COVID-19 April 07, 2020 Special Colorado Edition

With today's numbers hot off the press, the range of possible Gompertz model outcomes has narrowed.  Baring some strange event, I feel reasonably confident with the Colorado predictions.  What does "reasonably confident" mean?  I think they are upper limits and the actual numbers will not be more than about 30% higher at most, probably lower. Maybe I'll eat some crow later. We'll see.  Maybe I'll eat some crow later.  We'll see.

So what do the numbers say?  The Gompertz model (see previous posts about this model) predicts 20,000 total infections.  The number of new daily cases will peak at 441 on April 11. Colorado will reach 10,000 cases--halfway to the final number--on April 18.  Final deaths will be 800.  The peak number of daily deaths will be 18 on April 15.  Colorado will reach 400 deaths--halfway to the final number--on April 21.

What's interesting about these predictions is that the Gompertz growth rate for both infections and deaths is the same: 0.06.  Yet, the curve fits for both data sets were arrived at independently.  The fact that they have the same growth rate provides a good amount of confidence in the predictions.  Assuming that deaths are proportional to the number of infections, you'd expect to see the growth rate be about the same.  Furthermore, note that the peak in deaths and the halfway point for deaths lags the infection case milestones by about four days.  This delay between the official infection report and death is reasonable.  Thus, the two model results for infections and death are consistent.

COVID-19 April 06, 2020

Yesterday, I made a point of describing how screwed up the weekend data appears to be.  Several of you chimed in with anecdotal information that was in concurrence.  Thank you! 

I thought I'd follow up with a little bit more analysis and data.  I'll let you decide.  Below is a plot of the number of new infections each day.  The reported data is in blue.  The orange data represents predicted numbers based on a Gompertz model.  I've plotted this on a linear scale rather than the log scale I've typically been using.  The linear scale lets you see variations easier than a log scale.  See those last two points?  I've circled them in red. Those are the data that were reported on Sunday and Monday.  To my eye, the Sunday data is way, way out of family with every other data point up to that point.  The Monday data is a little better in that it is at least increasing again.  I've seen similar aberrations previously.  I've circled those in green.  The earlier times aren't as obvious as the ones this weekend, but that's because all the numbers were much smaller back then.  But, the blips are there.  There are similar patterns in many, but not all, of the 50 states.  Anyone want to lay odds that we'll see the same pattern next weekend?

It will be interesting to see if we get a flood of data coming in later this week to make up the difference.  It's possible (likely?) that agencies and labs are not working on the weekends or are at least running with minimal staff.  Let's also remember that testing is still not able to keep up with demand even when labs are going full bore.

In the meantime, what I believe to be spurious data is affecting the trends.  So, for now, those low numbers have slightly slowed the doubling rate.  I've seen a lot of press and reports stating how good the numbers were this weekend. I would exercise caution. Those numbers could be real.  Or not.

COVID-19 April 05, 2020

Are you an optimist or a pessimist?  The optimist will take the view that yesterday's numbers are good. The number of new cases dropped dramatically, and we failed to reach 10,000 deaths due to a drop in the number of daily deaths.  The pessimist will look at the numbers from yesterday and find them improbable.  I'm an optimistic pessimist.  I hope the numbers are real, but they look very suspicious.  I saw a similar pattern in the data from last Sunday, and I suspect the low numbers are a result of numbers either not being officially tallied by the state health departments or the state health departments not reporting a complete tally.  Last week, a spike in numbers came in on Monday and Tuesday after seemingly low numbers on Sunday.  Let's see what happens this week.

Look at the Daily Rate of Change graph and the Daily Death Rate graph to see how different yesterday's numbers are from the historical trend.

COVID-19 April 04, 2020

I'm going to make this short today because it's a beautiful spring day here in Colorado. I want to get outside, work in the garden, and go for a nice walk/run wearing a socially responsible mask. The usual charts are below.

 As a reminder, each day's data is valid for the end of the day before. So, today's report represents the final numbers from yesterday. Tomorrow will tell us how today ended, and so forth.

 Yesterday's total cases for the U.S. came in at 307,876. The doubling time is currently 4.8 days and decadaling time is 16.0 days. Both of these times have generally been increasing in length very slow each day. That's a good trend, but it would be nice if it were faster. At the current rates, we will hit 1,000,000 total cases in a little over 10 days.

 Total deaths came in at 8,360. At the current rate, we will hit 10,000 sometime today, or come very close. We'll see the reality of that in tomorrow's report. The doubling time is now 3.1 days and the decadaling time is 10.4 days. That would mean we will reach 100,000 deaths in less than two weeks from now if current trends continue.

COVID-19 April 03, 2020 and Colorado Special Edition

All data continues to proceed as it has for many days now.  The initial high exponential growth has given way to more moderate but still exponential growth.  Doubling time for infections is now at 4.5 and doubling time for deaths is 3.0 days.  The forecast for reach 10,000 dead will happen by the end of the day on the 5th.  The decadaling time for deaths is currently 10.2 days, so if the trend continues we will reach 100,000 dead a little over 10 days later.  The graphs are below.

I have taken a little time to look at using more advanced models to fit the data.  Why am I familiar with all this stuff?  At the age of about 13 years I received an Apple II clone computer on which I taught myself the BASIC programming language.  After mastering the language, one of the first "real" programs I wrote was a predator-prey model, and continued to add complexities to that model over time.  My first foray into numerical computing and modeling was in the area of population biology.  Eventually, that gave way to an interest in modeling of other natural systems, and I have been modeling fluids and atmospheres for the last several decades.

The next level of model complexity beyond the exponential model is arguably the Gompertz model .  There are three parameters that are set in that model: the initial population, the final population, and the growth rate.  I've applied this model to both the US and Colorado.  I've set the initial population of infections to the observed value at the start of recorded infections in the US or Colorado, as appropriate.  I then adjusted the growth rate and final population to obtain a good fit, as determined by eyeballing the Gompertz curve compared to the observed data.  There is a wide variation of parameters that give a reasonable fit to the US data, and that tells me that the data is not yet sufficiently constraining to obtain much quantitative information.  All that can be said is that the exponential model has broken down (which is already self-evident).  Where the US data goes is difficult to predict in the longer-term.  Short-term trends can be used for short-term predictions.

The spread of reasonable parameters for Colorado was more limited.  The Gompertz model compared to the data is shown in two figures below.  One uses the semi-log scale and the other uses linear scales.  The fit, judging by eye, is good, and here's what the model indicates.  The total number of infections in Colorado will hit approximately 70,000.  We will hit the following milestones on the following days: 10,000 on April 12; 20,000 on April 22; a maximum number of new daily cases (1,133 new daily cases totaling 26,476) on April 27, at which point the number of new cases will finally start to decrease each day; 30,000 on April 30; 40,000 on May 10; 60,000 on June 8.  The 70,000 cases are reached asymptotically, which is to say it takes an infinity amount of time to get there.  Note that the plots below only go through early May.

I want to emphasize this model is extremely uncertain, and I would not put great faith into the exact number or dates.  But, unlike the exponential model which grows unrealistically forever, the Gompertz model exhibits a realistic growth and then flattening.  I will update this model periodically as new data comes in and we'll see how it does.  As a very rough estimate, it does predict that the worst for Colorado is yet to come, but that by mid- to late-May, the worst will be behind us.

COVID-19 April 03, 2020

The slightly flattened curve continues with, perhaps, ever so slightly more flattening.  The 10-day orange trend has now clearly departed from the initial exponential rise (blue) and is currently indicating a 4.3 doubling rate.  At that rate we will reach 1,000,000 total cases on or about the 10th, one week from now.  The three day average (yellow dots/gray line), which captures most strongly the current data is just below the 10-day line.  That slight difference isn't great, but it can have a big difference as time goes on due to the nature of exponential functions.  I caution, however, that three days of data is probably not statistically significant, so extreme caution should be exercised in the interpretation.  If the three-day trend stays that way consistently for more days, the confidence of further flattening increases.  And, as I feel obligated to state almost every day, testing sucks!  As my colleague, Dr. Scott Anderson, who provided the Colorado data the other day, pointed out, tests are growing linearly in Colorado whereas cases are growing exponentially.  Testing is getting farther behind every day.

Total deaths continue their rise unabated.  The three-day trend in death rates shows some flattening, but as discussed above, I wouldn't get too excited yet.  Let's see if that sticks.  Let's hope that it sticks. Using the 10-day trend, we will hit 10,000 deaths in the next couple of days.  100,000 deaths will be a little over week later.

COVID-19 April 02, 2020

Today's total infection rate graph, created from the tallies at the end of the day yesterday, continues to show a consistent trend with the new, slower rate of exponential growth that was established about 5 days ago.  I want to emphasize that these data are very likely influenced by poor reporting and testing.  For example, this recent article from CNN reports that there are up to 160,000 tests backlogged at a single lab.  I think it is likely that at least part of the flattening is real--the rate of new infections really has gone down.  Nearly 80% of the country is now under some kind of quarantine.  That is going to have an effect, and most of those policies have now been in effect long enough to show, even if there is a lag in testing and reporting.

What is troubling is that the daily rate of deaths, and therefore the total number of deaths, appears to be rising without much of a break in slope.  Note that the 10-day daily death rate in orange is just barely below the rate for all the data (blue).  In principle, deaths should be correlated with the number of cases, but with a time lag.  If the number of cases has slowed, why haven't deaths?  We'll have to keep an eye on total deaths and especially the death rate to see if it starts to flatten more.  If not, that suggests that the testing data is significantly biased and problematic.

Another concern is that with so much of the country already sequestered, why hasn't the rate of new infections begun to stabilize more quickly?  Or, stated another way, if we wish to reduce the rate of new infections further, it seems like it's going to take even more draconian measures to do so. Either that, or we are going to have to live with this semi-flattened curve for a loooonnnnnggggg time.  The curve is not going to flatten naturally until a significant portion of the population has been infected and become immune, or, possibly, until warmer weather begins to affect the stability of virions.  The latter idea is only a possibility.   No one is sure how the virus will respond under warmer conditions.

Finally, I'll point out that the population is not uniformly distributed across the country.  Much of the rapid growth we saw was due to New York.  Other cities could be next to experience this growth, and that could cause the curve to steepen again.  The social distancing policies that are in place across much of the country should help mitigate this risk, but it is still there.  Florida only recently put in place relatively weak orders that are going to take at least 10 days to have an effect.  The major cities in Florida could spike dramatically within that period of time and set the US curve as a whole along a steeper path in the same way that New York did.