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.

COVID-19 May 03, 2020

This will be a fairly short Sunday post. The usual figures are below. My friend pointed out that it might be possible to remove some of the signal from varying testing rates by normalizing the cases by the number of tests. Not sure I'm going to go down that rabbit hole for a variety of reasons, the primary reason being that this is just a temporary hobby and I do still have a day job. However, a quick, back-of-the-envelope calculation shows that to compare current numbers to those of mid-ish April, I should multiply today's numbers by about 63%. With 30,000 new cases, 63% would be about 18,750. And, if you look at the daily case graph, that's only a few thousand off from where the Gompertz curve predicts it should be. So, the number of tests can be important.  Deaths continue to follow the model.  I still consider deaths to be the most robust metric. I may need to make a minor adjustment mid-week as the model seems to overdoing the numbers ever so slightly.

COVID-19 May 02, 2020

New day, same data trends. The daily new cases are not dropping, but holding fairly constant if you smooth out the weekly oscillations. I've added a new linear trendline based on the last 7 days (Edit: changed from 10 days) worth of data. The slope of this line is 26,445 (Edit: changed from 28,501) new cases/day. That's effectively the 7-day (Edit: changed from 10-day) average. I'll keep an eye on that number to see if it changes substantially in one direction or the other. I'm still plotting the red Gompertz model, which is clearly no longer valid.

Deaths, on the other hand, remain consistent with the model. They are inconsistent with growing infections. We will know in the next week whether the daily death curve starts to move sideways in concert with infections, or whether it will decrease, as would be expected on the slow downward glide slope of a waning pandemic. Based on the public displays of idiocy I've seen reported in the press, I expect the deaths to start moving sideways. And then up again.

Edits: I decided to use a 7-day linear fit since the oscillation has a period of 7 days. A 10-day fit is likely to bias the result. The numbers above have been updated, and I've updated the cumulative case figure below.

COVID-19 May 01, 2020

And so another month begins.  Where will the month end?  Unlike April, where the trajectory was mostly predictable, May is likely to be much more uncertain.  New daily cases are not falling.  They have more or less leveled off with a very, very large weekly oscillation superimposed.  On the other hand, the death rate is falling, slowly, on average but also with a large weekly oscillation. 

For now, the infections and deaths are inconsistent unless the mortality rate is changing substantially over time. If the mortality rate is invariant, then deaths should track infections with a lag.  Will deaths start to level off over the next week?  If not, that suggests that the behavior of the infection data is due to changes in testing and reporting.  It means that that data becomes almost useless without substantial analysis to remove the spurious signal.  It means that, for now, the death rate will be one of the few easily accessible and analyzable data for tracking this pandemic. 

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.