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 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.
********************
Figure I – Shape and Characteristics of the Unscaled Pulse
Function
Figure II – Shape and Characteristics of the Unscaled
Accumulation Function
Figure A – CDC Data for U.S. Accumulated Cases of Covid-19
Infections as of 8 April 2020
Figure B – Recreated CDC U.S. Daily Case Rate Data as of 8
April 2020
Figure C – Comparison of Raw U.S. Data Versus 3 Different
Moving Averages
Figure D – Comparison of Model and Data for U.S. Accumulated
Cases
********************
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.
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.
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.
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.
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.
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 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.
*********************
Figure I – Increasingly Erroneous First Model
*********************
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.
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 money, or
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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