COVID-19 April 04, 2020

I'm going to make this short today because it's a beautiful spring day here in Colorado. I want to get outside, work in the garden, and go for a nice walk/run wearing a socially responsible mask. The usual charts are below.

 As a reminder, each day's data is valid for the end of the day before. So, today's report represents the final numbers from yesterday. Tomorrow will tell us how today ended, and so forth.

 Yesterday's total cases for the U.S. came in at 307,876. The doubling time is currently 4.8 days and decadaling time is 16.0 days. Both of these times have generally been increasing in length very slow each day. That's a good trend, but it would be nice if it were faster. At the current rates, we will hit 1,000,000 total cases in a little over 10 days.

 Total deaths came in at 8,360. At the current rate, we will hit 10,000 sometime today, or come very close. We'll see the reality of that in tomorrow's report. The doubling time is now 3.1 days and the decadaling time is 10.4 days. That would mean we will reach 100,000 deaths in less than two weeks from now if current trends continue.

COVID-19 April 03, 2020 and Colorado Special Edition

All data continues to proceed as it has for many days now.  The initial high exponential growth has given way to more moderate but still exponential growth.  Doubling time for infections is now at 4.5 and doubling time for deaths is 3.0 days.  The forecast for reach 10,000 dead will happen by the end of the day on the 5th.  The decadaling time for deaths is currently 10.2 days, so if the trend continues we will reach 100,000 dead a little over 10 days later.  The graphs are below.

I have taken a little time to look at using more advanced models to fit the data.  Why am I familiar with all this stuff?  At the age of about 13 years I received an Apple II clone computer on which I taught myself the BASIC programming language.  After mastering the language, one of the first "real" programs I wrote was a predator-prey model, and continued to add complexities to that model over time.  My first foray into numerical computing and modeling was in the area of population biology.  Eventually, that gave way to an interest in modeling of other natural systems, and I have been modeling fluids and atmospheres for the last several decades.

The next level of model complexity beyond the exponential model is arguably the Gompertz model .  There are three parameters that are set in that model: the initial population, the final population, and the growth rate.  I've applied this model to both the US and Colorado.  I've set the initial population of infections to the observed value at the start of recorded infections in the US or Colorado, as appropriate.  I then adjusted the growth rate and final population to obtain a good fit, as determined by eyeballing the Gompertz curve compared to the observed data.  There is a wide variation of parameters that give a reasonable fit to the US data, and that tells me that the data is not yet sufficiently constraining to obtain much quantitative information.  All that can be said is that the exponential model has broken down (which is already self-evident).  Where the US data goes is difficult to predict in the longer-term.  Short-term trends can be used for short-term predictions.

The spread of reasonable parameters for Colorado was more limited.  The Gompertz model compared to the data is shown in two figures below.  One uses the semi-log scale and the other uses linear scales.  The fit, judging by eye, is good, and here's what the model indicates.  The total number of infections in Colorado will hit approximately 70,000.  We will hit the following milestones on the following days: 10,000 on April 12; 20,000 on April 22; a maximum number of new daily cases (1,133 new daily cases totaling 26,476) on April 27, at which point the number of new cases will finally start to decrease each day; 30,000 on April 30; 40,000 on May 10; 60,000 on June 8.  The 70,000 cases are reached asymptotically, which is to say it takes an infinity amount of time to get there.  Note that the plots below only go through early May.

I want to emphasize this model is extremely uncertain, and I would not put great faith into the exact number or dates.  But, unlike the exponential model which grows unrealistically forever, the Gompertz model exhibits a realistic growth and then flattening.  I will update this model periodically as new data comes in and we'll see how it does.  As a very rough estimate, it does predict that the worst for Colorado is yet to come, but that by mid- to late-May, the worst will be behind us.

COVID-19 April 03, 2020

The slightly flattened curve continues with, perhaps, ever so slightly more flattening.  The 10-day orange trend has now clearly departed from the initial exponential rise (blue) and is currently indicating a 4.3 doubling rate.  At that rate we will reach 1,000,000 total cases on or about the 10th, one week from now.  The three day average (yellow dots/gray line), which captures most strongly the current data is just below the 10-day line.  That slight difference isn't great, but it can have a big difference as time goes on due to the nature of exponential functions.  I caution, however, that three days of data is probably not statistically significant, so extreme caution should be exercised in the interpretation.  If the three-day trend stays that way consistently for more days, the confidence of further flattening increases.  And, as I feel obligated to state almost every day, testing sucks!  As my colleague, Dr. Scott Anderson, who provided the Colorado data the other day, pointed out, tests are growing linearly in Colorado whereas cases are growing exponentially.  Testing is getting farther behind every day.

Total deaths continue their rise unabated.  The three-day trend in death rates shows some flattening, but as discussed above, I wouldn't get too excited yet.  Let's see if that sticks.  Let's hope that it sticks. Using the 10-day trend, we will hit 10,000 deaths in the next couple of days.  100,000 deaths will be a little over week later.

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.