Team Toad: Data Groupies Blog

Team Toad Data Groupies - Skew-Normal Epidemic Modeling

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February 18, 2023, the final post?

We are up to twelve (12) distinct waves of infection, the last being the Omicron XBB.1.5 subvariant, with a hint of a thirteenth wave of cases. The CDC has a variant tracker showing the estimated proportion of each subvariant in the US. The current dominant variant is XBB.1.5, which is the latest variant of Omicron.

Now that each new wave is really a small ripple on top of other small ripples, there really isn't much structure to model.

Case fatality rate

If you look at the right side of the death graphs, you'll see estimates of the case fatality rate (CFR), which is the percentage of patients who test positive for Covid and then die.

At the beginning of 2020, the CFR was around 5.0%, but that dropped to 2.0%, then 1.3%, and the current estimate is 0.48% (based on the cases and deaths since April 2022).

Daily new cases 7-day average 17-Feb-2023

Daily deaths 7-day average 17-Feb-2023

Variants and vaccine rates

CDC Variant Proportions 17-Feb-2023		Americans with one dose 17-Feb-2023
Americans fully vaccinated 17-Feb-2023		Americans with current booster 17-Feb-2023

November 19, 2022, the model stops working

We are up to eleven (11) distinct waves of infection, the last being the Omicron BQ.1 subvariant. The CDC has a variant tracker.

In October 2022 it looked like a new wave was starting, but the numbers of cases simply froze at about 40,000 new cases per day. The sum of normal curves assumption breaks down in this case. Similarly deaths also became linear, although they dropped steadily from 400 per day in early October to 300 per day in mid-November.

This behavior could be a sign that the virus has become endemic, regularly found in normal people. This article from Johns Hopkins from February 2022 discusses what that looks like.

Daily new cases 7-day average 11/18/22

Daily deaths 7-day average 11/18/22

Americans with one dose 11/18/22

CDC Variant Proportions 11/18/22

July 23, 2022, two more variants and two more waves

We are up to ten (10) distinct waves of infection, the last being the Omicron BA.5 subvariant. The CDC has a variant tracker, that shows that BA.2 has replaced the BA.2.12 and BA.2 subvariants of the original Omicron.

While there are clear visual peaks showing ten waves of cases, the death data the last two months is much flatter. The curve fitter just applies least-squares and happily fits as many curves as you like. There is also not such a clear correspondence of waves to BA.2, BA.2.12.1, and BA.4.

Daily new cases 7-day average 07/22/22

Daily deaths 7-day average 07/22/22

Americans with one dose 07/22/22

CDC Variant Proportions 07/22/22

April 26, 2022, an eighth wave starting

I am using a larger screen to create this week's charts, so they are significantly higher resolution.

We are up to eight (8) distinct waves of infection, the last being the Omicron BA.2 subvariant. The CDC has a variant tracker, that shows that BA.2 has replaced the original Omicron variant.

It is not yet clear how big this eighth wave will be, but so far the data does not show this wave becoming larger than the seventh wave, Omicron, that had more cases than any other wave.

Daily new cases 7-day average 04/26/22

Daily deaths 7-day average 04/26/22

Screen shot of the CDC vaccine tracker
as of April 27, 2022

According to the CDC vaccine tracker, 219.5 million Americans are fully vaccinated.

February 12, 2022, Omicron has peaked

I am using a larger screen to create this week's charts, so they are significantly higher resolution.

We are up to seven (7) distinct waves of infection, the last being the Omicron variant, which has now peaked. The CDC has started tracking a sub-variant of Omicron designated as BA.2, but it's not clear yet whether that will be very different from the original B1.1.529 Omicron line.

Daily new cases 7-day average 02/11/22

Daily deaths 7-day average 02/11/22

Screen shot of the CDC vaccine tracker
as of February 12, 2022

According to the CDC vaccine tracker, 204.6 million Americans are fully vaccinated (61.6% of the whole population).

December 22, 2021, here comes Omicron, US cases and deaths

I've been publishing weekly charts on Facebook since September, but have not been including them here, because it's simply more work. But at the request of a reader, I'm updating this week here so you can view the charts uncompressed.

Since September, we've seen the start of a sixth wave starting in October, comprised 99% or more of the Delta variant. Within the past week, however, the CDC has announced that Omicron has become the dominant variant, so the sixth wave will be a mixture of Delta first and then Omicron.

Daily new cases 7-day average 12/21/21

Daily deaths 7-day average 12/21/21

Screen shot of the CDC vaccine tracker
as of December 21, 2021

According to the CDC vaccine tracker, 204.6 million Americans are fully vaccinated (61.6% of the whole population).

The sixth or "Delta and Omicron" Wave

My last blog entry confirmed a fifth wave. Since then we've had a sixth wave almost solely due to the Delta variant first found in India (The Darden Report). These graphs are just the same as above, but with the X axis expanded to show this new wave.

The latest variant of concern is Omicron, and as of last week the CDC announced that 73% of all new cases were Omicron, which is much more transmissive than Delta. (CDC, December 20, 2021).

Detail of the sixth wave of cases

Detail of the sixth wave of deaths

As always, please note that this is a retrospective data model, and can neither predict nor prohibit any future waves of virus outbreak. See the about page for data sources, disclosures, and disclaimers.

September 4, 2021, the Delta wave, US cases and deaths

I spent the summer recovering from my hip replacement surgery at the end of May, and although I continued to publish Covid charts on Facebook, I did not take the extra steps to publish them here on my web site.

Daily new cases 7-day average 9/04/21

Daily deaths 7-day average 9/04/21

Screen shot of the CDC vaccine tracker
as of September 4, 2021

According to the CDC vaccine tracker, 206,461,869 have received at least one dose of vaccine as of May 8, 2021.

The fifth or "Delta" Wave

My last blog entry confirmed a fourth wave. Since then we've had a fifth wave almost solely due to the Delta variant first found in India (The Darden Report). These graphs are just the same as above, but with the X axis expanded to show this new wave.

The Delta variant is estimated to have an R0 of 6 to 7, making it much more transmissible that the original Wuhan strain (NPR, 2021). Without that mutation, our level of vaccination of over 70% of adults would likely have stopped this wave.

And within the last week we've seen cases continue to rise in spite of the August 14th model predicting a peak. It's too soon to tell whether the model of the 5th wave is wrong, or that a "sixth wave" of cases is actually starting.

Detail of the fifth wave of cases

Detail of the fifth wave of deaths

May 8, 2021, US cases and deaths, India cases and deaths

A fourth wave in cases and deaths in the US is confirmed, but much smaller than the third wave.

Daily new cases 7-day average 5/08/21

Daily deaths 7-day average 5/08/21

Screen shot of the CDC vaccine tracker
as of April 17, 2021 at 9am CDT

According to the CDC vaccine tracker, 151,315,505 have received at least one dose of vaccine as of May 8, 2021.

Included in that 151M are 112,626,771 people who are fully vaccinated, having either received both doses of the Pfizer or Moderna vaccine, or one dose of the J&J vaccine.

Latest graphs for India, May 8, 2020

India has recently experienced a huge leap in cases (there called a second wave, although I'm modeling it as the fourth).

Within the last week the growth has gone sub-linear, allowing the skew-normal model to predict a peak.

India Daily new cases
India Cases

India Daily deaths
US Deaths

April 17, 2021, US cases and deaths, County Heat Maps

I updated the US cases and US deaths using data posted today for 4/15/2021. The April 16th data was not available, so Friday is not included. Since the last blog entry, a fourth wave in cases is confirmed, and seems to be at least as large as the first two waves of cases combined, but so far is not nearly as large as the third wave of cases over the Winter of 2020/21.

Although deaths are trending slightly upward, there is not enough data to confirm a fourth wave in deaths.

Daily new cases 7-day average 4/16/21

Daily deaths 7-day average 4/16/21

Last 21 days of cases and deaths as of April 14, 2021

These heat maps shows where the most new cases and were confirmed in the last 21 days on a county by county basis. This chart is normalized by population.

Note that Oklahoma restated their numbers on April 7 adding 1,800 cases from previous months, making them appear to be much worse for recent deaths on this map.

Average daily new cases per million last 21 days, by county
US CPM21

Average daily deaths per million last 21 days, by county
US DPM21

Screen shot of the CDC vaccine tracker
as of April 17, 2021 at 9am CDT

According to the CDC vaccine tracker, 127,743,096 have received at least one dose of vaccine as of April 16, 2021.

Included in that 127M are 80,609,818 people who are fully vaccinated, having either received both doses of the Pfizer or Moderna vaccine, or one dose of the J&J vaccine.

April 3, 2021, US cases and deaths

I updated the US cases and US deaths using data posted today for 4/02/2021. Cases are rising again, and the likelihood of a fourth wave seems even higher than last week. Since I have been modeling the case-bump last September as its own wave, that means we need five waves to model cases right now.

I also ran a new model fit for deaths, but as you can see the curve us very nearly identical to the Christmas Eve model run. There was no sign of a small wave of deaths in September to match the mini-wave in cases then. There's just a hint of a fourth wave in deaths this week, but the numbers are still declining.

Daily new cases 7-day average 4/02/21

Daily deaths 7-day average 4/02/21

Here is this week's spaghetti plot of possible fourth wave curves.

Daily new cases spaghetti plot 4/03/21

Screen shot of the CDC vaccine tracker
as of April 3, 2021 at 9am CDT

According to the CDC vaccine tracker, 101,288,138 have received 1 or 2 doses of one of the two vaccines as of April 2, 2021.

Included in that 101M are 40,699,456 people who have received both doses.

March 28, 2021, US cases and deaths

I updated the US cases and US deaths using data posted today for 3/26/2021. Cases have started going sideways, and this raises the possibility of a fourth wave. And since I have been modeling the case-bump last September as its own wave, that means we need five waves to model cases right now.

Deaths continue to drop.

Daily new cases 7-day average 3/26/21

Daily deaths 7-day average 3/26/21

It's clear that the cases numbers have started a new trend, so I ran a new series of genetic searches to the case data through March 26, 2021. The plot on the left superimposes all eight model runs together, and you can see that there is no real agreement from run to run.

This is typical when modeling the upside of a skew-normal curve, as there is rarely enough date to predict the "peak".

The thick red line is the run with the best R-square measure.

The chart on the right shows the actual trailing 7-day average, the sum of the five waves, and the individual waves for the best fitting of the eight genetic searches.

The press will likely call this the "Fourth wave", and we can see that the magnitude of this wave is already as large as the second wave last summer.

We do not yet see a corrsponding bump in daily deaths. It is possible that either a younger demographic or the vaccination of over two-thirds of the elederly population will prevent this case-bump from increasing the numbers of deaths. Time will tell.

Daily new cases spaghetti plot 3/26/21

Daily new cases, five waves 3/26/21

Screen shot of the CDC vaccine tracker
as of March 28, 2021 at 5pm CDT

According to the CDC vaccine tracker, 91,417,268 have received 1 or 2 doses of one of the two vaccines as of March 27, 2021.

Included in that 91M are 39,273,886 people who have received both doses.

March 20, 2021, US cases and deaths

I updated the US cases and US deaths using data posted today for 3/19/2021. Cases continue to drop sharply, and I ran a new version of the genetic algorithm curve fitter. Deaths continue to drop.

Daily new cases 7-day average 3/19/21

Daily deaths 7-day average 3/19/21

Screen shot of the CDC vaccine tracker
as of March 20, 2021 at 5pm CDT

According to the CDC vaccine tracker, 79,175,241 have received 1 or 2 doses of one of the two vaccines as of March 20, 2021.

Included in that 79M are 37,193,728 people who have received both doses.

March 6, 2021, US cases and deaths, new Case model

I updated the US cases and US deaths using data posted today for 3/05/2021. Cases continue to drop sharply, and I ran a new version of the genetic algorithm curve fitter. Deaths continue to drop.

Daily new cases 7-day average 3/05/21

Daily deaths 7-day average 3/05/21

Screen shot of the CDC vaccine tracker
as of March 6, 2021 at 2pm CDT

According to the CDC vaccine tracker, 57,358,849 have received 1 or 2 doses of one of the two vaccines as of March 6, 2021.

Included in that 57M are 29,776,160 people who have received both doses.

Tuesday update, February 23, 2021, US cases and deaths, new Case model

I updated the US cases and US deaths using data posted today for 2/22/2021. Cases continue to drop sharply, and I ran a new version of the genetic algorithm curve fitter. Deaths continue to drop.

Daily new cases 7-day average 2/22/21

Daily deaths 7-day average 2/22/21

Since cases have been dropping rapidly for weeks, I ran a new model using data through last night. The output is groups of four parameters, and since we are modeling four waves, there are four groups of four parameters being optimized.

Peak parameters are : skew, scalex, peakx, peaky
Skew parameters are : skew, offsetx, scalex, scaley

The 1313-mCmCmmC indicates that the best solution had 1,312 generators of parents, and it was produced by a small mutation of a cross between a small mutation of a cross of a small mutation and a cross... I use that predigree to evaluate which genetic operators are giving better solution.

You can also see a listing of the percentage of the time that each operator gave a better solution that the parent genes. For both round 19 and round 20 the small mutation operator was the only source of improvement.

The score being optimized is the modified least squares difference, and small scores are better.

The output on the right converts the peak coordinates to skew coordinates and calculates an R-squared fit value. Even though the optimizer is using a modifed R-squared measure to find a curve that skims the top of the data, the reported R-square value is the standard definition.

Genetic algorithm run for daily cases

Weekend update, February 20, 2021, US cases and deaths

I updated the US cases and US deaths using data posted today for 2/19/2021. Cases continue to drop sharply. Deaths are finally dropping, and yesterday the 7-day average dropped below the pre-Christmas model. From here I'd expect to see deaths dropping very rapidly.

Note that the death chart includes a new model run from February 6, 2021 with data through February 5th. The skew for the third wave of this model increased from 1.244 to 2.038. That would seem to show more deaths in January than expected.

But this week's data ended well below that model, just touching the pre-Christmas model.

Daily new cases 7-day average 2/19/21

Daily deaths 7-day average 2/19/21

Screen shot of the CDC vaccine tracker
as of February 20, 2021 at 10am CDT

According to the CDC vaccine tracker, 41,977,401 have received 1 or 2 doses of one of the two vaccines as of Feb 19, 2021.

Included in that 41.98M are 17,039,118 people have received both doses.

Weekend update, February 10, 2021, US cases and deaths

I updated the US cases and US deaths using data posted today for 2/09/2021. Cases continue to drop sharply. Deaths are dropping slowly, but not as rapidly as we'd expect from the case numbers.

Daily new cases 7-day average 2/09/21

Daily deaths 7-day average 2/09/21

Here are three graphs showing the trailing 7-day average cases and deaths, with varying amounts of lag added to the case number and a scaling factor applied to the death numbers to try to bring these two lines together. Knowing the right lag time allows the calculation of an estimated case fatality rate (CFR), which is the ratio of deaths to cases a certain number of days in the past.

The challenge is that the data is so choppy around the Christmas and New Year's time that it's difficult to determine the amount of lag. The best fit seems to be 28 days with a multiplier of 66, as show in the first graph.

During the summer a lag of 18 days and a multiplier of 60 worked well, as shown in the second graph.

But the peaks and troughs in the data also suggest a smaller 4-day lag which would give a multiplier of 77.

Deaths x 66 versus Cases with a 28-day lag

Deaths x 60 versus Cases with an 18-day lag

Deaths x 77 versus Cases with a 4-day lag

Weekend update, February 5, 2021, US cases and deaths

I updated the US cases and US deaths using data posted today for 2/4/2021. Cases are down sharply, far below both the pre-Christmas model and the mid-January mode. Deaths are steady week-over-week, but given the January 11 peak in cases, and the 18-day lag seen between cases and deaths, we'd expect deaths to start dropping from here.

Daily new cases 7-day average 2/05/21

Daily deaths 7-day average 2/05/21

Screen shot of the CDC vaccine tracker
as of February 5, 2021 at 6pm CDT

According to the CDC vaccine tracker, 30,250,964 have received 1 or 2 doses of one of the two vaccines as of Feb. 5, 2021.

Included in that 30.2M are 8,317,180 people who have received both doses.

Weekend update, January 30, 2021, US cases and deaths, and CFR

I updated the US cases and US deaths using data posted today for 1/29/2021. Cases are down sharply, far below both the pre-Christmas model and the mid-January mode. Deaths are up week-over-week, but given the January 11 peak in cases, and the 18-day lag seen between cases and deaths, we'd expect deaths to start dropping from here.

Daily new cases 7-day average 1/30/21

Daily deaths 7-day average 1/30/21

Screen shot of the CDC vaccine tracker
as of January 30, 2021 at 9am CDT

According to the CDC vaccine tracker, 22,858,318 have received 1 or 2 doses of one of the two vaccines as of Jan 29, 2021.

Included in that 22.9M are 4,780,888 people who have received both doses.

The estimated case fatality rate (CFR) is 1.262% or roughly one death per 79 confirmed cases.

Estimate CFR as of January 29, 2021 at 9am CDT

Mid-week update, January 20, 2021, US cases and deaths

I updated the US cases and US deaths using data posted today for 1/19/2021. Both cases and deaths were down significantly, and there is no sign of a post-holiday surge that many feared.

Daily new cases 7-day average 1/20/21

Daily deaths 7-day average 1/20/21

Note that Monday, January 17th was Martin Luther King Day, and as a federal holiday may be delaying some case and death reports.

According to the CDC vaccine tracker, 13,595,803 have received 1 or 2 doses of one of the two vaccines as of Jan 19, 2021.

Included in that 13.6M are 2,023,124 people who have received both doses.

Going forward, I expect the vaccinations to significantly reduce the total number of cases and deaths below models from Christmas and last week.

Screen shot of the CDC vaccine tracker
as of January 20, 2021 at 9am CDT

New model run, January 16, 2021, US cases and deaths

Daily new cases 7-day average 1/15/21

I updated the US cases and US deaths using data posted today for 1/15/2021, and ran new curve fits for both.

The new case model (left, red) is 11% higher than the pre-Christmas model (left, green), with a peak about one week later (Jan 1 at 223,655 before and Jan 8 at 247,792 in the new model).

The new death model (right, red) is actually 4% lower than the pre-Christmas model (right, green), with a peak one day earlier (Jan 16 at 3,404 before and Jan 15 at 3,269 in the new model).

Daily deaths 7-day average 1/15/21

I note that the pre-Christmas model nailed both the timing and height of the third wave peak. The trailing 7-day average of deaths has been dropping for three days, now.

Because of the likelihood of underreporting cases during the Christmas and New Year's holidays, I used an upper-bound least-squares scoring function for today's curve fitting run. Basically the actual squared error was used if the model under-predicted the data point, but only 25% of the distance was used if the model over-predicted the data point. This modification was only used for data after November 16th, to allow for missing data from Thanksgiving on.

This change in scoring causes the fitted curve to skim the top of the data points, and to ignore the valleys.

The R-squared measures are still reported using standard least-squares (in this case 0.9682 for cases and 0.9633 for deaths).

The CDC is reporting that 10,595,866 people have received their first dose of the vaccine and 1,610,524 people have received two doses (CDC vaccine tracker).

My hope is that as more people are vaccinated, the model curves will start to over-predict and the skew will reduce in future model runs.

As always, please note that this is a retrospective data model, and can neither predict nor prohibit any future waves of virus outbreak. See the about page for data sources, disclosures, and disclaimers.

Graph update, January 13, 2021, US cases and deaths

Daily new cases 7-day average 1/12/21

Added US cases and US deaths using data posted today for 1/12/2021.

Cases (left) are above the trend line from the pre-Christmas model run, but they are steady for the last 5 days (using a 7-day trailing average).

Deaths (right) are higher, but are also growing less rapidly, and have just reached the pre-Christmas model trend line.

Daily deaths 7-day average 1/12/21

Graph update, January 9, 2021, US cases and deaths, vaccinations

Daily new cases 7-day average 1/08/21

Added US cases and US deaths using data posted today for 1/08/2021.

Cases (left) are above the trend line from the pre-Christmas model run. That probably indicates a new acceleration in the spread of the virus.

Deaths (right) are higher, but are not yet above the pre-Christmas model trend line.

Daily deaths 7-day average 1/08/21

I will run a new model at the end of next week, but there's still not enough good data to offset the numbers that were clearly depressed by the Christmas and New Year's holidays.

The CDC reports that 6,688,231 have received at least one dose of the vaccine so far (CDC Covid Data Tracker > vaccines).

I will be curious to see how vaccinating the population will affect these curves. My expectation is that we'll start to see decreasing skew for the third wave, and a much smaller or non-existant fourth wave.

As always, nothing predicts or prevents a new wave.

Graph update, January 7, 2021, US cases and deaths, CFR

Added US cases and US deaths using data posted today for 1/06/2021.

Cases are back to the trend line established before the Christmas holiday. The public expectation is that cases will climb sharply from here, assuming lots of family interaction over Christmas. Remember this model just fits the historical data, and has no knowledge of social interaction rates per se.

As always, nothing predicts or prevents a new wave.

Also added a new estimate for Case Fatality Rate (CFR) showing 1.224% for the last 21 days (shown at right). That corresponds to one death per 81 cases, an improvement over 1 death in 70 cases last month.

Estimated US Case Fatality Rate (CFR)

Also added charts showing the top six (6) states compared by daily deaths per million, and a heat map showing new confirmed cases by state the last 21 days. See the graphs page for explanation.

Deaths per million per day, top 6 states
Detail chart showing Fall 2020

Deaths per million per day, top 6 states
Same chart but showing the whole year 2020

New cases last 21 days, by state
not normalized by population
US CFR

Graph update, December 31, 2020, new outlier in Turkey

Update the case and death charts for the United States, the World ex-China, and I added a new pair of charts for the World ex-China ex-USA. See the Graphs Page

Notes: the World charts now have an outlier on 12/10/2020 from Turkey.

"Turkey has recorded nearly 1.75 million cases since the start of the pandemic, the health ministry said on Thursday, after releasing a significantly lower toll including only patients requiring medical care.

Officials had previously only published the number of coronavirus "patients" but last month returned to publishing the total number of cases." (Medical xPress, 12/10/2020).

Graph Update, December 26, 2020

The first version of the Data Groupies web site is up. The first thing to check out is the Theory page, which describes how I use the skew-normal distribution to model the cases and deaths in the coronavirus epidemic.

Facebook Posts

Before Christmas all the Data Groupies content was on Facebook. Here are all the links about coronavirus to all the posts I've made since January 29th.

January 29, 2020

February 14, 2020
February 24, 2020
February 26, 2020

March 4, 2020
March 14, 2020
March 17, 2020
March 17, 2020
March 22, 2020
March 28, 2020

April 3, 2020
April 8, 2020
April 14, 2020
April 22, 2020
April 22, 2020

May 2, 2020
May 5, 2020
May 9, 2020
May 14, 2020
May 16, 2020, part 1 of 3
May 16, 2020, part 2 of 3
May 16, 2020, part 3 of 3
May 17, 2020
May 19, 2020
May 22, 2020
May 22, 2020
May 29, 2020

June 6, 2020
June 13, 2020
June 22, 2020
June 27, 2020

July 4, 2020
July 11, 2020
July 18, 2020
July 25, 2020

August 1, 2020
August 8, 2020
August 9, 2020
August 29, 2020
August 30, 2020
August 31, 2020

September 5, 2020
September 10, 2020
September 14, 2020, fitting double wave
September 19, 2020
September 27, 2020
September 28, 2020
September 28, 2020

October 3, 2020
October 3, 2020
October 4, 2020
October 4, 2020
October 7, 2020
October 12, 2020
October 17, 2020
October 18, 2020
October 18, 2020
October 25, 2020
October 27, 2020
October 31, 2020
October 31, 2020

November 4, 2020
November 8, 2020
November 14, 2020
November 28, 2020

December 9, 2020
December 11, 2020
December 12, 2020
December 13, 2020
December 19, 2020
December 24, 2020