COVID-19 April 23, 2020

I will be updating the Gompertz model projections at the end of the day after today's numbers come in.  Based on what I see so far, I don't expect there to be substantial changes.  The Gompertz curve is tracking the data rather well with the greatest uncertainty coming from the large weekly oscillation in daily new case and death reports.

I've seen media reports over the last couple of weeks that the IHME model, which is widely used by the government and policymakers, revised their death toll down to 60,000 by August.  I'm usually loathed to criticize other modelers, especially when they are almost certainly experts, whereas I am not.  But, I don't see how the IHME has any basis in reality.  I think it's fair to criticize a weather forecaster that predicts warm and sunny skies when a simple look out the window shows that it's pouring rain.  And I say this as a weather forecaster. We are now at over 42,000 deaths in the US.  Just 18,000 short of 60,000.  And we aren't even through April.  The daily death rate is over 2,000 per day.  Even if it drops to half that, we will hit 60,000 in May, not August.  A lot can change by August.  I don't know what the numbers will be three months from now, especially as social distancing policies change.  One thing I am nearly certain of, however, is that the death toll is going to be over 60,000 within weeks, not months.

For clarity, I want to distinguish the modeling I'm doing from what most of the other experts are doing. (Again, I am not an expert modeling infectious diseases).  I am simply fitting a mathematical function to the historical data and using that to extrapolate out into the future.  I am not trying to explicitly simulate the rate of spread or how contagious the disease is (i.e., the R0 factor).  The advantage to this method is that it relies upon the underlying data itself with the implicit assumption that the infection spread is governed by the laws of nature, that the future is a function of the past, and that the laws of nature follow some smooth mathematical function. I have chosen to use the Gompertz curve, because it has been shown to empirically reproduce the historical rise and fall of many biological populations.  The Gompertz curve is a solution to a very simple population biology model.  Thus my "modeling" implicitly rather than explicitly incorporates a very simple numerical model.  But I don't really care about that.  From my perspective, I'm just fitting a curve to the data without care of WHY that is happening.  It just is.  And for now, it seems to be doing a rather good job.


In contrast, most of the models discussed in the media and which are used by government officials are solutions to a bunch of complex mathematical equations that are explicitly trying to simulate the growth and decay of infection based upon quantitative measures of infection.  These models require specific inputs, many of which are poorly constrained.  For example, the R0 is a key parameter and the solutions are very dependent on small changes to the poorly constrained value.  The models also require knowledge of when an infection started, which even now is uncertain by as much as a month or more.  Population density and a measure of the interaction between people can also be important in these models.  The advantage of these models is that you can plug in a range of values and see what happens. You can understand how sensitive the system is to changes in R0 or other parameters. You can UNDERSTAND how the system behaves.  These models answer a lot more of the "why", which I completely ignore.  Most of these models should work very well if the input values are reasonably constrained.  But they aren't, and thus there is a very wide range of outcomes predicted between models and within individual models.

COVID-19 April 22, 2020

I'm posting today's numbers tonight rather than waiting until the morning as usual. This good doctor has a busy day tomorrow, so I want to clear this off my to-do list tonight. Be prepared tomorrow to hear the press talk about big jumps in deaths and infections over the last day. If you've been following my posts, you'll know that this is not reality. Instead, it's the usual weekly oscillation in numbers. If you couldn't see it in prior data, it's plain as day now. Take a look below at the daily new cases or the daily deaths. Count the blue dots and note how cleanly they rise and fall over a 7-day period. Up and down like clockwork. Today was the predictable big jump up, just like in previous weeks. This is almost certainly the catch-up bookkeeping of weekend numbers.

The Gompertz model fits continue to look pretty good. I'll provide an update toward the end of the week, but based on what I'm seeing so far, I don't expect any major changes. By all accounts, we are past the peak of new daily infections. This means that, on average, there will be fewer infections each day. The overall number of cases is still going up, however. But, we are no longer in exponential growth. For now. The loosening of restrictions in some states could change that. And, don't forget that all of these statistics are strongly weighted to the New York metropolitan area (including New Jersey), where numbers are falling nicely.

Nearly the same story is found in the death rates and total deaths, but deaths lag infections. We should see a peak in the death rate tomorrow or thereabouts. After that, we should start seeing, on average, fewer deaths each day. This assumes that the loosening of restrictions doesn't send us back in the wrong direction. That scenario is possible.

COVID-19 April 21, 2020

Yesterday's numbers remain more or less in line with expectations.  As I mentioned yesterday, I will only update predictions roughly once per week because of the noise and oscillations in the daily data make it pointless to try and adjust the predictions more frequently.

Many states are starting to loosen restrictions to varying degrees.  It will be interesting to see how that affects the data going forward.  With a mean incubation period of about 5 days, delays in testing, and the lag between symptoms and hospital admissions, it may take a while before we know the impact.  Of course, just like the light from distant stars, the data we see on a given day is actually the reality as it was some time in the past.  I'm still not sure politicians and policymakers fully understand this.

COVID-19 April 20, 2020

The usual weekend data aberration was in full effect this weekend.  Both new daily cases and daily deaths took a dip.  The daily deaths, in particular, are starting to show a very, very large oscillation around the mean.  Over the last five days, there was a low yesterday of 1631 and a high of 2438 on April 15.  It was 1462 just two days prior to that.  So, there's an oscillation of about +/- 500 around a mean of about 2000, or 25%. That's huge, and it makes it very hard to pin down a curve that best fits the data. The variations in new daily cases are also similar, ranging from the mid-20,000s to the mid-30,000s.

Due to the very large oscillations, I will only be adjusting the best-fit curves roughly weekly.  If I try to adjust them daily, I will end up chasing the oscillating data up and down. For now, the curves fitting the cumulative statistics for total cases and total deaths are reasonably close to the historical data.

Some states, like Florida, are starting to reduce restrictions on social isolation.  It will be interesting to see how this affects the data.  At present New York and New Jersey still account for something like 40% of the numbers across the US.  These two states disproportionately affect the statistics, and the trend from those two stats has been overwhelmingly downward.  If states like Florida began to see an increase in cases and deaths, they could start to bend the curve back upward.  If that happens, it will invalidate the use of a Gompertz model, which assumes a relatively uniform infection process across the population.

It may very well be that to model the US it is necessary to model each state individually and then add the results together to obtain the solution for the country as a whole.  This is far too time consuming for me to undertake, but even if I was able the uncertainty in the models would likely increase beyond utility.  The certainty of statistical calculations often strongly depends on the number of data in the calculation.  As I already showed, the oscillations in the data aggregated across the US are already so large that the curve fitting is becoming problematic.  These oscillations are often worse at the state level and the number of data in each state is much smaller than the country as a whole.

COVID-19 April 17, 2020

Keepin' it short and sweet today.  Data from yesterday is plotted in the figures below.  I'm using only a linear scale today.  The Gompertz model continues to do a good job of representing the data.  The ups and downs of new cases and daily deaths continue their fairly repeatable pattern.  Yesterday was an up day.  I will likely not provide a report over the weekend unless something unexpected appears.  I do expect the weekend numbers to fall, followed by a rebound next week. 

COVID-19 April 16, 2020

Good morning from snowy Colorado.  Another fresh foot on the ground.  Normally, it would have likely been a snow day for the schools.

The adjustments I made to the model curves yesterday seem to be consistent with the data that came in yesterday.  Today, I made an additional very small adjustment that shifted things inconsequentially.  The curve fit is now very good.  Excellent, actually.  In the figure below, I have plotted the Predicted number of cases from the Gompertz curve vs. the actual number observed.  It's a beautiful straight line with a slope of 1.01, which is to say that the predictions are off, on average, by about 1%.  A perfect model would have a slope of 1.0.  The correlation is actually much higher: 99.99%.

The rest of the standard plots are shown below with a mix of linear and log axes.  I will call your attention to the daily case numbers and daily death numbers. The weekly oscillations have grown rather larger with a minimum around the weekend and a maximum around mid-week.  This makes curve fitting a challenge because rather than trying to get the model to reproduce the data, the goal is shifted to trying to match the average of the data.  As you can see, the model more or less goes right down the middle. The weekend data is too low, but yesterday's data pops up above the model.  There's little doubt that yesterday's data includes backlogged data from over the weekend.  I expect today's data to be a bit above the model, and then I expect things will start to fall again over the weekend.

COVID-19 April 15, 2020

The US numbers continue to be problematic from the standpoint of statistical analysis.  The expected weekend dip and subsequent rebound appeared, as predicted.  The magnitude of both the dip and rebound varies from week to week, and the varying data makes it difficult to determine the best-fit curve.

In the first panel below, the number of new daily cases, plotted on a linear axis shows the weekend alterations. The weekend before last there was a large, single-day drop followed by four days of upward growth.  This last weekend, there was a more continuous drop followed by a smaller rebound yesterday.

If I try to fit a curve to all the daily new case data (the blue dots), I find that there is no good solution.  In particular, the early data has a different behavior than the latter data.  This is seen most clearly in the second figure, which has the data plotted on a log scale.  The data in the shaded box has a different trend than the latter data.  I have decided to exclude these data from the curve fit and use only the latter data.  When I do that, I find the Gompertz curve shown by the red crosses.  Using this new fit generates different estimates of peaks and total cases than previously speculated.  The peak is now estimated at 32,300 cases, which already happened on April 7.  The analysis suggests we are on the downward slope.  Note that the data actually exceeded this estimate on several occasions, which is the nature of noisy data compared to a smooth statistical curve.

A similar issue was found when looking at the total, cumulative number of cases as shown in the next two figures.  Again, the early data shown in the shaded box has a different trend than the latter data. This should not be surprising since the it is the sum of daily cases that comprises the cumulative total.  Since the daily data has this behavior the cumulative sum shows it as well.  With the new curve fit, the total number of infections has been cut nearly in half compared to the previous estimate: 1.1 million.  On April 12, the halfway point of 550,000 infections was reached.

The cumulative deaths and daily death rate curves underwent a smaller adjustment.  Note, however, that the data has similar weekend noise problems, as specifically noted on the daily death rate chart.  The total projected deaths at this point is 120,000.  The halfway point of 60,000 is still ahead of us on April 29.  This is tragic, but compared to the trends of data a few weeks ago, the numbers are substantially lower.  Social distancing has saved lives.  Quite likely hundreds of thousands of lives and perhaps a million have been saved. 

It may very well be that our large, diverse, and patchworked-policy country cannot be modeled as a single unit.  An alternate and far more complex and intensive effort of modeling regions, states, and cities independently could produce better results.  On the other hand, the data from some states are extremely noisy, not just on the weekends.  One of the lessons in all of this is that few if any of the State Health Departments were ready to archive and distribute data.  This goes hand in hand with the lack of testing.  This same criticism goes ten-fold at the federal level.  My impression is that the states are making tremendous progress while the federal government continues to flounder.  I'm quite sure everyone is working overtime to do the best they can.  This is not a failure of people, but a failure of leadership.  Something to remember at the next election.

UPDATE:

The following unconfirmed story suggests that testing has suddenly dropped by up to 30%.  If true, this might (emphasis on might) explain why I am no longer able to fit a single curve to the data.  If the recent data has suddenly suffered from a 30% drop, the apparent stabilization and slight decrease in cases may be fictitious.
https://www.politico.com/news/2020/04/14/coronavirus-testing-delays-186883