ dominate the total statistics, they can mask trends in other states.  Places like Florida, Texas, and Louisianna have exploding numbers.  Even in Connecticut, the numbers are growing, no doubt due to their proximity to New York and the delay in issuing "stay at home" orders from the Governor. The slow roll out of orders in nearby Pennsylvania is also likely behind the increase in that state.  None of these states' numbers are anywhere near NY yet, but given exponential growth they can get there very quickly.  There are many other states with similar patterns.  If immediate steps are not taken in these other areas, it is likely that a second surge of new infections will emerge in the next week. This will drive the growth curves back up.  Time will tell.

In the meantime, it looks like the grim 100,000 total has been delayed by perhaps a day thanks to the leveling of numbers in the NY area.  It still appears nearly unavoidable that we will get there.  The flattening of the curve even a little bit, however, is incredibly important for how things unfold in the long-term.  Whereas the 10-day best fit puts us at 1 million cases in the first few days of April, the three-day best fit pushes that milestone out a week further or more.  That's hugely important for the medical and health facilities, many of which are already at the breaking point.

One last thought for the day.  Don't forget that the actual number of infections is perhaps up to 10 times what's being reported.  That means we may very well be approaching 1M infections in the next day, not 100,000.

COVID-19 March 25, 2020

Some good news? We're still closing in on the 100,000 mark in just a couple of days from now, BUT take a look a the second graph below--the daily rate of change. Yesterday's total showed a drop in the number of infections compared to the day before: 9,905 vs. 10,408. It's hard to say if that is real, which would be good news, or a change in testing, which would not be good news. A closer inspection of the numbers indicates that the drop is largely due to a drop in new cases for NY. The number of daily new cases in NY fell by almost 1,000. Are the social distancing and lockdown policies in NY showing? At the same time, other states are exploding. For example, MA went from 131 to 382. Is that due to increased testing or is it real? And then some states still seem wonky, as I indicated yesterday. Did LA really drop from rising numbers up to 335 the day before down to 216? Seems unlikely. So, the nationwide total could be good news, or it could just be bias in the measurements. We'll have to wait a few more days to see how things go. I do find it interesting that I haven't seen the drop in numbers in the press.

Nerdgasm alert for what follows:

There is no need to be a medical professional or epidemiologist to understand and then predict what is happening. It's simple mathematics that emerges in countless natural systems. If you tell me that the change in some quantity is proportional to the amount of that quantity, I can write down a simple ordinary differential equation (ODE), the solution to which has been known since Newton invented calculus: an exponential function. I could be totally ignorant of the origin of the data I've been plotting, but if you told me that the next number in a sequence of numbers was proportional the current number, I'd know exactly how to model that series without any knowledge whatsoever of the physics behind the numbers. In this case, the number of new infections is proportional to the number of existing infections, at least initially. Thus, it can be described by an exponential function. Similarly, the change in the intensity of light passing through an absorptive medium is proportional to the amount light (a.k.a. Beer's Law) and is described by the same exact ODE with the same exponential solution. It's just math.

Eventually, other things will begin to influence the growth rate besides the existing number of cases. For example, the number of people that have died could start to matter. Those people cannot pass on the infection nor get reinfected (because they're dead). For now, we can ignore them with only a small error, because they are a small fraction of the total population that is alive. But, that may not always be the case, and thus the error of using a pure exponential function will grow with time. And the number of people that have been infected but are fully or at least partially immune to reinfection can also come into play. And there can be subtleties, like having some fraction of the population be more susceptible than others or more likely to pass an infection than others. Or the virus could mutate and change its lethality, viability, or transmissibility. Each of these effects, and others, can be described by other ODEs, all of which then become a coupled system of equations. But, in the end, it's still just math, and there are analytical and numerical ways to solve those coupled equations. My area of expertise requires me to solve even more complex coupled partial differential equations (PDEs), but the techniques for solving ODEs are a simplified subset of the PDEs. It's all just math.

For the analyses I've been doing, I'm only comparing the data to the simple exponential model. The data fits. It works. When the data starts to diverge from an exponential function, as it eventually must, I can try to include the additional complexities that are making it do so, and then project those data forward (I *may* do that, but it's more time-intensive), or I can just note the degree to which the exponential model fails, which is still an informative metric. We aren't there yet. I hope the data will diverge from the exponential model sooner rather than later, and in the correct direction (downward). The sooner the simple exponential model breaks the better.

One of the flies in the ointment (there are many) is that the exponential model can appear to break when it shouldn't. That can happen when the input data is bad. If consistent reporting fails or testing protocols change that will cause a change in trends that is due entirely to how the data is obtained; the data is no longer purely a metric of how infections are changing over time. Unfortunately, this particular fly seems to be increasingly bathing in the ointment.

/Nerdgasm

COVID-19 March 24, 2020

More and more cities, counties, and states are (finally) starting to take action. I find it incredibly frustrating that it took so long. All data, for weeks, has pointed precisely to the current situation with incredible accuracy in the growth rate of infections. We will surely hit 100,000 on the 26th, as predicted weeks ago, for example. Boulder, CO where my workplace is, has now issued Stay at Home Orders. Denver has done the same. My city of residence (Longmont) has not and neither has the State of Colorado as a whole.

Some of the individual numbers coming in from the states are starting to look a little wonky. I suspect it's because of changes in testing protocol more than anything else, but I can't be sure. In Colorado, tests are now only being done for those admitted to the hospital. I know of several cases in my community that were likely COVID-19, but were not tested. Those cases, and surely the majority of others, are not making it into the daily roundup provided by the State Health Departments. One example of inconsistency is in Illinois. The number of new cases over the last number of days was: 134,163,174,296, and 230. There is no reason for yesterday's number to fall. The inconsistent measures taken in Illinois are too recent for them to have an effect. In California, the series of numbers for a longer number of days were 80, 126, 77, 331, 218, 268, and 265. Again, it's hard to see how this is a realistic reflection of the actual situation. The effect on the slope of yesterday's particularly odd tally is notable in the second figure below. Nevertheless, in aggregate across all 50 states, the noise in the individual numbers is swamped by the totals, which continue upward along the exponential curve.

I would expect the increased social distancing measures taken early on in some cities and states to start to show towards the middle to end of next week. Note that the total number of cases will be 1 million by April 3. Perhaps that date can get pushed out slightly, but it seems almost certain that we will cross that threshold. The measures currently in place are unlikely to reverse the trend, only to slow it; the curve is likely to start flattening a bit sometime next week. As I posted the other day, however, it will be hard to detangle what is real flattening and what is due to changes in testing protocol.

COVID-19 March 23, 2020

Nothing new to add today. Just the stark numbers with continued exponential growth. Still on track for 100,000 sometime on the 26th.

I've been taking the dog for a walk every day, and I make sure to maintain a distance of 50 to 100 feet from anyone that might also be walking around. On each day I've noticed young kids on the playground with their parents socializing at the table. That's not how this works. That's not how any of this works. It's why we are going to be stuck in our houses for a lot longer than we should be, and it's why these half-measures and gubernatorial proclamations are insufficient. The reality is that people are selfish and/or stupid. You shouldn't expect them to behave appropriately. Make them.