This is another post for those of you who don’t quite know what to say when your kids ask you what math is good for. I think this is at a pre-algebra level but I’m not sure. It’s about exponents and logarithms. But anything I do here, probably a child who’s a few years from that, but good with a spreadsheet, can do it that way.
Christopher Hogan, Ph.D. firstname.lastname@example.org
Daily percentage case growth is about 10% in both Fairfax and in Virginia. What you see below is a plot of the current percentage rate of growth in COVID-19 cases in Fairfax (orange) and Virginia (blue). Here’s a plot of the average daily percentage increase in cases.
But, of course, because the base for that percentage increase keeps growing, that number (count) of new cases seen each day continues to go up. And that’s what you see below, again for Fairfax (orange) and Virginia (blue).
What we are all hoping to see, in the near future, is an “inflection point” in the growth of coronavirus cases. In practical terms, that’s the point where we reach the top of the hill on those last two graphs, and the daily count of new cases begins to fall. When that occurs, it will be the first clear signal that we have started to get a grip on this. The first simple indication that this is behaving qualitatively differently from an unchecked epidemic.
That inflection point will tell us, for sure, that we are moving away from an unchecked epidemic. Because raw epidemics, at least at the start, almost always show exponential growth.
You see a lot of overly-complicated definitions of exponential growth. It’s really just the same idea as compound interest. Next year, the money in your CD will be whatever it is now, plus 2%. And then the year after, it’ll be whatever it is next year, plus 2%. And so on.
Constant percentage growth. That’s exponential growth. Looking at the graphs above, if conditions don’t change, tomorrow’s case count will be today’s count, plus 10%. And the day after will be tomorrow’s count, plus 10% of that.
Unchecked epidemics tend to spread exponentially for the simple reason that each person tends to infect N others, who then go on to infect N others, and so on. So the math for this really is just a straightforward reflection of the biology.
For example, suppose that with Binary Syndrome, you’re only sick for a day. But in that day, you infect two others, who go on to become sick the next day. And repeat. After you start the epidemic rolling, the count of active cases then goes 2, 4, 8, 16 and so on. On the Tth day, the case count is 2T. Two-to-the-power-T. Two-to-the-exponent-T. Hence, exponential growth.
And the originator of the epidemic? The day when there’s only one case. Call that day zero.
If there had been 10 of you, at the start, instead of just one guy, the case count would simply by 10* 2T. And if this were Trinary Syndrome instead of Binary Syndrome, and each person infected three each day, and you started with 10, you’d get a daily case count of 10*3T.
Actual epidemiologists don’t use different bases for this case-count figure. They wouldn’t use 2 for one disease, and 3 for another. Instead, they put them all on a common basis using Euler’s Number (not to be confused with Euler’s Constant), or “e”, the basis of natural logarithms.
If you knew that “e” was in honor of Euler, pat yourself on the back for an excellent education. Heck, if you so much as remembered that Euler was a mathematician, you get full credit.
You can do that — convert everything to base “e” — because there really aren’t an infinite number of different exponential curves, one for every base. There’s only “the” exponential curve or formula, the one that uses a base of “e”. Anything you can express with a base of 2, 3 or any other number, you can express with a base of “e”. You just need to add a parameter — one number in the formula that reflects the exact curve you want to represent.
And it takes just a bit of work to get from the obvious formula for the Binary Syndrome case count — using a base of 2 — to the professional standard formula, using a base of “e”. Solve for a, in this equation.
2T = ea*T
ln(2T) = ln(ea*T) Take natural logs (ln = natural log)
ln(2)*T = ln(ea*T) Logs convert exponentiation to multiplication
ln(2)*T = a*T That, my friend, is the definition of a natural log.
a = ln(2) Divide out the T’s
a = eh, about 0.693 or so.
So instead of a case count of 2T on the Tth day, an epidemiologist would say a case count of e0.693*T . Take a spreadsheet and check that, and you’ll see that on the 4th day, you’re expecting 15.99 cases using the exponential curve. Close enough.
The beauty of this is that now you have a common way of comparing all epidemics, based on the rapidity of their spread. Just compare that exponent. And you can easily handle cases in the real world, where the average spread rate isn’t going to be a whole number.
As I said earlier, exponential curves are ones that have a constant percentage growth rate. And if somebody hands you a formula, like the one above, it’s easy enough to get from the factor “a” in that formula, to the growth rate. The ratio of cases on day T+1, to cases on day T, is always:
ea*(T+1)/ ea*(T) = ea*(T+1)* e-a*(T) = ea*(T+1) -a*(T) = ea
Again, plug that into a calculator to make sure that’s right. And e0.693 is 1.9997. Again, close enough. We started from a series that doubled every day, and we have a series that (within rounding error) still doubles every day.
But back in the real world …
That’s all well and good. But that growth can’t go on forever. That simple exponential growth curve assumes you have an infinite pool of people, ready to be infected. So this really only characterizes the start of an epidemic. At some point, even with no intervention at all, eventually enough people have had the disease and are now immune to it that you begin to deplete the pool of persons able to contract the disease.
So, even without intervention, at some point the spread of Binary Syndrome would slow. It would no longer double each day, due to the simple fact that you run out of people to infect.
And, even more realistically, if you get your two infections in because you touched two (and only two) randomly chosen individuals, then the rate of growth doesn’t just suddenly hit the wall when all the uninfected people are gone. When you start out, you touch two, and infect two. But as time passes, there’s a greater and greater chance that one (both) of your two touches is (are) already immune. So the rate of growth slowly declines. As you begin to deplete the pool, your average daily infection rate drops below 2.
Obviously, there’s a lot more to it than that in the real world. Some viruses are hard to spread in summer, presumably because the higher humidity destroys them faster. And that reduces the spread. Sometimes people wise up, and (e.g.) stop shaking hands during a flu epidemic. And so on.
But that’s the basics. In a “natural” epidemic — where nobody tries to stop it — early-stage growth is exponential. And, if nothing else happens, that only slows down when the disease runs out of targets.
And at the end, the remaining uninfected become so few and far between that a lot of people begin infecting less than one additional person. And at that point, the epidemic winds down. Each day you get fewer new cases than the day before.
And that’s still exponential “growth”. But with a negative exponent. Estimate what that factor “a” above would be, if every current infection yielded just 0.5 new infections (instead of the 2 we started with): a = ln(0.5) = -0.693 (or so).
And if that number looks familiar, great. And for many, it will eventually dawn on you that the natural log of (1/2) has to be the negative of the natural log of 2. Which is even better. If not, don’t worry about it, just use a calculator.
And the result of all of that is what epidemiologists and others refer to as a sigmoid curve or logistic curve. For a classic epidemic, the number of people starts small, and grows like crazy. Then something happens, one way or the other. You begin to run out of uninfected people. You begin to vaccinate people. Something. Growth in new cases slows down (in percentage terms). Then, at some point, it actually slows down to the point where the daily count of new cases starts to fall — the “inflection point”. And then, eventually, it stops. The count of total infected people levels off. No new cases.
Why exponential growth is dangerous.
More than two weeks ago, Virginia Commonwealth University began converting some of their dorms to overflow hospital space. That’s not because they are over-reacting. Nor did they do it out of blind fear.
Rather, it is a calculated step. Literally. They did it because somewhere, a statistician or epidemiologist made a projection of the likely number of peak cases for this epidemic, based on exponential growth and the sigmoid/logistic curve. Possibly tempered with some real-world experience of how such pandemics are most likely to behave.
It’s a classic example of what I call the “Panic Early and Often” approach. Somebody looked at those projections and said, we’re going to need some more beds. And they set about getting it done. Far better to figure that out sooner than later.
But they aren’t setting up as many beds as they possibly can. That would be hugely expensive. They’re setting up as many as they think they might need. Plus, probably, a margin for error.
(Oh, would that NoVA toilet-paper shoppers had done the same.)
In this case, a whole lot of money, and maybe a whole lot of suffering, relies on somebody, somewhere, being able to do that calculation and projection right. Or, as right as it can humanly be done. Too high, and you waste a lot of money redoing these rooms. Too low, and you run out of hospital beds.
So if your kids ever ask you what math is good for, this is a dandy example.
It’s tough to make predictions, particularly about the future. And I hardly have the depth of understanding that the government’s own epidemiologists do. On the other hand, once we’ve settled on doing a logistic curve, and have a ready source of the data (us-counties.csv, on this page), even something as simple as Excel will fit that logistic curve to the data.
So we can do a simple-minded check on this. Do the data look like they are following an exponential growth curve? And then, how will things look, three weeks away, if nothing changes and that exponential growth continues?
I want to be clear that, per the first blue graph above, we are seeing daily growth rates slowly declining in Virginia. And its far too soon to see the effect of the latest CDC guidance to wear a mask when you are in public.
But for those who might think that VCU over-reacted, we can certainly take the situation just as it stands, right now, and see what things will look like in three weeks. If nothing changed from the current growth rate. In four easy graphs.
And what’s our criterion? Right now, the Virginia Hospital and Healthcare Association says there are 5953 available hospital beds in Virginia. How does that compare to the likely total coronavirus case count (times hospitalization rate per case), three weeks from now, under constant exponential growth.
This first chart fits an exponential trend line to the last 28 days of total case counts in Virginia. And it shows that the fit is lousy, but in a good way. And that’s because the growth rate has fallen significantly over this period. So an exponential fit to the last 28 days isn’t usable, because the situation changed too much over that time.
Just because you can fit a curve to some data doesn’t mean the curve is right. Or even useful.
Now let’s try fitting a curve to 14 days and to 7 days. If we’re lucky, and the situation is stable, we’ll get something close to the same curve either way. And we do. And because I know we do, I’m also going to include a 7-day projection, based on those fitted curves.
Using the last 14 days of data, we’d project 14,000 cases by the end of next week. If nothing changes, and it just follows the same curve.
But we know that the growth rate has been falling, from the graphs at the very top of this post. So how does it look if we use just the last seven days? Sure enough, lower growth leads to a smaller projection. Only a little over 10,000 cases, based on using the slower growth rate seen in the last seven days.
But here’s the danger of exponential growth. Let me see what this looks like, three weeks from now. Same curve, just looking a little further ahead.
And now, that same curve, if left unchecked, generates a case count of 50,000 three weeks from now. I’m not going to show the math, but given the current COVID-19 hospitalization rate, that would in fact consume just a bit more than every empty hospital bed in Virginia. (The only fact not in your possession is the fraction of diagnosed coronavirus cases that get hospitalized. Currently, that’s 14.2%).
Was VCU justified in going to the expense of building overflow capacity. Yeah, two weeks after they made that decision, using the lowest “static” growth rate estimate we can justify, it looks like they’re going to need those before three weeks pass.
A more sophisticated model would note that the growth rate hasn’t been “static” but has been falling, and would work that in. But this is more than enough to say that, however this turns out, VCU clearly made the right call.