Progress? Doubling time and second derivatives

26 March 2020

Today, I’m going to focus on a quantitative way to tell if we’re making progress. There’s a bit of maths, but you can skip it; everything is explained in plain English as well.

While the tide might be turning in Italy …

Over the last two weeks Italy has moved from a doubling time of 2-3 days to 7 days. France, Germany, and Spain all show early signs of a slowing of the doubling rate as well.

That leads to an interesting question I’ve been debating with a friend: how would we know if we’re making progress? It seems strange to say that we’re making progress when large numbers of new cases are announced every day, and sometimes even increasing numbers of new cases.

But in an exponential growth epidemic like COVID-19, that’s what the earliest sign of progress would look like: a slowing of the rate at which confirmed cases are increasing.

A little high school calculus is helpful here.

Many charts plot the cumulative number of confirmed cases for a given country. Those charts look scary, and they should scare us:

These curves are all increasing sharply.

What is the rate of change of cumulative confirmed cases? It’s the number of new confirmed cases daily, the first derivative (or equivalently, the slope) of the curve of cumulative confirmed cases.

Most, but not all, of these curves are also increasing. (The daily confirmed rate can be quite volatile for a number of reasons including lags in reporting, “catch-up” reporting, changing rates of testing, even changing definitions of confirmed cases.). You can see examples where the number of daily new confirmed cases ticked down and then soared upwards again.

Helpfully, the rate of change in cumulative cases is strictly inversely proportional to the doubling time. (See below for the maths.) That is, if the rate of change in cumulative cases goes up by a factor of two, the doubling time halves; and vice-versa.

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But if you look at another metric, harder to illustrate, there are signs of progress; that’s the rate of change in which new cases are appearing. This is the first derivative (or slope) of the number of new daily confirmed cases, or the second derivative of the cumulative number of confirmed cases. Put in plain English, this metric measures whether the rate at which new cases are being confirmed is increasing or decreasing.

I haven’t seen anyone plotting this curve. It’s easy enough to do, but I don’t have time right now to download the data and plot it. But helpfully, because of the relationship between the rate of change of daily new confirmed cases and the doubling time, there’s a simply way to observe this second derivative, and that’s through the change in the doubling time.

If the doubling time for a given country is increasing, the second derivative (again, the rate of change in which new cases are appearing) is decreasing; and vice-versa.

That’s why the first sign that we’re making progress is likely to be a sustained increase in the doubling time. Your best daily source for this data is (no surprise!) the great table in Our World in Data (OWiD).

I downloaded the OWiD data, which mostly comes from the ECDC, and took a quick look at how doubling time is evolving:

China, the top line, has seen doubling time steadily increase, meaning the rate at which infections are increasing is decreasing — a lot! South Korea and Japan show the same trend. Unfortunately, the World (all cases in the world) doubling time is decreasing significantly, meaning the rate at which cases are increasing worldwide is increasing rapidly.

Let’s look at that same chart with China, South Korea, Japan, and World removed to make it easier to see what’s happening in Europe and the US. (I made the lines curvy, which they’re not–these are calculated daily–as it’s easier to see the individual countries.)

It might be easier to see this in a table:

Good news: Italy looks like they’re making progress! And there are early signs of progress in Spain and France too.

Of course, it will take many more days of ratcheting up the doubling time to begin to feel more confident. And with a mean lag time of 3-4 weeks in fatal cases between infection and death, the number of deaths will continue increasing rapidly even when the rate at which of confirmed cases is growing is slowing. And then there’s the questions of what happens when we begin to relax control measures–could there be a second wave? Those are all questions that merit further reflection.

... the US is really, really scary

The doubling time in the US has been around 3 for more than two weeks now. Let’s put that in mathematical context: if the US continues to double the number of cumulative confirmed cases ever 3 days for another 30 days, the number of cases will increase 500-fold.

`A 500-fold increase in the number of confirmed cases would take us from 55,000 cases on 25 March to more than 27 million cases!

Now, for a lot of reasons, that’s not going to happen. First, some of the rapid increase in the US must be the result of more testing identifying a higher proportion of cases. Second, a number of cities and regions have rolled out, and are increasing the intensity of, control measures. Finally, no epidemic can increase exponentially forever: at some point, the virus runs runs out of new people to infect (even if we don’t develop a vaccine), and the evidence is starting to confirm that there should be at least partial / temporary immunity (and likely better than that) after recovery.

Still, I am really, really worried about the US for reasons that go beyond the numbers. The response so far has been piecemeal: state-by-state and city-by-city. Unfortunately, SARS-CoV-2 doesn’t care if you live in New York or Florida. And unlikely countries, it’s not really clear that states or cities have the option of closing their borders to their neighbours, or what the consequences would be of trying.

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