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,
ex 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.
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