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

Thu May 28 17:58:10 PDT 2020

Or even simpler:
111 days from 1=10^0 to 100,000=10^5=10*10*10*10*10
So, each 10x would be 111/5 days, or about 22 days.
So, from 100,000 to 1,000,000, just add 22 days:
2020-06-18 = 2020-05-27 + 22 days

> From: "tom r lopes" <tomrlopes at gmail.com>
> Subject: Re: [conspire] 100,000, bc(1), & miscellaneous
> Date: Thu, 28 May 2020 01:32:01 -0700

> 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