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.

Again, my numbers should not be taken seriously. Like I said, this is just an old blackboard exercise on the natural logarithm and not a proper analysis of the data on COVID-19 infections.

The new number of US infections, according to the CDC, is 140904. This represents, now, 13 days of growth. So, like before, I take the natural logarithm of 140904 and that’s approximately 11.86. I subtract the natural log of our beginning number, 4226 (8.35), and get 3.51 for the 13-day period. Divide 3.51 by 13 and you get 0.27 per day. That’s 27% growth, and that’s a bit lower than the 30% we got last time.

Sure enough, if I divide the natural logarithm of two (ln 2 = 0.693) by 0.27, just like last time, I will get an estimate of the doubling time. This calculation results in 2.56 days.

So, the doubling time has increased from 2.31 to 2.56 days. That’s good. You want to see the rate of growth slow down. The number of infections is still growing rapidly, but not as fast as before. You’ve experienced this driving your car. You hit the freeway on-ramp at 35 mph and power up to 65 mph in a few seconds so you can merge with traffic. That’s acceleration—an increase in your rate of speed. Later you make a gradual increase from 65 mph to 75 mph in order to pass someone. That increase of 10 mph happens in about the same amount of time as your increase from 35 mph to 65 mph, which was a 30 mph increase. So you are still accelerating, but the change in your speed, over the same time period, is slower.

According to this site, the doubling time in the US is now 5 days.

I reported four days the last time, so my rough math reflects the same thing!

Of course, the number of infections may not be a particularly useful or even a robust number. You need widespread testing to get a handle on the true infection rate and we don’t have widespread testing in this country so we are still a bit in the dark. Compare Iceland, for example, which has tested 3.5% of their population. We’ve tested less than 0.2% of our people! That reflects very poorly on our political leadership, of course. But more than that it means we are making public policy decisions based on incomplete information. Perhaps after we get through this mess we will be better prepared for future disease outbreaks.

If we stick to the plan—social distancing, sheltering in place, etc.—we can get out of the dangerous growth phase and get a handle on this pandemic. We can see the doubling time continue to increase (which means the infection rate will be decreasing) and give our health care system a chance to cope.

So, do your part.

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/

I was an HIV/AIDS educator for a time. I remember the phrase “it ain’t love without a glove” running around. It was a reference to condom use. Our trainers told us that if person A had unprotected sex with person B it would be like person A having sex with everyone person B ever had sex with! Unprotected sex was not just sex with someone but with someone’s entire sexual history.

It was a graphic depiction of the nature of disease transmission.

COVID-19 is of course quite a bit different than HIV. But what’s being asked of us is the same. With HIV education we asked young people to protect themselves but we also made it clear they needed to protect others! Taking precautions, communicating honestly, and abstaining from certain behaviors takes effort. But if you care about yourself and the other people in your life you will put forth the effort.

If we want to reduce the threat of this virus we have to stop interacting with people. It is the best and most effective thing we can do.

This is hard. We need each other. We need close contact with friends and family. We need a healthy society that we can work and play in. We need goods and services. But we have to delay gratification. We have to inhibit our natural spontaneity. We have to isolate ourselves, as best we can.

We used to tell our students that you had to assume your partner had a sexually transmitted disease, that way you’d certainly protect yourself. And we reminded them that they could be carrying a disease and unwittingly infect someone if they were unprotected. They didn’t want to be that person, did they?

If you assume you are infected with COVID-19 you will take precautions not to spread the virus, like proper social distancing and self-isolation. This protects other people. And guess what? It protects you, to.

Isolation and social distancing are acts of respect. You are saying to your neighbors “I want you to be safe.” And at the same time you are looking out for your own health and well-being. Who can argue with that?

Remember:

“It ain’t love without a glove.”