Saturday, April 4, 2020

On the Covid-19 Pandemic

Update 7-14-2020 Well,  we opened up too quickly,  with too many people behaving like there is no disease problem.  There is no reason why businesses and other things cannot be open,  as long as careful attention is paid to stopping disease transmission.  But too many people are either too stupid to understand the science,  or too incapable of critical thinking not to be deceived by internet lies and conspiracy theories.  The worst of which keep coming from our president and his ardent supporters,  instead of actual leadership in a crisis.  November is coming -----

Update 4-10-2020:  see modeling analysis appended to the end of this article.

Update 4-11-2020:  see also recommendations for how and when to end the quarantines appended below.

Update 4-23-2020:  added second model;  see update appended at end below.

Final Update 5-2-2020:  evaluation of second model appended below.

Update 4-11-2020:  a version of this basic article appeared as a board-of-contributors article in the Waco "Tribune-Herald" today.

The current pandemic is a disease about which we know little,  for which we have no vaccine,  and for which we have no real treatments.  After this is over,  we will know more,  but for now,  the only thing we can do is to use the same thing we have used for centuries:  quarantining at one level or another,  to slow its spread.  Calling it "social distancing" makes no difference,  it is still a simple quarantine.

Here is what we do know,  as of this writing,  learned the hard way as the epidemic sickens and kills people.  It seems similar to,  but not the same as,  the 1918 "Spanish Flu" pandemic.  We have not seen this dangerous a disease since then.  It is a once-in-a-century event.

Covid-19 seems to be at least as contagious as,  and perhaps more contagious than,  the 1918 flu.  It seems to have a similar death rate (the number who die compared to the number thought to be infected),  which is somewhere around 10 to 20 times higher than ordinary influenzas.  Those are seriously-dangerous characteristics.

There seems to be another unusual characteristic that combines with the other two to make Covid-19 a truly dangerous threat.  It seems to be more generally spread by people showing no symptoms,  than by people who are just getting sick and beginning to run a fever. 

That makes all of us potential "Typhoid Mary" carriers of the disease.  It also makes taking temperature rather useless as a screening tool to determine who might be infected,  and who might not be. Without massively-available testing,  one must presume that all other persons are contagious,  which argues for using stricter levels of quarantine.

So far,  it is thought that the Covid-19 virus is spread within the moisture droplets ejected by sneezing or coughing,  or even by talking.  5 minutes talking spews the same droplet numbers and size distribution as one cough.  A sneeze just spews a lot more.  The Covid-19 virus does not seem to be able to remain airborne outside of those droplets,  the way a chickenpox or measles virus does.

Masks vary in their effectiveness against particle sizes.  It is hard to breathe through a mask that stops particles the size of a large bacterium.  No mask stops a virus particle.  But even a simple cloth bandana will stop most of the moisture droplets from coughing or sneezing,  as does about 6 feet of space (the droplets quickly fall to the floor).  See Figure 1 at end of article.  

What that means is that the new CDC recommendation to wear masks in public is not to prevent the infection of the mask wearer,  but to stop the mask wearer from infecting others.  It would protect the wearer only when someone got right in their face to sneeze,  cough,  or talk at very close range.  The 6 foot distance rule already stops that effect.

The recommendation to wear a mask is actually based on this uncomfortable reality:  that many seemingly-well people are actually infected,  just not showing symptoms,  and are walking around spreading the disease.  This "Typhoid Mary" effect is not common,  but may well be the case with this particular virus.

As already indicated,  a simple bandana will work.  Leave the real surgical masks for the health professionals.  They need them.  We ordinary citizens do not.  When you go to the store,  wear a bandana or a home-made mask.  That's all you need,  to protect others.  The 6 foot rule protects you.

And for Heaven's sake,  quit panic-buying toilet paper and other supplies!  There is plenty being made,  and plenty in the supply chain,  for everybody's needs.  The shelves are bare because so many folks panicked and took far more than their share (their "share" being what they really need).  Shame on you!

Predictions about this pandemic are still guesswork.  The CDC figures show a peak of 100,000 or more deaths in about another month.  Maybe a month or two after that,  it will be more-or-less over,  and we can safely re-open our lives and businesses.  But that's a guess,  and it will likely change.  See Figure 2 at end of article.  

Had we started with the quarantining a month or two sooner than we did,  the death totals would have been lower,  but the time to the end of this gets longer.  Time spent shut down costs all of us money and jobs. 

That is the inevitable tradeoff:  lives versus money. And it is quite the serious effect,  make no bones about that.  Job losses are already beginning to resemble those of the Great Depression of the 1930's.

But almost all of your mothers and churches taught you to value lives over money,  that valuing money over lives was evil!  Think about that,  when you vote.  Not just next time,  but from now on.

 Figure 1 -- Data on Particles Versus Filter Pore Sizes
Figure 2 -- How Quarantining Works,  and What It Does

Update 4-5-2020:                                                                       

The best numbers I have seen on Dr. Fauci’s curves and predictions,  as of end-of-March,  say that with “social distancing” quarantining in place,  US deaths may accumulate to 100,000 to 240,000 people lost.  That death rate trend should peak out somewhere in early May.  Without the quarantining measures,  something like 2 million deaths would be expected.  Maybe more.

Just to “calibrate” the threat of this thing,  the US lost 407,300 soldiers in WW2,  for a 1939 population of 131 million.  That’s 0.31% of the population dead from war

With Covid-19 at a population of 325 million today,  it is 0.03-0.07% of the population dead with quarantining,  and something like about 0.6% of the population dead without quarantining.  You don’t credibly compare this pandemic to yearly traffic deaths or the H1N1 epidemic.  You compare it to the casualties of a major world war.

Based on the numbers published in the newspaper,  the US death rate appears to be near 2% of known cases of infection.  For Dr. Fauci’s predicted death accumulation numbers,  that corresponds to something like 5 to 12 million accumulated known infections.  That’s about 1.5-3.7% of the US population infected,  and 0.03-0.07% of the population dying of it.  These numbers are clouded by uncertainty,  because without widespread testing,  we cannot know the real number of infections.

Using the rough-estimate 2 million deaths for no quarantining,  and the same 2% death rate of those infected,  the accumulated infections would be about 100 million,  which is 31% of the US population.  Quarantining is thus very,  very important,  by about a factor of 10 on the total infections,  and on total deaths.  So,  those who deny or ridicule the risk are dead wrong,  if you will forgive my choice of words.

According to Wikipedia,  the 1918 Spanish flu killed something like 1-6% of the world population.  The same article gives these statistics for the US:  about 28% of the population became infected,  and about 1.7% of those infected died of it. 

The death rate among those infected is quite comparable between Covid-19 and the 1918 flu.  The number of expected Covid-19 infections is lower,  probably because of our quarantining efforts,  despite our delay getting started.  The estimate of infections without quarantining is actually quite comparable to 1918. 

The Covid-19 pandemic really is an event comparable to the 1918 flu pandemic.  We have not seen such a thing in 102 years.

This is quite serious,  so I reiterate the recommendations I gave above:

#1. Stay away from crowds and gatherings,  and when you must go out,  stay at least 6 feet apart (which is what protects you from infection,  not any mask you might wear).

#2. If you must go out where 6 feet apart is not feasible,  wear a bandana or home-made mask to protect others in case you are unknowingly contagious (save the real masks for the health care folks who need them).

               Corollary:  if you are sick in any way,  DO NOT GO OUT.

#3. Stop panic-buying and hoarding supplies,  there is no need for that.

#4. Watch what your public leaders do (not what they say) to judge whether they values lives over money,  or not.  Then stop re-electing those with the wrong priorities. 


Update 4-10-2020:

Pulsed events like the daily infection rate for Covid-19 are actually well-modeled by the mathematics of something called the “logistic distribution”,  which is similar to,  but numerically a little different from,  the “normal distribution” in statistics.  The daily rate corresponds to a pulse function f,  and the accumulated total follows a S-curve shape corresponding to the F-function.  F is the integral of f (which means f is the derivative of F).  F is defined from 0 to 1,  so you have to scale it to apply it in the real world.   The following mathematics were obtained from Wikipedia under the article name “Logistic Distribution”.

Derivative (like a probability density):
f(x, µ, s) = exp(-(x - µ)/s) / s[1 + exp(-(x - µ)/s)]2

Cumulative function (S-shaped accumulation curve):
F(x, µ, s) = 1/[1 + exp(-(x - µ)/s)]

Variable definitions:
x is the independent variable,  usually time in applications,  a real number,  from – infinity to infinity
µ is the location variable,  center of the f-distribution,  and location of half the total accumulation
s is the scale variable > 0,  a measure of the distribution width;  a bigger s is a flatter and longer pulse
F varies from 0 to 1;  you have to scale it by the total accumulated T

The shape of the pulse function is shown in Figure I for multiple values of the 3 model parameters.  The smaller s is,  the “peakier” the pulse.  The location parameter µ merely moves the shape left or right on the graph.  The larger T is,  the taller the pulse,  as a direct scale factor.

 Figure I – Shape and Characteristics of the Unscaled Pulse Function

The shape of the unscaled S-curve accumulation function is shown in Figure II.  The smaller s is,  the steeper and shorter-in-time the S-curve shape is.  The location parameter µ merely shifts the shape left or right,  same as with the pulse function.  The T factor merely scales the shape from its 0-to-1 variation to whatever numbers your data are.  

 Figure II – Shape and Characteristics of the Unscaled Accumulation Function

To model a pulse of something that eventually totals to T instead of 1,  scale up both the cumulative and the derivative with the factor T.  Thus:
Derivative (pulse function)
Pulse rate vs time = T f(x, µ, s) = T exp(-(x - µ)/s) / s[1 + exp(-(x - µ)/s)]2 
Cumulative (S-curve function)
Accumulated total vs time = T F(x, µ, s) = T/[1 + exp(-(x - µ)/s)]

At the peak of the pulse in your real-world data,  there is a max rate with time,  and a location in time,  and in the accumulation function at that same location,  the total is exactly half the eventual total.   The model parameters can be calculated quite easily from those three pieces of data,  if you can confirm you have actually seen the max of the pulse.  Here is how you do that:

Where in x it peaks is µ
T = 2 times the accumulated total-at-peak
s = T/(4 times the peak rate)

I went on the CDC website 4-9-2020 and retrieved their posted Covid-19 Infections data as of 4-8-2020.  They had posted both accumulated cases,  which I used,  and a daily case rate vs time from a presumed infection date,  which did not share the same time scale.  I did not use the daily case data because of differing assumptions,  and because they said in no uncertain terms that the final week or so was clouded by as-yet unreported data.  Here in Figure A is the CDC’s own accumulated case data (as of 8 April 2020) vs time from 12 January 2020,  as plotted from the spreadsheet in which I put it.

 Figure A – CDC Data for U.S. Accumulated Cases of Covid-19 Infections as of 8 April 2020

The accumulated data are simply summed from one day to the next,  from the daily reported infections data.  So,  I simply recreated their daily case rate data by differencing the accumulated data from one day to the next.  That way everything shares the same sources and assumptions.  See Figure B.

 Figure B – Recreated CDC U.S. Daily Case Rate Data as of 8 April 2020

Early on,  the numbers are small,  and even a large-percentage inherent scatter is not significant.  Later,  as the numbers climb,  a large-percentage inherent scatter becomes very significant,  actually to the point of obscuring the trend.  There would seem to be a suggestion of the daily case rate bending over a peak value,  but picking a number for the peak point would be difficult indeed.  This some sort of averaging is needed to actually “see” the peak in the data well enough to quantify it.  I tried the moving-average technique,  with a 2-day average,  a 5-day average,  and a 3-day average,  as seen in Figure C.

 Figure C – Comparison of Raw U.S. Data Versus 3 Different Moving Averages

The 2-day moving average did not fully suppress the up-down scatter variation,  but showed very little “lag” in its trend behind the raw data.  The 5-day average suppressed the scatter,  but lags the data trend by about 4-5 days,  which is too much.  The 3-day average showed the peaking behavior the clearest,  and with about a 2-3 day lag behind the actual data trend.

The daily case rate seems to peak at about 32,400 cases per day at day 86 in the 3-day moving average data.  Its rise trend is seemingly 2 days too late in terms of the rise before peaking.  So revise the peak date to day 84,  keep the 32,400 cases,  and read the accumulated case data at day 84 (not 86) for about 331,000 accumulated cases at the peak point.  The corresponding T,  s,  and µ data for the model are T = 662,000 final total cases,  a scale parameter s = 5.108 days,  and a location parameter µ = 84 days (on the graph time scale). 

That model matches the accumulated case data quite well,  as shown in Figure D.   If you do not compensate for the lag of the moving average technique,  and use the wrong µ,  your model fails to match the initial upturn (near day 60 to 70).

The same choices of parameters do a good job matching the daily case rate data,  as long as you compare it to the moving average that reveals the peak.  Like with the accumulated data,  if you do not compensate for the lag of the moving average,  then the model fails to match the initial upturn in the data (again near day 60-70).  This is shown in Figure E.  

 Figure D – Comparison of Model and Data for U.S. Accumulated Cases

Figure E – Comparison of Model and Data for U.S. Daily Case Rates

This case study illustrates the inherent difficulty in choosing the “right” model parameters T,  s,  and µ unless you have already reached the peaking daily case rate in your data.  This is due to the mismatch around the initial upturn if you do not compensate for your moving average lag.  There are multiple combinations of the parameters that might match up the tail of the daily case rate distribution over time,  but most of these will not correctly predict the peak.  And the inherent scatter problem forces you to use a moving average to “see” the peak,  which inherently introduces the lag that causes the error,  if not compensated.

All that being said,  with a peak you can “see” and quantify,  this modeling technique becomes very accurate and very powerful for pulsed events like epidemics,  as the plots above indicate.  As a nationwide average,  the U.S. Covid-19 epidemic seems to have peaked just about 5 April 2020 (day 84 in the plots),  at about 32,400 cases per day reported,  and an expectation of being “over” by about 11 May at ~100 cases/day at the earliest,  or at worst about June 1 with 2 cases /day in this model,  and with about 661,000-to-662,000 total accumulated cases.

You do NOT release the quarantine restrictions until the event is actually effectively over!  Relaxing just after the peak pretty-much guarantees a second pulse of infections just about as bad,  and just as long,  as the first.  To propose doing so is a clear case of valuing money over lives,  instead of lives over money.  Valuing lives over money is what your mothers and your religious institutions taught to most of you readers!  I suggest that you use it as a criterion to judge your public officials.

A final note:  this same pulse model has been used to predict resource extraction and depletion.  The most notable example was geologist M. King Hubbert trying to predict “peak oil”.  One of his two best models came very close to predicting peak oil from data he had long before the peak actually occurred,  something very difficult at best.  Since then,  the model has diverged from reality and thus fallen into disrepute. 

Since that peak,  the development of then-unanticipated cheap fracking technology has not only made fracked oil and fracked gas available,  their simple availability has vastly increased the total recoverable resources available.  Those are very large and very fundamental changes in the assumptions underlying the formulation of any predictive model.  That’s analogous to the situation of a second wave of infections in the epidemic application. 

A new peak fracked oil/gas model would be the right thing to do to respond to these developments.  And if recovery technology improves much past the current 2-3% recovery rates,  yet a further new prediction would be warranted.  That’s just the nature of prediction models being sensitive to the assumptions underlying them.


Update 4-11-2020:

These predictive models cannot tell you when to lift the quarantine!  Period!  These exponential functions never,  ever predict when the daily case rate goes to zero.  Mathematically,  they cannot. 

Instead,  you have to watch (1) the daily infection rate field data   and (2) you have to determine from experience during the epidemic,  what the actual incubation time really is.  You CANNOT lift the quarantine,  until the daily infection rate has been zero for an interval longer than the observed incubation time.  There is NO WAY around that requirement!  To do otherwise is to value money over lives,  an evil according to the morality you were taught as a child.

If the observed incubation time is 7 days,  then 8 or 9 or 10 days of zero infections ought to do the trick.  If the observed incubation time is 10 days,  then 11 or 12 or 13 days of zero infections ought to do it.  If the observed incubation time is 14 days,  then something like 15 or 16 or 17 days ought to do.  There is simply no way around such a criterion,  if you intend to be moral and value lives above money!

A caveat:  this needs to be on a regional basis,  not nationwide.  That is because the infection pulses did not all start at the same time around the country.  They will not last the same interval,  nor end,  at the same time.  A national edict to end the quarantine by this or that date is just wrong technically,  and demonstrably immoral by the criterion I have offered.  Regions can de-quarantine,  but travel between them should stay restricted until the last region is past the crisis. 


Update 4-23-2020:

As time went by,  the first model I set up looks ever poorer.  It was clearly not “right” in the sense that the predicted ultimate total accumulated cases of infection were quite demonstrably wrong by 4-21-2020,  using the published CDC data for the US.  A lot of that can be attributed to the wild scatter in daily rates about the peak point,  making it hard to quantify that peak point.  This is shown in Figure I.

 Figure I – Increasingly Erroneous First Model

So,  I repeated the process as of 4-21-2020,  obtaining a second model.  It has about the same peak daily rate,  just a bit lower and later in time,  with a larger accumulated case number at that later time.  This led to a larger scale factor with a broader peak,  which matched the peak data quite well. 

However,  out in the initial “tail” the match is not as good,  as can be seen in both the accumulated and daily rate data in Figure II.  And,  the time interval is longer.  We will see as time goes by whether this is really significantly better than the first model.  I think it is,  but I’d also bet it will be “wrong”,  too.

Figure II – Second Model at its Inception

This just emphasizes the point I tried to make in the article:  that these models are quite uncertain even if you have peak data.  If you don’t,  this is even more uncertain.  Exact predictions from the model are not the point of doing this.  The trend shapes and behavior are the real goal. 

This modeling process gives you only a crude idea what the time interval will be from peak daily rate to no more new infections.  That would be half the width of the predicted pulse of daily rate data.  It’s only crude,  but it’s far better than nothing.  My first model’s pulse was about 70 days wide.  The second model’s pulse is about 130 days wide,  ending well into June.   

It would appear from my experience here that predicting the ultimate infection total (where the accumulated curve levels out) is even more problematical than modeling peak behavior.  That seems significantly more uncertain than identifying the peak in the daily rate curve. 

Clearly,  ultimately,  one must live through the epidemic event,  and just use the actual data after it is all over,  if one wants accurate statistics.  You won’t get that from this modeling activity.

But,  the other thing I want to point out is that the peak curve shape in the daily rate data is symmetrical.  There is a fall-off over time after the curve peaks,  it does NOT immediately crash to zero!

This is modeling a very real effect:  after the peak,  the daily infection rates are not zero for some time interval,   meaning there are still infectious people walking around out there and spreading the disease!  Ending the quarantine during this time guarantees a resurgence,  a second wave that starts everything again from scratch.  You have to start your quarantine all over again!  And the longer that goes on,  the more jobs and money everyone loses.  There is no way around that!

This model behavior,  which matches real-world experiences,  is exactly why I say you do not end the quarantine until your daily infection rate has zeroed,  and has been zero for longer than the microbe’s incubation time.   The health professionals would agree with me,  not the politicians who want to end it too soon,  just so that so very much money is not lost.

Once again,  I submit that you should use a simple criterion to judge whether public officials have your best interests in mind:  they either value lives over moneyor they don’t.  If they don’t,  then you don’t want them making decisions for you.  Simple as that.

Where did I get that?  From the moral teachings just about all of us got from our mothers and our churches.  It’s a question of moral fitness.  That has to take precedence.  And nearly every one of you readers knows that,  somewhere deep down.

Final Update 5-2-2020:

The CDC quit updating the national database that I was using as my data source.  They said modeling communities was more appropriate than the entire nation.  And I believe that may be correct,  as the totals for states show mostly steady or still-rising daily cases,  strongly at variance with each other.  As a result,  I am no longer able to update the 4-21-2020 model posted in the previous update.  The last reliable data I have are for 4-27-2020. 

However,  as can be seen in the figure,  the daily infection rate appears to be defying the previous interpretation of a peak in daily cases about 4-13-2020.   There was a bit of a downward trend,  but in the last few days it has trended sharply upward again.  Possibly this is related to some states ending quarantine measures,  trying to reopen for business,  especially without adequate testing and contact tracing.  Or it may just be the inherent variability in the data.  There is no way to know. 

From what I have read about the 1918 flu pandemic,  here in the US there was the initial pulse of infections,  followed by two resurgence peaks in daily rates,  for a total of three.  I added that qualitatively to the figure. 

This Covid-19 disease epidemic seems similar in many ways to that earlier epidemic.  It is both highly infectious (meaning easily transmitted),  and has a higher death rate than most other flus.  The most dangerous aspect seems to be “asymptomatic carriers”,  meaning “Typhoid Marys” who have the virus,  are contagious,  but do not know it because they are not sick.  For most of you,  the mask you are asked to wear in public is to protect the public from you!  You may well have the virus and not know it. 

Multiple pulses and “Typhoid Mary” transmission may be why some authorities are now warning that the pandemic may be with us for as much as 2 years yet.  We are going to have to figure out how to get back to business while at the same time interrupting the transmission of this disease.  It would appear that we do not yet know how to do that successfully!  All that we do know is that the pre-pandemic status quo is NOT it!  So for now,  wear your mask and keep your distance,  when in public. 

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