COVID-19 April 30, 2020

As each day passes, the meaningfulness of infection numbers seems to diminish.  The number of tests continues to ramp up and this strongly affects the statistics.  The more tests you conduct on symptomatic people or people that have likely been exposed to the virus, the greater number of positives you'd expect.  Is this why the number of new cases confirmed each day leveled off?  It may be possible to answer this question at some time in the future when a clever epidemiologist is able to sift through the chaos that is the states' testing programs, but I don't think it's possible to do this in real time anymore.  That's a real problem because the number of new cases each day should be an important metric for determining the effectiveness or consequence of policy.

There are other metrics.  The number of daily deaths remains an important number.  I've said the dead are almost always counted.  The use of "almost" is appropriate.  Some dead are not counted.  Or, rather, some dead are counted but not attributed to COVID.  Sometimes they are counted long after they've died.  And, based on the unconfirmed news coming out of Florida, sometimes the dead are intentionally miscounted.  Perhaps this metric will soon become as unreliable as the number of new cases.  After that, we may have little left but to monitor hospital load.

Below are today's graphs.  The number of new daily cases is not decreasing as would be expected.  That has caused the cumulative number of cases to continue to rise at a greater rate than would be expected.  It is not quite an exponential rise, but rather a near linear increase with time. Total deaths and daily deaths are still well behaved with respect to the model.

COVID-19 April 29, 2020

Just a short post today with the graphs.  Daily new cases have settled down a bit, but the trend over the last week still does not match with the prior trends.  Testing is really ramping up in many states and social distancing policies are in flux.  All of these can lead to a departure from the prior trend.  We'll just have to keep an eye on it.

Deaths continue to match with the model predictions.  If the behavior of infection rates is indeed changing, then that should start to show up in the death statistics.

COVID-19 April 28, 2020

After reaching the highest number of new infections just a few days ago, numbers have plummeted to the lowest they've been in nearly a month.  Yesterday's 22,000 new cases are just a little over where the model predicts they should be. The data variability makes no sense in terms of a natural system.  The behavior has to be due to some combination of reporting cycles over the period of a week combined with an overall increase in testing ability.  Superimposed on that is almost certainly a signal due to changes in the policies of individual states.  Depending on which data set you believe, we are about to cross or have just crossed 1,000,000 recorded cases.  This is an artificial number and the number of actual infections is almost certainly 10 or 20 times greater.

Daily deaths have also dipped to lows that have not been seen since early in the month.  The last two days have been substantially below the model trend, but I expect the numbers to rise, as they typically do through the weekdays.  Deaths should not be strongly tied to testing availability, but the data still suffers from reporting inconsistencies.  We have crossed the 50,000 dead mark.

COVID-19 April 27, 2020

The number of new cases dropped significantly yesterday.  This is usually the case for the weekend, which has historically been the low point of the weekly oscillation cycle.  The question for the coming week is whether the numbers will begin to take off again or whether the numbers will relax back to the model trend.  If the numbers increase again, it will be interesting to see whether they do so exponentially, which is shown by the orange curve I've added back since yesterday.  Perhaps we are starting to see the rise of the foolish states.

The number of daily deaths plummeted yesterday to one of the lowest values in weeks.  This is following the high point roughly a week ago.  If the weekly oscillation is still operating, the numbers should increase over the next few days.  The question for deaths is whether they will rise back towards the model trend.

What is interesting is that while daily new infections are climbing daily deaths are dropping.  The rise in new infections could be consistent with the increase in testing.  The number of deaths is independent of testing.  If deaths don't rise within the next week or so, that means that any continued rise in the number of infections is mostly the result of increased testing and not an actual surge in the pandemic. 

Overall, the quality of the data has become increasingly poor. The amplitude of the weekly oscillation has grown, the reporting criteria have changed from state to state, and the way that earlier cases just now discovered are added to the pile contaminates the data.  If the data collection and documentation protocol were rated by scientific standards it would be a fail minus.  It really just goes to show how poor (absent) the coordination and standards are at the federal level. 

COVID-19 April 26, 2020

Daily new cases reported on Friday departed substantially from the trend.  Yesterday's numbers are lower, but still substantially different.  I am inclined to think that a good amount of that is due to states adjusting their numbers and an increase in testing.  This is just a hunch, but as an example, the number of tests given each day in Colorado is shown below.  In the last two days, the numbers have increased by more than 2.5 times the average.  Not surprisingly then, the number of positives have increased proportionally.   I can't say whether this is true for other states, but the numbers of many other states do look suspicious.

At the same time, I do think that some states are about to undergo an increase in the rate of infections.  In principle, this increase will be balanced by a decrease in other states that have peaked.  In principle.  If too many states take their foot off the break too quickly, then it is almost inevitable there will be an acceleration of cases again.

The best metric continues to be the number of deaths.  Even these numbers, however, can be subject to uncertainty as states go back and reevaluate the cause of some deaths.  It would be acceptable if historic deaths were logged in on the day they occurred, but many states lump them into the daily statistics.  As long as those numbers aren't too great, it's not a big problem. But if New York goes back and decides that there were 8,000 additional deaths that should be attributed to COVID, and if they lump them all into next Tuesday's numbers, that's clearly going to cause problems.

Below are the graphs for today.  I've readded the exponential best fit line on the cumulative cases.  I want to see if the data begins to take off on an exponential trajectory again.  That would be bad.

COVID-19 April 25, 2020

The data reported yesterday are...interesting.....

Death statistics, both the daily deaths and the cumulative, were as expected.  The new daily infection cases, however, broke the model.  They were far higher than expected, even considering the known, weekly data oscillations.  Over 36,000 new cases were reported.  That's the largest number since...ever!  The next highest daily infection was 33,578 on April 10.  Prior to yesterday, all the data and the model indicated that we had passed the peak infection on or about April 9.  You can see the effect of this large daily increase in the cumulative data where the last point jumps above the best fit model curve.

There are really only two choices here.  Either the data has some problems and inconsistencies or the infections in the US have started down a more troubling trajectory.  The first choice is possible. There has been quite a bit of chatter about states adjusting their statistics: counting cases that had previously been unknown.  Also, testing is ramping up, so maybe the sudden increase is just a reflection of that increased testing.  The second choice is also possible.  As states relax restrictions, the number of daily infections would be expected to increase.

It may be possible to disambiguate the possibilities with time.  If it's simply a matter of a data adjustment and an increase in testing, then we might expect to see the data jump up for the next few days (or even over a week given the oscillation issue).  After that, the slope of the curves should then return to where they were.  That is, there will be a discontinuity and offset between the historical data and the new data, but the curves will be roughly parallel to each other.

If the behavior is due to a real increase in infections, then the curve will continue to steepen, and we may be able to identify the states that are most responsible.  Remember that NY/NJ account for a disproportionate number of daily cases and cumulative cases.  As those two states settle down, other states are rising and coming to dominate the statistics more strongly. Perhaps the most telling signature will be if the number of deaths begins to rise.  Death statistics lag infections.  If infections really are increasing dramatically, deaths should follow within a week or so.

COVID-19 April 24, 2020

I've readjusted the Gompertz curve to the data acquired up through the end of the day yesterday.  The difference between these updated numbers and those I previously presented are rather small and almost inconsequential.  Overall, the curve fits are very good. 

The data from new daily cases is showing what might be an unexpected increase, or at least leveling over the last few days.  There could be several causes.  First, it could be a normal statistical variation.  Data over the next week will be the judge of that.  Assuming it's real, then the explanation could be increased testing and reporting.  Or, it could also mean that infections are starting to accelerate in some locations.  With the easing of restrictions in an increasing number of states, this wouldn't surprise me. We'll also have to see if the death rate begins to follow, which would be expected if the daily death rate really is leveling off rather than continuing to decrease.

If the data continues to deviate from the Gompertz curve then it means two things.  The first is that the pandemic is beginning to sweep into new locations and fertile ground within the US and/or the easing of restrictions is going to result once again in a growing problem where it is already established.  The second is that I will no longer be able to use the Gompertz model to make predictions, except to say that the final result will likely be worse than what the Gompertz model predicts.

I'll provide another update in about a week from now unless conditions warrant an update sooner.

For now, here's the rundown:

• 1.2M confirmed and presumptive positives predicted by the end.
• Both the data and model indicated we reached the halfway point of 600,000 on April 14.
• Peak daily new cases were predicted at 33,467 on April 9.  Data has 34,243 on April 9.
• Predicted 100,000 dead by the end.
• 50,000 dead predicted on April 26.
• Peak death rate of 2,170 predicted on April 20.  Data has a peak of 2,627 on April 21.

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.