Numbers, part 3

As of yesterday, 4/8/20, this site reported 427,460 cases of COVID-19 in the US.

Let’s do some math. On 3/16 there were 4226 cases. That’s 23 days of growth and we see a ten-fold increase. When something grows by a factor of ten (that is, 10x) we say the increase is “one order of magnitude.”

The natural logarithm of 4226 (ln 4226) is roughly 8.35 and ln 427460 is roughly 12.97 and that difference (12.97 – 8.35) is 4.62.

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%.

We now take ln 2 and divide by 0.20 to get the doubling time. The result is 3.47 or about 3-1/2 days. That beats the 2.56 we got last time! Remember that I’m just running these numbers for fun, they are not meant to be a serious analysis.

According to this site the doubling time in the US is currently 8 days. That’s good news. We continue to see the rate of growth of new infections dropping.

One caveat: testing, as we know, is woefully inadequate. We won’t get a real handle on the numbers of infected people without widespread testing. There could be a lot of asymptomatic carriers and if we can’t test enough people then we can’t know for sure. But so far we have seen that aggressive social distancing measures appear to make a big difference. So, keep up the good work, people!


Manganese is an essential nutrient. You have to have it in your food in order for your body to function properly. That being said, instances of manganese deficiency are rare. This particular trace element is found in sufficient amounts in a wide variety of foodstuffs, everything from nuts to grains to vegetables and fruits. It would be hard not to get the 1-2 milligrams per day you need just from ordinary eating.

Manganese, like many metals, is toxic in larger quantities. People who are exposed to manganese in the workplace—mostly welders—have to take precautions. Manganese is a crucial ingredient in the production of stainless steel. In fact, there are few substitutes for manganese in its metallurgical applications.

So why should we care about manganese? Clearly it has enormous industrial importance. Without steel there is no modern world! The other reason we should care is that there is no domestic production of manganese in the United States.

We import ALL of the manganese we use in this country. It comes from places like South Africa, Brazil, Gabon, Ukraine, India, and China. There are probably billions of tons of manganese on the seafloor in the form of nodules, but no one has figured out how to mine that stuff. Known deposits here in the States are too low-grade and extraction costs too high to be an alternative.

That means we have to depend on other countries for a critical mineral.

It’s not the only one. Vanadium—another significant ingredient in steel—is entirely imported. So are tantalum, indium, gallium, cesium, fluorspar, asbestos, niobium, arsenic, rubidium, and several other important industrial materials.

We live on this great big rock. The spot we live has a lot of good stuff. But not all the stuff we need. That means we need other people.

There is no such thing as “self-sufficiency.” It is one of those nice notions that gets kicked around by romantics, back-to-the land types, politicos, and ill-informed pundits (are there other kinds?). Humans are a social species. We live in a web of inter-connections. Without each other, that is, without society, we cannot survive. Civilization may be a relatively recent thing in human history, but society is not. The first humans lived in groups and depended on each other, the last humans will as well.

It’s OK to re-watch Jeremiah Johnson and marvel at the independence and self-sufficiency of those old mountain men, but stop and think a little. All those guys had rifles and cartridges. It takes an industrial base to manufacture such things. All those guys had horses and pack animals. Those are the fruits of a well-developed agrarian economy. Jeremiah and his pals may have been tough and smart, but they couldn’t have done shit without steel and animal breeding. And those are just the first two things that come to mind.

If you want to “shop local” and “buy American” because you want to do right by your neighbors, I say good on you. I try to do that. Even if they are selling Japanese cars made in Mexico!

Numbers, continued

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:

I respect you . . . I don’t want to infect you

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?


“It ain’t love without a glove.”



If you didn’t already think you were living in a sci-fi world now you have no choice but to succumb to its inevitability.

A research team from several East Coast institutions (University of Vermont, Tufts University, and Harvard University) recently demonstrated “living” robots made from biological materials.

Their paper has a rather modest title: A scalable pipeline for designing reconfigurable organisms.

These folks aren’t as excited about what they created so much as they are about the process and what it suggests for future creations.

Stem cells were harvested from Xenopus laevis (African clawed frogs) as well as progenitor cardiac cells which were then manipulated mechanically and shaped into designs, creating “creatures” of about 1 mm in size. (The heart tissues are contractile and provide a crude locomotion.) The designs were done in silico and subjected to an evolutionary algorithm that winnowed out un-workable architectures and provided models for assembling the living-tissue robots.

The computational requirements to model the designs and test them in virtual environments were immense and done on the so-called “DeepGreen” supercomputer at the University of Vermont. The actual biological assembly was the least complicated part of the process. The “surviving” designs were further analyzed and improved in succeeding trials. The goal was to create novel organisms capable of four things: locomotion, object manipulation, object transport, and collective behavior. From the study:

Here, we demonstrate a scalable approach for designing living
systems in silico using an evolutionary algorithm, and we show
how the evolved designs can be rapidly manufactured using a
cell-based construction toolkit. The approach is organized as a
linear pipeline that takes as input a description of the biological
building blocks to be used and the desired behavior the manufactured
system should exhibit (Fig. 1). The pipeline continuously
outputs performant living systems that embody that behavior in
different ways. The resulting living systems are novel aggregates of
cells that yield novel functions: above the cellular level, they bear
little resemblance to existing organs or organisms.

“They bear little resemblance to existing organs or organisms.”

This is not Jurassic Park! This is something else entirely and the focus is on reproducibility, that is, industrial-scale applications.

What might such things be good for? Again, from the authors:

Given their nontoxicity and selflimiting
lifespan, they could serve as a novel vehicle for intelligent
drug delivery (28) or internal surgery (29). If equipped to express
signaling circuits and proteins for enzymatic, sensory (receptor),
and mechanical deformation functions, they could seek out and
digest toxic or waste products, or identify molecules of interest in
environments physically inaccessible to robots. If equipped with
reproductive systems (by exploiting endogenous regenerative
mechanisms such as occurs in planarian fissioning), they may be
capable of doing so at scale. In biomedical settings, one could envision
such biobots (made from the patient’s own cells) removing
plaque from artery walls, identifying cancer, or settling down to
differentiate or control events in locations of disease. A beneficial
safety feature of such constructions is that in the absence of specific
metabolic engineering, they have a naturally limited lifespan.

In the future—which is closer all the time—medicines will be customized to the patient. Extraction of materials from the earth, whether for remediation (waste cleanup) or resource mining, will be done without risking human workers.

I’m surprised this story didn’t generate more interest. I think it is pretty damn amazing and I hope I live long enough to see such schemes become economically feasible.

The Year Zero

You know, the year that was before AD 1 and after 1 BC.

Well, there wasn’t one.

There is no Year Zero. Our modern calendar starts with Year One. Anno Domini means “in the Year of Our Lord” so it marks the birth of Jesus of Nazareth. Whether Christ was actually born in that year is irrelevant. A 6th-century scholar named Dionysius Exigenus created the Anno Domini system and most of the modern world uses that marking point. Nowadays we call it the Common Era as opposed to the Christian Era, so we say CE 1 and 1 BCE (Before the Common Era), but the starting place is the same. The Romans would have called that year 754 AUC. That stands for ab urbe condita or “from the founding of the city.”

So if Year One was the first, and there was no Year Zero, when did the first decade end? Year Ten, of course. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. That’s ten years, that’s a decade.

So that means the second decade started in Year Eleven. And went for ten years: 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. It ended in Year Twenty.

I’m sure you see where I’m going. The first decade was AD/CE 1-10 and the second was AD/CE 11-20. Then 21-30, 31-40, 41-50, etc. There’s a rule: decades start on years with a “one” at the end and they stop with years with a “zero” on the end. That means this current decade started on 01 January 2011 and will end on 31 December 2020.

But people don’t like that scheme. We are much happier to start our counting at zero and end it at nine. Year 2010 starts the twenty-tens or the twenty-teens or whatever it’s called and Year 2019 ends it. Hence all the “Best of the Decade” lists coming out.

I’m not sure why people like that kind of counting better. Maybe it’s the digits. From 2010 to 2019 you only change the ones place. With 2020 you have to change the tens place as well. Perhaps it is more intuitive to visualize a decade that way, flipping over one number at a time until you run out.

There’s nothing wrong with starting at zero when counting. It’s the same number of leaps, the same number of things, you are just using a different numeral to represent the stopping point.

So if people want to count decades from 0-9 instead of 1-10 that’s OK. The only confusion is for those folks from the first decade (CE/AD 1-10). They have to be a nine-year decade, a nonade or novemade or something. Since there is no Year Zero they go 1, 2, 3, 4, 5, 6, 7, 8, 9 and call it good. In Year Nine they would publish their “Best of the Nonade” (the one-and-only nonade) lists. Then we could get on track and call Year Ten the start of the second decade (10-19) and it will all dovetail nicely with our 2019 end-of-the-decade stuff.

Whether you are a pedantic scholar and refuse to celebrate the End of the Decade until next year, or a party animal who loves those lists and can’t wait to ring in The New Decade, I hope you have a wonderful New Year in CE/AD 2020!