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

COVID-19 April 14, 2020

Today should be a very interesting day.  Will the low weekend numbers bounce back as they have in the past? If so, by how much? Today's numbers should be highly indicative of when the "peak" will occur.  If the numbers today and tomorrow don't give a substantial upward bounce, I will have to readjust the model curves (and that's a good thing). And so we wait!

COVID-19 April 13, 2020

Nothing all that new today.  The weekend number bias was in full force, as you can see in the data below.  The daily death rate chart and the daily new cases chart really show it well.  I suspect we'll see things increase over the next few days. 

I have tweaked the Gompertz parameters a bit.  I'll wait a few more days before I report on that in detail.  I'm currently finding that we will have a peak in daily deaths of 3420 per day around April 27.  A peak in new daily infections will occur around April 18 at 38,623.  As usual, these numbers are highly uncertain, so take them all with a large grain of salt.

COVID-19 April 12, 2020

Happy Zombie Day everyone. I'm only going to post the current charts. No real analysis today. The weekend data glitch seems to have arrived on schedule. I expect artificially depressed numbers today and probably on Monday as well. Then numbers will increase as the weekend backlog is unjammed.

COVID-19 April 11, 2020 Special Colorado Edition

Below is the analysis specific to Colorado.  All the caveats and warnings in my post from earlier today apply to this and then some.  You'll note that the data is even noisier than the US.

The Gompertz model predicts:
18,000 total cases
800 total deaths
Peak daily cases of 410 on April 11 (Today!)
9,000 cases (halfway to the total) on April 18
Peak daily deaths of 18 on April 14
400 total deaths (halfway to total) on April 20

All the above are HIGHLY SPECULATIVE.

COVID-19 April 11, 2020

It's the weekend!  I expect bogus numbers to flow in over the next couple of days based on history.  Today, I thought I'd focus on some highly speculative projections.

**********WARNING -- HIGHLY SPECULATIVE *****************

See the warning above?  Read it.  

The exponential models no longer fit any of the data.  In particular, the data have fallen below the exponential model predictions.  This is what we hoped, and is without a doubt due to social distancing and other measures that have been put in place with the exception of a few states.

Instead of an exponential function, I have been using a Gompertz model.  This model seems to do reasonably well for where we are in the overall evolution of the pandemic.  I expect that it may breakdown as things start to wind down.  Still, the Gompertz model can provide a bit more insight than an exponential model that already deviates substantially from observations.

The Gompertz model has three adjustable parameters (see my previous posts for an explanation).  The first parameter is the initial number of infections.  This number and the day on which it happened is set to 15 on the 57th day of the year (February 27, 2020) and is based on data.  The second parameter is the growth rate of infections.  The third and final number is the final number of infections at an infinite time later.  I adjust the second and third parameter to obtain a good fit to the observed data as judged by eye.  I can then extrapolate the Gompertz function out into the future and see what it predicts.

There are a couple of things to consider before looking at the results.  First, the observational data has problems. I've pointed these out multiple times and previously discussed these in detail.  Since the observational data is of poor quality, even a perfect model will not be able to match the data exactly.  Thus, there are inherent errors in fitting the model to the imperfect data.  Second, the Gompertz model assumes a natural progression of infection, whereas we know that we (humans) threw a wrench in the natural progression by instantiating social distancing programs.  We changed the calculus, and thus the model may not be able to fit both the early observational data (before social distancing effects) and later data (after social distancing effects).  To predict the future, it might be better to emphasize a good fit to the most current data rather than the oldest data from early in the pandemic.

Now, let's get to the plots and the predictions.  The first two figures below show the cumulative number of cases.  The data are the same, but one is plotted on a semi-log plot and the other using linear axes.  The blue dots are observations. The dashed orange line is the 10-day trend based on an exponential.  The red crosses are the data obtained from the Gompertz model.  

The Gompertz model overpredicted the observations early in the pandemic; the red crosses are above the blue dots.  The Gompertz model actually predicted that infections would grow faster than they actually did.  But, we must also consider that the number of observed infections may be biased to the low side due to the clusterf*ck of testing.  From about day 85 (March 26) and onward, the fit looks pretty good.  

If I look at the parameters from the Gompertz model I get a total number of infections of 3,200,000.  According to the model, we reach this number at time equals infinity, which is not entirely helpful.  Instead, consider that the model does give a specific date for when we reach the halfway point of 1,600,000.  That happens on May 3.  We reach 2,000,000 on May 11.  For documentation purposes, the growth rate is found to be 0.044.  

Recent press reports have quoted professional epidemiological predictions (n.b. that's definitely not me!) that say the US has reached its peak.  My model says otherwise.  As a meteorologist, I'm used to going out on a limb with predictions.  And, I'm also used to be wrong!  I'm also used to being right!  I'm sure to be either right or wrong, that I'm sure of.  Time will be the judge.

Before you run off with the numbers above, please go back and read the warning near the top of this post.

I can also use the Gompertz model to look at the daily number of infections.  This is shown in the two plots below.  From the model, I find that the peak number of new daily infections will be 51,781 on April 24.  For reference, we are at about 40,000 cases per day right now.  According to the model, we will still be at over 20,000 cases per day by the end of May.

You can really see the noise in the observational data in these plots, particularly when using a linear scale.  The green circles highlight dips that seem to appear over the weekend.  Does the model (red crosses) fit these data?  Not perfectly, that's for sure, but it does more or less go right down the middle.

I can also fit a Gompertz model to observed death data.  Cumulative deaths and the daily death rates are shown below.  Deaths should be more accurate than the data on infections, because the dead are almost always reported.  Yes, some deaths may slip through the cracks because those deaths can sometimes be attributed to other causes.  It also turns out that in some cases deaths don't hit the books until well after they occurred.  So, while the deaths are recorded, they can show up well after they happened and can bias the data.  

As with the infection cases, the observational data departed from the exponential model quite a while ago.  The Gompertz model follows the observed data much better.  The best fit parameters show a total death of 320,000 after infinite time.  We will reach 160,000 on May 7.  The peak death rate will be on May 4 with 5179 deaths per day.  Again, these numbers are in conflict with at least some of the pros that say we have reached (or will very shortly) reached a peak in deaths.  

Going forward, I'll be showing these plots and will mention and major changes in parameters and predictions.  

COVID-19 April 10, 2020

I have no charts today...too busy with actual work!  I did look at the numbers, and they are mostly as expected.  Colorado had a large increase in cases and deaths, but the note on the State Health Department website indicated that was due to the incorporation of older data that had not yet been recorded.  This just points to the ongoing problem of data reliability.

I hope to be able to post more analysis tomorrow.