COVID-19 May 10, 2020

The very slight, mean downward trend in daily new cases continues.  I consider this good news, especially with the increase in testing.  Note that this week's peak was much lower than in the past two weeks.  If the trend starts to slow, or if the weekly peak starts going back up, that could be a sign of trouble ahead.  As a result of the gentle decrease in new cases, the slope of the cumulative cases has eased very slightly to an average of 25,061 new cases over the last week.  That's about 2,000 less than in the previous week.  BUT, remember, it is still increasing.  25,000 per day equals 100,000 every 4 days.

Yesterday's daily death mark was not as low as I would have liked.  We are near the bottom of the weekly cycle, but it may be difficult for this week's low to beat last week.  If that happens, it's an indication that the death rate is starting to level off rather than continuing to decrease.  This would be a strong indication that the near constancy of the new daily cases that started several weeks ago is real and not entirely due to an increase in testing.  As I said yesterday, this week's death statistics will be the truth-teller.

COVID-19 May 09, 2020

For the past few days, I've had little time to do more than post daily plots.  Let's catch up on where things stand, starting with infection cases and then moving to the death statistics.

The number of new cases each day is shown below.  Note that I have added a 7-day running average (orange) to help smooth out the oscillations with a period of 7 days.  Per numerous previous posts, the data followed the Gompertz model (red) right up until about the peak in early April.  Then, rather than starting to slowly decrease, the number of daily cases remained more or less constant.

In a post about one week ago I discussed how the increase in testing could partly explain the constancy of the new daily cases.  In short, the more testing you have, the more new cases you might expect to find.  A normalization of the data taking into account the change in testing suggested that at least some of the constancy was an artifact of the testing effect.

If you look at the data since around April 22, the 7-day average has been very slowly decreasing.  I cannot say whether this is statistically significant.  If it is, it's probably on the hairy edge of significant, because it's clear that there are other past variations that are of nearly comparable magnitude.  Don't put too much (or any) faith in the trend.

Whether decreasing or remaining constant, it is clear that the number of new infections is not increasing despite an increase in testing.  This fact may be helpful because if we start to see an increase without a corresponding increase in additional testing it might suggest that infection rates really are increasing.  This is something to keep an eye on as more and more states open up and the stupid, gullible, misinformed, and conspiracy loons are left to their own devices.

Since the number of new daily cases has remained nearly constant since early April it should be no surprise that the total number of cases has been increasing almost linearly at the rate corresponding to the orange 7-day average in the daily new cases plot. The slope of the line in the total daily cases below is exactly equal to the average number of daily new cases, which currently stands at 24,481 cases per day on average over the last 7 days. (See Algebra is useful.  Calculus, which relates slopes to curves is also useful!)

I have continued to leave the Gompertz model on the cumulative case plot even though it is clearly of little utility other than as a comparison to a by-gone era.  Had we stayed on the Gompertz model, we'd still be below 1.2M cases.  We've blown through that with cases increasing by 20-some thousand every day (or over 100,000 per week!).  At least it's a linear increase and not exponential.  At least not yet.

Now let's turn to the death statistics. These, in principle, should be more robust against influences like testing.  The dead are almost always reported, but the cause of death is not always properly categorized.  It is now coming to light that deaths early in the pandemic were falsely attributed to things like pneumonia or influenza.  Post-mortem testing has revealed that many of these were actual COVID-19-related.  Also, those that die at home from the coronavirus are not always properly added to the statistics.  Some states require testing before those dead are added to the total.  That process can take days to weeks.  Further, once a cause of death is confirmed as COVID-19, that is usually recorded on or after the testing date not when the death occurred.  Bottom line: deaths should be more robust than the case numbers, but it is still a messy metric.

As just a couple of examples of the problems with the death statistics, I bring you New York and Pennsylvania.  On May 6, New York added 720 deaths on top of the 232 deaths reported for that day.  Where did the 720 come from?  It was an accumulation of prior deaths that had not been properly accounted for either due to misdiagnosed causes of deaths or deaths that had simply slipped through the reporting cracks.  So, not only does this adjustment screw up the historical data, it causes a large, spurious bump for New York (and to some extent for the whole US) on May 6.  On the other hand, Pennsylvania removed 201 deaths on April 23. These deaths were reclassified as probable cases that needed further investigation in order to be confirmed.

The number of daily deaths is below, including all the idiosyncracies like those mentioned above PLUS the vagueries of the weekly reporting oscillation.  I have also added a 7-day running average to try and at least smooth out the oscillations and some of the noise.  You'll note that, like the daily cases, the number of daily deaths followed the Gompertz model rather nicely up until about April 15.  At that point, it went approximately constant.  I find this interesting because that breakpoint lags the new cases by approximately 9 days: New cases went constant (on average) on about April 6 and the number of daily deaths went flat (on average) on about April 15.  That lag is approximately what you might expect between the time of diagnosis at a hospital (i.e, the case is severe enough to warrant an actual test) and when the patient dies.

The lagged correlation between daily deaths and daily new infections suggests that the flattening of both has some basis in reality and is not entirely due to testing.  None of this is a sure thing--far from it.  I'm just trying to infer what I can out of very messy data and statistics.  Do not take any of this as being anything close to definitive.  I can say that none of this would pass peer-review muster in a reputable scientific journal.  It's not that it's necessarily wrong, it's just that a much deeper and careful analysis of the raw data and the statistics are needed.  I don't have the time (or the inclination) for that.  I am rather sure that many future epidemiological Ph.D. theses will spring out of these data.

Finally, the cumulative number of deaths is shown below.  For now, the data is still following the Gompertz curve. The data had sagged slightly below the model for the last week or so.  This is consistent with the number of daily deaths falling below the modeled number, as seen in the daily death plot.  However, with the daily deaths now trending toward a constant, the data has caught up to the model, and may very well exceed the model.  The Gompertz model predicts a continued fall in deaths and, therefore, a slowing in the rate at which deaths accumulate.  The next week should be the truthteller.

If deaths do not continue to fall that suggests that the pandemic has indeed plateaued and is no longer decreasing.  After all the social distancing (and even give the handful renegade states and individual miscreants), the best we've been able to achieve is a plateau, not a decrease.  We've flattened the curve to a constant rate of increase rather than exponential.  What do you suppose will happen as restrictions are lifted?  It's not hard to make an educated guess at the answer to this question.  Ultimately, the death rate will let us know.  

Unfortunately, there is a substantial lag between deaths and policy.  I've likened that lag to the light travel from distant stars.  When we see a star in the night sky, we are seeing that star as it was in the past when the light first left the star's surface.  That could be 100s or 1000s or millions of years ago!  It tells us nothing about what the star is doing now.  Death statistics are a time machine that tells us about reality a week or two weeks or perhaps three or more weeks ago.  People that die today are the result of the policies and the pandemic state put in place prior to them first becoming infected.  It takes time before changes in policy and the pandemic show up in the death statistics.  But they will.

COVID-19 May 08, 2020

Sorry folks, but once again I have no time to post much commentary.  The usual plots are below.  Total infections continue on the linear upward trend.  Death statistics continue to follow the model.  I'm waiting to see if they break away from the model, but that hasn't happened yet.  Note that I've added a 7-day moving averaging to the death statistics in order to smooth out the daily noise that makes up the one-week oscillation. 

COVID-19 May 07, 2020

Another super-busy day today.  I have a few things to say, but no time to say it.  Maybe tomorrow. Here are the data. Better late than never!

COVID-19 May 06, 2020

I'm running very short on time today, so I'll just drop the usual plots without commentary. I hope to be back with more discussion later this week.

COVID-19 May 05, 2020

There has been a fair amount of press about the predicted number of dead and revisions thereof.  The curve fitting that I've been doing has undergone changes with time as well.  During the early stages of the pandemic prior to much of any social distancing the model suggested deaths over 300,000.  On April 13, I revised that number down to 186,000.  Social distancing measures had started to kick in by that date.  Just a few days later, I was able to revise that number down to 120,000 as the curve flattening really kick in.  On April 24 I further revised the number to 100,000.  It's been there ever since.  In summary, before social distancing: 300,000+.  Since social distancing, about 100,000. 

The 60,000 number that was bantered around earlier last month was absurd.  It was never realistic. I wrote about that here http://freethoughtranch.blogspot.com/2020_04_23_archive.html.  When looking at the data, it was hard to understand how the numbers could be substantially less than 100,000. 

Looking at today's numbers, I may have to revise the total deaths down again very slightly.  That assumes the same conditions in the future as we've had in the recent past.  Clearly, that assumption is increasingly being violated.  As social distancing relaxes I expect the death rate to rise.  I haven't seen that happen yet.  What the data tells me though is that total deaths will be 100,000 if we maintain the strict policies that are currently in place.  If we lift most restrictions, we will set ourselves back on the exponential growth curve.  The data from early in the pandemic suggests that could result in many 100s of thousands more dead.  Policies in between these two will result in deaths that are in between. 

COVID-19 May 04, 2020

The usual plots are shown below.  There is no significant change from yesterday.

Today, I'd like to focus on the question of how increased testing may be affecting the number of reported positive cases. Many thanks to a friend (you know who you are), who motivated this.

As previously noted, the daily number of cases has leveled off rather than decreasing as would be expected by the Gompertz curve.  One possible explanation for this behavior is that because testing has increased, the number of positive cases has increased.  For example, consider the following situation: Suppose on March 20 you test 100 people and find that 20% of those tests are positive, you have 20 new cases. Now on April 10, you test 200 people and you find another 20 new cases, which is equivalent to 10% of the tests.  If you plot just the number of positives, you see a constant 20.  But, if you plot the percent of people infected, you get a descending trend from 20% to 10%.  The constant number of cases might lead you to believe that the infection rate has remained constant with time. Clearly, the more you test the more positive results you might expect.  That does not mean the number of infections has increased.  It just means you're doing a better job of finding them.  This sampling bias can be corrected to get a consistent metric.  It works in the same way that you adjust the cost of products for the cost of inflation over time.   More on that in a moment.

The total number of tests conducted in the US is shown in the figure below.  Note that this data is pulled from worldometer.com.  I don't know how accurate this information is, but it's probably close enough for the analysis here.  Testing increased slowly up through about April 10.  Then it stabilized for about another ten days.  But, beginning around April 20, it started to increase rapidly. It's now
well over 250,000 tests per day.

The percent of tests coming back as positive as a function of time is shown below.  The percent of positive cases increased up through March 31 to a maximum of approximately about 25%.  It's been slowly decreasing since, and now stands between 10% and 15%.  So, roughly speaking, as testing increased the percent of positives decreased.

The number of positive cases is shown in the standard plots I've been presenting daily.  It increased through time and then leveled off.  There is also a strongly weekly oscillation imposed. Also superimposed on the daily cases is some imprint of testing numbers.  Let's see if we can figure out roughly what that is.

I am going to pick April 1 as a reference date to which I will adjust all numbers.  As I previously stated, this works just like inflation adjustment.  If milk was $2/gal on April 1, and if inflation is 1% over 10 days, then I'd expect the milk to be $2.02 on April 11.  If I find the milk actually costs $2.05 then I deduce that the cost of milk has increased by $0.03 after taking inflation into account.

On April 1, I find that there were 107,582 tests taken and 26,756 tests came back positive.  I can then compare the number of tests taken on any other day, and adjust the results from that day to April 1.  For example, on April 10 there 157,745 tests taken.  Since there were more tests, I'd expect more positive results, and there were: 33,578.  How many positives would I have expected if the same 107,582 tests taken on April 1 were administered on April 10?  It's simple.  I just multiply by 33,578 by the fraction 107,582/157,745.   And the answer is 22,900.  This is less than the number of positives recorded on April 1!  So, even though the raw numbers have been increasing, we might have found that had testing capacity remained at the April 1 level the number of cases found on April 10 would have gone down.  Testing capacity matters.

If I normalize all the data to the April 1 reference, I get the data shown in the "Normalized Testing" plot.  The blue dots are the daily data.  I've added a green line which is a 7-day moving average and helps to remove some of the noise so that the trend can be seen.  I've added this same data to the usual daily case plot.  Nevertheless, we can see that the number of cases normalized to the April 1 date has been decreasing since early April.  If we were using the same testing capacity on April 1 throughout the pandemic, we'd likely be seeing fewer positives today.  The relatively steady number of positive cases we're currently observing can likely be attributed to the increase in testing capacity and not due to an increase in the rate of infection. The relatively steady number of positive cases we're currently observing may be attributable, in part, to the increase in testing capacity. Other factors besides an increase in the rate of infection are likely also in play. [Edit: thanks to the comment below, I have revised the cause-effect of testing to be more speculative].

What can we make of all this? I think the most important message is to be very, very wary of numbers that are bantered about by the media and especially by government officials who may wish to use biased or distorted numbers to their advantage.  There are lots of subtleties in all these data.  I've only scratched the surface of the veneer to reveal the truth behind the data.  A deeper and far more detailed analysis is necessary to really understand what's going on.  Had we been able to hit this pandemic with quality testing and reporting from the get-go, then the reality behind the numbers would be more transparent.  But we didn't.  So it isn't.