Bite Point

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.}

Using the Natural Logarithm to Forecast COVID19 Progression

In our last post, we explored how to predict the COVID-19 epidemic progression with the irrational, transcendental and may I say supernatural number e.

e is the base rate of growth shared by all continuously growing processes. e merges growth rate and time. If we know the growth rate of a process and we wish to know how it will grow in a certain specific period of time, e computes the answer.

What if we want to know the time? What if we have a certain amount in growth in mind, and we want to know how long it will take to reach that growth?

Ah, the answer is as beautiful as e itself. e has an opposite, a mirror. This is the natural logarithm, ln.

In other words,
e^x lets us plug in time and get growth.
ln(x) lets us plug in growth and get the time it would take.

Let us move right into COVID-19 epidemic forecasting examples. As we saw in my last post, over the course of March 2020, the epidemic grew at different rates in the U.S., depending on the day. I will present these rates to you and ask the two questions:

(1) How long would it take, at the given rate, for the number of infected patients to double?
(2) How long would it take, at the given rate, for the number of infected patients to increase ten times?

We will answer these two questions for each rate with the natural log.

Here are the epidemic growth rates we've noted in March, minus the early outliers, and the time calculations.
On March 31 there were 180,000 U.S. COVID-19 cases. For each rate, we will ask, how long will it take to reach 360,000 cases, and how long will it take to reach 1,800,000 cases, assuming that growth rate continued?

2% Growth Rate per Day

2x, to 360,000 cases:
ln(2) = 0.69 ...
0.69/.02 = 34 days
10x, to 1,800,000 cases:
ln(10) = 2.30 ...
2.30/.02 = 115 days

4% Growth Rate per Day

2x, to 360,000 cases:
ln(2) = 0.69 ...
0.69/.04 = 17 days
10x, to 1,800,000 cases:
ln(10) = 2.30 ...
2.30/.04 = 57 days

8% Growth Rate per Day

2x, to 360,000 cases:
ln(2) = 0.69 ...
0.69/.08 = 9 days
10x, to 1,800,000 cases:
ln(10) = 2.30 ...
2.30/.08 = 29 days

20% Growth Rate per Day

2x, to 360,000 cases:
ln(2) = 0.69 ...
0.69/.20 = 3.5 days
10x, to 1,800,000 cases:
ln(10) = 2.30 ...
2.30/.20 = 11.5 days

22% Growth Rate per Day

2x, to 360,000 cases:
ln(2) = 0.69 ...
0.69/.22 = 3 days
10x, to 1,800,000 cases:
ln(10) = 2.30 ...
2.30/.22 = 10 days

And so. We end up with case-doubling times that range from 17 days down all the way to just 3 days, depending upon human behavior.
And we end up with ten-fold case increase times that range from 57 days down all the way to just 10 days, depending upon human behavior.

I find it chilling that the March 30--March 31 interval returned us to a 22% growth rate of new COVID-19 infections. We need to (if we even can) return that to 2% and less, as these ln forecasts show.

Wash Hands, Social Distancing. They matter even more than you think.