[conspire] 100,000, bc(1), & miscellaneous

Thu May 28 01:32:01 PDT 2020

If you are trying to see how many days to 1,000,00 deaths (assuming
exponential)
there is an easier way.

Using your notation: b is the exponential rate per day.
And now after 111 days we have 100,000 deaths.

so
(*)  b^111 = 100,000

100,000 is 10^5 and 1,000,000 is 10^6

Raise both sides of equation (*) to 6/5th power

b^(111*6/5) = 1,000,000

111*6/5 = 133.2

133 days.  So 3 weeks from today (June 18th)

Thomas

---------- Forwarded message ----------
> From: Michael Paoli <Michael.Paoli at cal.berkeley.edu>
> To: conspire at linuxmafia.com
> Cc:
> Bcc:
> Date: Thu, 28 May 2020 00:41:34 -0700
> Subject: [conspire] 100,000, bc(1), & miscellaneous
> 2020-05-27 100,000 US dead; less than 4 months ago we were at 1 US dead
> Rate may not have been consistent (variations in
> lockdown/shelter-in-place/reopening/...), and theoretically more
> logistic than exponential, but still far from saturation / herd
> immunity, so if we approximate using exponential,
> and do gross approximation of consistent rate, we have ...
>
> First US confirmed fatality ...
> was earlier thought to be 2020-02-28 in Seattle, WA, but (autopsies)
> later confirmed first was:
> 2020-02-06 in in Santa Clara County, CA
>
> So ...
> $ echo \($(TZ=GMT0 date +%s -d '2020-05-27T12:00:00+00:00') - \
> $(TZ=GMT0 date +%s -d '2020-02-06T12:00:00+00:00')\)/3600/24 |
> bc -l
> 111.00000000000000000000
> 111 days ago - or
> 111 days from from 2020-02-06 (1) to 2020-05-27 (100,000)
>
> So, ... we'll make the (gross) approximation of consistent
> exponential growth rate throughout.
>
> So, then what per-day growth rate?
> b^111=100,000 = e^(111*ln(b))
> b=100,000^(1/111) (or 111th root of 100,000) = e^((1/111)*ln(100,000))
>   ~=
> 1.10928986489522280772, e.g.: $ echo 'e((1/111)*l(100000))' | bc -l
> Remember, exponentials get big/small fast - unless the base is 1.
> if the base is >1 they grow, if the base is <1 they get shrink.
> So, at ~1.11, we go from 1 to 100,000 in 111 days
>
> If we presume same exponential, when would we hit 1,000,000?
> b^x=1,000,000
> x=log-base-b of 1,000,000 = ln(1,000,000)/ln(b) =
> ln(1,000,000)/ln(e^((1/111)*ln(100,000))) =
> ln(1,000,000)/((1/111)*ln(100,000)) ~=
> 133.20000000000000014454, e.g.:
> $ echo 'l(1000000)/((1/111)*l(100000))' | bc -l
> So, 10x growth in ~22 (133-111) days
> $ TZ=GMT0 date -I -d '2020-02-06T12:00:00+00:00 + 133 days'
> So,
> 2020-06-18 for 1,000,000 (gross approximation of continuous unchanged
> exponential)
>
> And, when for 10,000,000, and 100,000,000 ...?
> The model would break down by/before then, logistic curve would be
> much more appropriate fit (and even then, if we do gross approximation
> of no factors changing along the way).
> Exponential doesn't take into account immune / no longer vulnerable
> to infection or any limits on pool available to be infected, whereas
> logistic does.
> Essentially no longer vulnerable to infection (at least to sufficient
> model approximation) happens one of two ways:
> o immune (vaccine, infected and recovered ... "immune" not necessarily
>    permanent immunity, but "long enough" to cover the period under
>    examination).
> o deceased (not the best way to get removed from pool of vulnerable to
>    infection)
>
> So, if we've got ~10x growth in ~22 days, and again, our crude gross
> approximation exponential modeling (presuming consistent rate of
> spread), how many days for, e.g. 2x (doubling), 10x, 100x, 1024x,
> ...?
> Let's call x our multiplier (e.g. for 2x, 10x, etc. growth factor).
> Let's use d for number of days.
>  From our earlier, we have base (call it b) of exponent,
> for our daily growth factor (b^1 = our 1 day multiplier)
> b=100,000^(1/111) (or 111th root of 100,000) = e^((1/111)*ln(100,000))
>   ~=
> 1.10928986489522280772, e.g.: $ echo 'e((1/111)*l(100000))' | bc -l
> We'll use (relatively) standard notation, and convert to bc(1)
> syntax/format:
> b=e((1/111)*l(100000))
> b^d=x
> d=log-base-b of x
> d=ln(x)/ln(b)
> d=l(x)/l(b)
> And, wee bit 'o shell:
> b='e((1/111)*l(100000))'
> for x in 2 10 100 1024 1048576; do
>    echo -n "x=$x d="
>    echo "l($x)/l($b)" | bc -l
> done
> And we have:
> x=2 d=6.68286590374038254157
> x=10 d=22.20000000000000002616
> x=100 d=44.40000000000000005242
> x=1024 d=66.82865903740382541642
> x=1048576 d=133.65731807480765083285
>
> "Of course" ... "reality" ... the base continually changes, and may be
> quite region/locality/country/... specific, altered by factors such as:
> o physical/social distancing
> o shelter-in-place / lockdown / reopenings / social and other
>    gatherings
> o hygiene and other relevant practices, especially as regards
>    SARS-CoV-2 --> COVID-19 infection/spread pathways
> Anyway, the more and longer the base can be pushed and further pushed
> below 1, the quicker the infections get to 0.  Until then, it
> spreads/grows, notwithstanding immunity(/deaths) numbers becoming so
> large the model starts to substantially differ from exponential and
> better approximates logistic.
>
> Bit more on bc(1) ... want a handy power (x^y) function in bc(1)?
> bc(1) goes relatively minimal in some ways, it give one the basic
> necessary functions needed to define most other functions/capabilities
> that may be needed.  E.g. it has natural logarithm/exponentiation, but
> no other bases, as those can be derived from natural base e.
> Likewise trig, it has sine and cosine functions, but no tangent
> function, as tangent can be calculated from sine and cosine.
> So, arbitrary base to arbitrary exponent:
> x^y = e^(y*ln(x))
> So, we can define a function, call it p (for power) in bc:
> define p(x,y){
>    e(y*l(x))
> }
> e.g.:
> $ bc -lq
> define p(x,y){
>    return e(y*l(x))
> }
> p(2,10)
> 1023.99999999999999992594
> p(144,1/2)
> 11.99999999999999999988
> quit
> $
> bc(1) also well handles relatively arbitrary precision, and other
> number bases.  E.g.:
> $ bc -lq
> scale=66
> 4*a(1)
> 3.141592653589793238462643383279502884197169399375105820974944592304
> obase=16
> 4*a(1)
> 3.243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA3
> ibase=2
> 110001011001001101101000010110100110000000000001011100000010011100000000
> C593685A6001702700
> quit
> $
>
> references/excerpts:
> http://linuxmafia.com/pipermail/conspire/2020-May/010792.html
> http://linuxmafia.com/pipermail/conspire/2020-March/010315.html
> bc(1)
> sh(1)
> https://www.youtube.com/playlist?list=PLIOESHELJOCnqaaUqq7AzTOGp-k-2KzKY
>
> And now for slightly cheerier note(s):
> France Musique Le Boléro de Ravel par l'Orchestre national de France en
> https://youtu.be/Sj4pE_bgRQI
>
>
>
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>
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