According to the CDC there were 4,226 cases of illness in the USA due to COVID-19 on Monday, March 16th. Yesterday, the 26th, there were 85,356 cases. That’s a lot of growth in ten days!
I like to round things off and make estimates. It helps me get a handle on the size of the problem. For example I look at the above numbers and round them off to 4,000 and 80,000 because I can see right away that is a twenty-fold increase. There were twenty times more COVID-19 cases yesterday than there were last Monday. (4000 x 20 = 80000). That’s an easy idea to grasp: 20x. (If I do the actual math, subtracting 4226 from 85356 and dividing by 4226 I get approximately 19.2, so 20 is a good estimate.) But it doesn’t really tell us the growth rate, that is, how fast these cases are accumulating.
Growth is continuous, not discrete. Fortunately we have math for that. Don’t run away, I’ll keep it simple. You may remember from high school algebra—fondly, I’m sure—the lessons on logarithms. Many a math student has been crushed by logarithms. This is too bad because they are slick and have many applications.
We can estimate the continuous growth rate by taking the natural logarithm of both 85356 (approx. 11.35) and 4226 (approx. 8.35) and subtracting them. That’s an easy one. We get three (11.35 – 8.35 = 3.00).
For you mathy-types (non-mathies can skip this), the inverse of the natural logarithm, the base e, raised to the third power (e^3) is just about 20!
We divide 3.00 by the ten-day period and get 0.3 and that tells us our continuous daily growth rate. Another way to say 0.3 is thirty percent (0.3 x 100 = 30). Percents are usually easier to grasp than decimals, and in this case they are a little more revealing. Imagine getting 30% on your investments! And we are talking continuous growth, like compound interest. I’d love to get 30% interest, wouldn’t you?
But the way to really grasp the speed of continuous growth is by calculating the doubling time. How long does it take for something to double? In this case, how quickly did the number of COVID-19 cases double in size? That is, how many days did it take?
If you take the natural logarithm of two (since we are doubling) and divide by the 0.3 we got earlier then you get that answer. The natural logarithm of two (written ln 2) is approx. 0.693 and that division yields 2.31, and that’s in days, so 2.31 days. My rough approximation of the rate of new corona virus cases is that they double every 2.31 days.
That’s fast. Now this trajectory is just a small snapshot of a big data set and there are far more sophisticated ways to analyze that stuff. I just wanted to play with simple math and see what it told me. I wouldn’t take my result too seriously. There are many smart professionals out there doing the real thing, and their numbers will be accurate. What I’ve got here is just an old blackboard lesson on logarithms, updated with some contemporary numbers.
According to the data on this site, the current USA doubling rate is FOUR days. Canada is currently experiencing a two-day doubling time, for example, and both New Zealand and South Africa are at three days. Other countries like Israel and Ireland are also at four days. According to the data* both China and South Korea have “flattened the curve” and pushed their doubling times to 46 and 25 days respectively. Japan is at 14 days.
That’s good news and I hope that we can do the same here at home.
Speaking of home, be sure to stay home! Be safe, my friends.
*The source for the data is called Our World in Data and the link is: https://ourworldindata.org/
Mark C. O'Connor 
I think the numbers are skewed by the virus testing available or lack thereof. California has now tested less than half the number of people that New York has, but the number of tests have increased greatly, probably picking up a lot of people with mild symptoms that would not have been previously counted. It might be better to consider the death rate, on the assumption that the death rate remains steady. That, too, has problems, because the virus may have mutated to some degree and the care for sick patients may have (hopefully) improved. I read this morning, however, that they now have developed a test that can give a response in 5 minutes, which sure beats 3 days. Of course, there are millions and millions of tests needed, and many people, like care-givers, will probably need to be tested often. Nevertheless, the caseload is certainly growing and probably will continue to do so.
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[…] The last time I posted I worked up a rough estimate of the doubling time for corona virus cases in the US. The CDC reported 4226 cases on 3/16 and 85356 cases on 3/26 and that represented a doubling time of 2.31 days. […]
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[…] Using the same math as before, I divide 4.62 by the 23 days of growth and get 0.20 which is 20% growth. We’d love 20% growth on our investments, right? That’s still a high rate. But it has come down from the 30% we got previously. Remember we had seen that figure drop to 27%. […]
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