For the corona virus, I've changed my estimate from one person in ten who goes into the hospital, to one person in 20 who gets ill ends up in hospital. The other 19 either get the virus, and have no symptoms, or maybe minor symptoms, or more, but not enough to end up in hospital.

Rather than a more difficult Gaussian curve, I found this one discussed somewhere:

tanh( at + b)

I saw a math forum where someone had a Gaussian problem, but it was too hard to solved (yup it is) and wanted a similar function, but easier to solve.

A hyperbolic tangent function. "a" and "b" are constants, t is time (in days) and the tanh() functions is -1 far far in the past, and around at+b =0 it gets larger and smoothly crosses over to +1 as time gets larter and larger. This function gets larger quickly in a brief time and has a similar quality to a spreading infection. A slight adjustment to this function:

EQN. 1: F(t) = ( 1 + tanh(at + b) ) / 2

F(t) is the fraction of the population exposed to a disease. It starts off near at much earlier times, and much much later, at large t everyone has been exposed.

The crossover area from where few have been exposed, then all the activity, then after that nearly everyone has been exposed is near (at + b) = 0. The width of the crossover is roughly 1/a days. The peak of the spreading is around when (at +b) = 0

The F(t) function has the right qualities for a spreading infection, nothing much for a long time, then it spreads like crazy, then it has finished spreading.

Making yesterday t2= 0 and the day before yesterday t1= -1

one can first solve for "b":

EQN 2: F(t2) = (1 + tanh( a*t2 + b) ) /2

We know that F(t2) was 502,876 cases in the USA total so far. Cases that made it to hospital. I'm assuming the real number is 20 times that. So 19 folks either have no symptoms, or have symptoms but not bad enough to end up in hospital. Population of the USA is about 327,000,000

So the fraction of people exposed in the USA up until yesterday:

EQN 3: F(t2) = 20 * (502,876) / 327,000,000 = 0.031 = (1 + tanh(b) )/ 2

That 0.031 number would mean about 3.1% of the US population has been exposed. Not precise, but a good guess.

A little manipulation and:

EQN 4: tanh(b) = -0.938

and

b = -1.72

Since "b" is negative, it means the peak in cases is coming in the future. With only 3.1% of the population exposed so far, it makes sense that the peak is still coming.

Now to figure out "a". If you take that equation, and take its derivative with respect to time (something from calculus):

EQN 5: d F(t) / dt = d (( 1 + tanh(at + b) ) / 2 ) / dt

This is the time rate of change of the quantity, how much does it change from day to day

For example, if F(t) is how much of the liquid is in bottle, d F(t) / dt is the rate that it is filling up, the rate of change of the quantity.

If you have had a calculus class then you know, or you can figure this out:

EQN 6: d F(t) / dt = a* ( 1 - tanh(at+b)*tanh(at + b) ) / 2

The rate of change on the left is known from 469,124 on April 9, to 502,876 on April 10, subtracting those, up 33,752 in a day. Assuming 20 times that is how many people got infected in one day (remember I assume only 1 in 20 made it into the hospital statistics)

So probably 20*33,752 infections up in one day, and the population of the USA is 327 million, so the rate of change of the fraction of people who are exposed today is:

EQN 7: d F(t) / dt = 20 * 33,752 / 327,000,000 = 0.002 = a* ( 1 + tanh(at+b)*tanh(at + b) ) / 2

Then plunking the value of tanh(at+b) today, which is in equation 4, it is -0.938

0.002 = a * ( 1 + (-0.938)*(-0.938) ) /2 = a * 0.06

EQN 8: a = 0.033

So what the heck does "a" mean, or this 0.033 imply?

So since the formula has (at + b), 1/a = 1/0.033 = 30 is the time scale or width of the peak in days, which is about 30 days. So you'll be close to that peak value for about 30 days, say getting "close" 15 days before, reaching peak, then coming down, but still being high for the next 15 days. This is not a surprising number ... people who recover from the corona virus do take a few weeks to recover and be out of hospital. 15 days, two weeks or so, yes a very realistic number.

Now to use the values of "a" and "b" to figure out when the peak itself is, and that is when (at + b) is zero

EQN 9: days to peak = -b/a = 1.72 / 0.033 = 52.12

OK Peak number of cases in the USA in 52 days, pretty much the end of May. The peak is about 30 days wide so high number of cases from mid May (15 days before peak) to Mid June (15 days after peak).

- - - - - - -

Not every area is impacted at exactly the same late May period, that is just the high point for the country as a whole.

- - - - - - -

I looked at the only two numbers I could find for New York City .... I really wanted New York State, .. maybe I'll find them, but I lost the link. Anyway I found two numbers of NYC, 60,850 on April 4, and 64,995 on April 5. Population of NYC 9,400,000

So still growing, and peak in the future. Now the good news for New York City, from those two data points, fitting that curve gives:

a = 0.071

b = -0.85

So width of the peak for New York City is about 1/a days, or 14 days. The peak is in the future - (a) / (b) days, which is about 12 days from April 5, so peak around April 17. So this week. The 14 day width of the peak means .... about a week to the peak 17, and a week on the other side. So about April 24 and we will be where we are now, but the number in hospital going down.

So yes, this is the sort of information coming from Gov. Cuomo , ... and I agree !!!

Edited by: -JAMES- at: 4/12/2020 (02:27)
James

Alberta, Canada

All time highest weight : 217 pounds

Starting weight : 195.0 pounds (June 7, 2012)

Final weight : 168.2 pounds (July 23, 2013)