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.

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.

COVID-19 March 22, 2020

The march toward 100,000 cases continues with excellent adherence to the exponential model. 23,640 was the total at the close of March 21, as reported by the individual state health departments. Both the short- and long-term trends predict that we will reach the shameful achievement of 100,000 about four days from now on the 26th. We only passed 10,000 a few days ago.

California and New York are now restricting testing to health workers and those that are hospitalized. This is likely to impact the total statistics, especially since New York currently has over 40% of cases. If the change in testing protocol results in a decrease of reports, it may be hard to detangle any actual decrease (i.e., flattening of the curve) from biased testing and reporting.

At some point, the curves will flatten out. They have to. Exponential growth cannot continue forever. If you take the exponential model I've been using and extrapolate those predictions forward in time, we will surpass 1 million cases in early April, 10 million about 8 days after that, 100 million yet another 8 days later, and 1 billion by early May. That exceeds the entire population of the United States. The exponential model will break down. When it does, there are slightly more complex mathematical models to represent the behavior of the data. These models are well known and understood and generally produce an S-shaped (sigmoid) curve. Once we start to deviate from the exponential curve, I may switch to those more complex models. The question is, when will that happen?

COVID-19 March 21, 2020

I'm a little late posting today's update. I took advantage of the self-imposed house arrest to sleep in.

New reported cases continue to closely follow the exponential model (first figure below). With testing continuing to ramp up, new cases have actually slightly exceeded the model predictions and have caused the curve to steepen, not flatten (second figure below). I do believe this is a bias due to increased testing and not an actual change in the true rate of transmission. Yesterday's numbers bring the short-term trend ever closer to the long-term trend, as discussed in previous updates. The doubling time is approximately 2.4 days now. Just the other day, I noted that we had smashed through the 10,000 mark. It took something like 22 days to reach that point, depending on exactly when you start counting. Some time today, we will almost certainly break 20,000. It took weeks to reach 10,000 and only two days to then double that number. The power of exponents!

A number of states have started to follow California's lead, filling the leadership vacuum at the federal level. New York is perhaps among the most important, as it accounts for approximatley 40% of all cases right now. Better late than never. Some other state leaders still don't get it. They don't think it's bad enough yet. They don't understand that they are one or two weeks away from being New York (see my post from yesterday). With New York, California, and few other states now on some sort of lockdown, the rate of growth in those places should start to slow, but it will take 10 to 14 days before we start to see the effect in the data (also see my post from yesterday). While the new restrictions should help, they may not be strict enough, as we've seen in Italy. Time will tell, but time is also the enemy.

COVID-19 March 20: Update and Some Soap-boxing

The final score from yesterday, March 19, was just shy of 12,000. Not only did the number of infections bust through the 10,000 mark, as predicted, but it did so decisively. This is not altogether surprising. The short-term trend was underpredicting reality because of inadequate testing. Now that the testing is starting to ramp up, the number of confirmed cases is increasing above the predictions that were based on the artificially low numbers. As seen in the graph below, the short-term trend continues to close in on the longer-term trend. Per yesterday's discussion, I expect the orange line to rise above the long-term blue line once testing really gets going. The simple exponential model still does a reasonable job of informing us of what to expect. The next stop is 100,000 cases, and that will be on or around March 26, one week from now. Of course, the large increase in cases over the last day did nothing to help flatten the curve. In the second graph below, you can see the slope not only getting steeper, but rising above both the short- and long-term trends. I'm now also showing the doubling time and decadaling time (increase by a factor of 10), which stand at 2.5 days and 8.5 days, respectively. As in previous posts, click on the images for hi-res.

Today, I'd like to focus on an important point that seems to be lost in the noise: Decisions being made today should be based on what the numbers will be two or more weeks from now. Today's numbers are only relevant in that they can provide a good estimate of what those future numbers will be. There is almost nothing we can do to change the numbers between now and two weeks from now. The cases that will show up over the next two week period are from people that are already infected or will be in the next day or two. The trajectory has been set. Our fate has been sealed. In two weeks, on April 3, we will be approaching at least 500,000 cases if you follow the short-term trend line, or nearly 1,000,000 based on the long-term trend. Given the increase in testing, the 1 million is likely an underestimate. There is nothing we can do about this. Previous decisions and lack of needed action were the recipe ingredients that baked the cake that will emerge in two weeks.

A huge problem is that the media, the public, and the politicians are far too focused on what happened between yesterday and today. How many new cases are there? How many more died? I have admittedly been part of this problem with my previous posts. The real story is that in two weeks we will have something like 1 million infected people. And, that's just the ones we will know about. The actual number is probably many factors higher, because not everyone is tested and some show little to no symptoms. Policy decisions are, unfortunately, being made based on current numbers. That's bad policy. "The numbers aren't bad enough to warrant a complete shutdown of the city or the state", say the mayor or governor. Wrong. Stop thinking about today's numbers, and base your decisions on what should be done if the numbers two weeks from now were staring you in the face, because they will be. And in two weeks, after having failed to grasp the idea of what time does to an exponential function, the same politicians will make the same mistake, failing to recognize that two weeks later yet again, the numbers will be another 100 times larger.

Governor Newsom of California is the first politician to quarantine an entire state. Not just any state, but the State of California. Perhaps he understands that his actions have a two-week delay attached to them. Unfortunately, the state-wide shelter in place probably should have happened two weeks prior. If it had, then the effects of that order would start to take effect today. Instead, we wait, helpless, for two weeks as people get sick and many die. Other leaders in every state need to follow Governor Newsom's lead. We can't wait for the federal government to do it; the person in charge is incapable of grasping the situation.

So, stop focusing on the numbers from yesterday, today, or tomorrow. Think about where we will be in two weeks, because we WILL be there and predictably so. Take actions today based on the future, not on the past and present. What should be done today knowing that in two weeks there will roughly 1 million known cases? What should be done knowing that there is nothing we can do to change the number two weeks from now? What should be done today knowing that hospitals will be well beyond capacity, that there will be insufficient ventilators, that the doctors and nurses we depend on to treat the sick will not have the personal protective equipment they need to stay healthy themselves, and that the people still working the basic services and infrastructure that we need to remain functioning begin to fail? Act today as if it is two weeks from now.

EDIT: I realize I was a little sloppy with my words and in the graph below. Actions taken today can have an immediate effect. But, we will not see that effect in the data for about two weeks. I shall strive to be more precise.

COVID-19 March 19, 2020

A couple of important changes with this update. First, going forward, I will provide an analysis using only the prior day's data. I've had to make this change, because the CDC releases info at noon (EDT) and those data are woefully incomplete and behind compared to the reporting from the individual state health departments. As soon as the day's numbers from the CDC hit the street, they are already obsolete. For example, yesterday's numbers ended up at 8,058 cases, whereas the CDC report from yesterday showed only 7,038.

Second, I am now using wikipedia as a data source, which in turn posts data directly from the state health departments. See here and here. These data are continually updated in near real-time, and are verifiable by going straight to the state health department websites that are given as references for the data.

So, today's update is really a reanalysis of yesterday, but using more accurate and complete data. There's no point in looking at what the CDC will say later today, because the data are still pouring in, and will be continuing to do so until late until the evening. I'll look at those final numbers and report on them tomorrow.

Below is the analysis based on data valid for yesterday (click on the image for hi-res).

As I indicated above, the actual number of cases (8,058) was far above what the CDC had reported at noon (7,038). Including this more accurate data drives the short-term trend line (orange) closer to the long-term trend (blue). I can't be certain, but I strongly suspect that effort to increase testing is starting to show. Still, the number of suspected cases far outweighs the current testing capabilities. As capabilities increase, I expect the short-term trend to approach and then actually surpass the long-term trend. It seems very likely that the slight dip over the last few days was due to systematic bias in testing, and not a true deceleration of the exponential growth.

The short-term trend line is predicting 10,129 cases by the end of the day. That is likely to be low compared to reality and indicates that the short-term trend is indeed suffering from systematic data bias. Using either the short- or long-term trend, the data predicts that we will hit 100,000 cases around March 26, plus or minus a day. This is 10x the current number. If things continue, we will hit 1,000,000 a week later. The 100,000 number is nearly a sure bet. There's nothing we can do about it because the people that will be diagnosed are already infected. If significant actions are taken, there's a chance that we can have an impact on the 1,000,000 two weeks from now. It will be interesting to see if the patchwork of half-assed local and state policies will show any effect; it's going to take more than a week to assess that.

Flattening the curve is a popular meme right now. Below is an analysis of the progress, or lack thereof, in doing so (click on image for hi-res). The graph shows the daily change in new cases. For algebra buffs, each data point shows the slope of the curve in the graph above for each day. For calculus buffs, it's the derivative of the curve above. If the number of cases was to remain the same as the day before, the line connecting two data points in the above graph would be horizontal. Horizontal lines have a slope of zero. Clearly, that's not the case. The line is not horizontal. It has a positive (and large) slope. Right now, the number of cases each day is increasing exponentially, and the slope of the line is getting larger and larger, which is why the line in the graph below is going up. In other words, the curve is not flattening at all. A flattening of the curve would mean that the data in the graph below would start to fall and head toward zero. A value of zero, however, indicates only that the number of cases isn't increasing or decreasing each day. Ultimately, we want the number of new cases each day to be lower than the day before. That would imply a negative value (below zero) for rate of change. We've got a loooonnnnggggg way to go before that happens.

The fate of all these curves depends exclusively on the collective actions of individuals. We must all do our part. Do not go out in public unless it is absolutely necessary. The CDC recommends no groups larger than 10. This is inadequate. Groups no larger than 1 should be avoided. In other words, no groups!

I continue to receive information from my doctor and nurse friends working at local hospitals. They are already running out of beds, supplies, personal protection equipment, and everything else that is necessary to treat the infected. We're only at 10,000. Next week it will be 10x worse, and the week after will be close to 100x worse. The system will likely have exceeded its breaking capacity. Nobody knows what will happen then. There is no plan. In Italy, patients were left to die. It's hard to imagine a different outcome here. Each of our individual actions can make a difference. DO IT.

EDIT:  CDC Website now says the following:
"In the event of a discrepancy between CDC cases and cases reported by state and local public health officials, data reported by states should be considered the most up to date."  The data above are from the state and local public health officials, as indicated.

COVID-19 March 18, 2020

CDC just put out their numbers based on reports from yesterday: 7038.

Today's graph hasn't changed much from yesterday. Note that yesterday's numbers have been adjusted slightly (upward) to account for reports that trickled in late. Today's 7038 will likely be adjusted slightly upward tomorrow.

Based on the orange trend line, we will come in just a little bit under 10,000 tomorrow. For reference, we crossed 1000 on the 11th. So, a 10-fold increase in one week. If the trend continues, we'll be at 100,000 at this time next week. The actual number of infections is probably 10 to 100 times larger.

Medical services are really starting to feel the pressure. I get this info directly from personal contacts I have working at the hospitals on the front lines. If that's what happens at 10,000, imagine next week at 100,000. PLEASE BE RESPONSIBLE and KEEP YOUR DISTANCE. Mortality rate from this is anywhere from 1-5% on average, and much higher for older and higher-risk persons. I'll let you do the math. (What is 5% of 10,000? What is 5% of 100,000? What is 5% of 1,000,000?)

All data except for today is coming from: https://en.wikipedia.org/wiki/2020_coronavirus_pandemic_in_the_United_States Today's data is from the CDC: cdc.gov/coronavirus/2019-ncov/cases-updates/cases-in-us.html

COVID-19 March 17, 2020

Today's numbers are in. I've updated the analysis a bit. I'm now showing all the data (blue) and the data for the last 10 days (orange). The trend line from the orange data will be more responsive to shorter-term changes. The good news is that the orange line is below the blue, although that could be a statistical aberration or the result of systematic issues with testing and reporting (likely). The bad news is that either line is bad news.