Using the Number e to predict COVID19 Progression

The number e is one of the most powerful, profound, mysterious, mystical things I've ever encountered. I mean in all of life. Let's use its power to explore the growth rate of the COVID-19 epidemic in the U.S.

From my very favorite explanation of e:
https://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

"e is the base rate of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit."

"e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and more. Even jagged systems that don’t grow smoothly can be approximated by e."

Let's get right to COVID-19 now. For those with interest in math, explore that chap's  delightful and intuitive exploration of e and its magical properties.

Starting on March 18th, I've taken the total number of cases of COVID-19 infection in the U.S. and divided by the previous day's total. Here are the results:
1.46    March 18th
1.50
1.40
1.25    
1.39
1.30
1.25    
1.24
1.23
1.21
1.18
1.15
1.13, on March 30th
1.14, on March 31st

Now let's convert those numbers to percentage growth:
46 % (on March 18)
50 %
40 %
25 %
39 %
30 %
25 %
24%
23 %
21 %
18 %
15 %
13 % (on March 30)
14 % (on March 31)

Now, e merges rate and time. That's one of its amazing powers.
The exponent to which we raise e signifies the rate of change and then the time passed.

We humans tend to think in terms of how we measure time. Hours, days, weeks, months, years. Let's see what the spread of COVID-19 would look like if we forecast several of those growth rates over one month--30 days of epidemic progression at the daily growth percentage that occurred on any of those days.

The formula is this: e^{x
x will be, in each case, the percent growth for the day times 30 days. We will start with one patient.
Thus, in these projections, we will see how many patients would be infected with COVID-19 one month after just one patient started off infected, at the growth rate for that day. Rounded to nearest whole number.}

^{13% growth}    e^{.13 x 30    49-fold number of cases at 30 days.}

^{18% growth}     e^{.18 x 30    221-fold number of cases at 30 days.}

^{25% growth}   e^{.25 x 30    1,808-fold number of cases at 30 days.}

^{40% growth}   e^{.40 x 30    162,755-fold number of cases at 30 days.}

^{If we were fortunate enough to slow this pandemic to 2% growth per day this month:}

^{2% growth}   e^{.02 x 30    1.8-fold number of cases at 30 days. How different than the others!}

^{We see how wildly different the month-end projections are given the different growth rates. Early in an epidemic, the growth rate is generally faster than later because at the beginning, the virus spreads far more rapidly than it could later. Put simply, that's because the virus is in a "target-rich environment." Nearly everyone is un-infected at the start and no one yet knows the dangers, or has changed their behaviors from normal.}

^{Now let's use e to come up with a useful projection. We start with today's known U.S. COVID-19 infections. (Many more of us likely have asymptomatic cases).}

^{On March 31, as I write this, there are 180,000+ known COVID-19 cases in the U.S.}

^{Starting with that, at the recent daily growth rate of 2%, by April 30 we end up with:
327,981 cases.}

^{At a 4% daily growth rate, by April 30 we end up with:
597,621 cases.}

^{At a 13% daily growth rate we recently experienced, by April 30 we end up with:
8,820,000 cases.}

^{At a 25% daily growth rate we recently experienced, by April 30 we end up with:
325,440,000 cases. Nearly all Americans.}

^{In real life, exponential virus infection curves steepen and they flatten. They usually flatten as the epidemic progresses because the number of un-infected "targets" decreases. Host behavior may change as well.}

^{Still. We now see, through the splendid magic and power of e, the importance of hand washing and social distancing. 327,981 cases by April 30 versus 325,440,000 cases by April 30. That's a 1,000-fold difference.}