# Epidemic Doubling Times

## Data in the Era of the Coronavirus

What do bacteria, human populations, and the Coronavirus epidemic have in common? Short answer: exponential growth. We often use this term as a synonym for growing really fast. But in fact it has a precise, mathematical meaning, one that is vital in understanding and responding to epidemics like Coronavirus. But before the math, a parable:

In a far away land, there is a magical pond. One the first day of Spring, at precisely 7 am, there appears one lily pad, fully formed. On the second day, again at 7 am, there are two lily pads, also fully formed. On day three, there are four. And so on. On day 48, the pond is completely covered.

The question: on which day is the pond half covered?

The answer is day forty-seven. To see why, observe that the fraction of the pond covered tomorrow is twice the fraction covered today. Equivalently, the fraction covered yesterday is half the fraction covered today.

The population of lily pads is our archetype for exponential growth. Exponential growth is characterized by the time it takes for the population to double, aka the doubling time. What holds for lilies also holds for bacterial populations, human populations, etc. Consider, for example, the current human population of planet Earth. It is 6.5 billion, growing at an annual rate of 1.14%. If one does the math, this works out to a doubling time of 61 years. Therefore, assuming current fertility and mortality rates remain the same, there will be 13 billion people on Earth in the year 2081.

One more question about the lilies: what fraction of the pond was covered one week before it was completely covered? One week ago is seven doubling times earlier, so we divide 100% by 2 seven times to get 100/128 = 0.7%. Thus a pond with a sprinkling of magical lilies can be completely overgrown a week later. Surprise!

These days, with the Coronavirus epidemic underway for some time, people still ask, What’s the big deal? There are not many cases in my city | state | country? Well, suppose that the number of cases is growing exponentially, as is typical in the first phase of an epidemic. Suppose, to be specific, that there are 100 cases today, and the doubling time is two days. Two weeks, or seven doublings later, there will be 12,800 cases. Eight days, just three doublings later, there will be 102,400 cases. With exponential growth, a sprinkling of cases can grow to a huge number in a very short time. In the case of Coronavirus, this could mean that hospitals are overrun and cannot handle the patient load. Doctors then have to make difficult decision of whom to treat and whom to let die. Not at all good.

Conclusion: the “big deal” argument is completely wrong. Everything seems fine until suddenly it is not. Fortunately, measures like social distancing, if taken early enough, can slow down an epidemic, “flatten the peak,” and thereby save lives. See this article for much more.

How do we measure doubling times? One way is to use an app. But let’s think about this a bit. In the Parable of the Lily Pads, the doubling time was given to us. Here is a another example, a daily count of who-knows-what:

10, 12, 14, 17, 20, 24, 29,  35, 42

The doubling time seems to be about four days, since on day 1 we have 10 things and on day five we have 20. To test, we try other data points: on day four we have 17 things, on day eight we have 35, which is pretty close to 2 x 17. So a doubling time of 4 days seems about right. No math needed for this. But the next set of numbers is not as easy to analyze:

13, 18, 38, 57, 100, 130, 191, 212, 285, 423, 613, 949, 1126, 1412, 1784, 2281, 2876

These are the number of reported Coronavirus cases reported in France for the period February 25 to March 12, 2012. Here is what the data look like:

Assuming exponential growth, (and ignoring the change occurring on March 7) how do we find the doubling time? There is a standard method: plot the logarithm of the number of cases against time. This makes exponential growth look like linear, or straight-line growth. Now find the slope m of the line that best fits the logarithms of the data, either by using a transparent ruler and your best guess (the seat-of-the-pants method), or by using standard statistical formulas (the method of linear regression). Then

log 2
Doubling time = -----
m

Doing this, we find a doubling time of 2.1 days. If we do the same calculation for the last seven days, we find a somewhat longer doubling time of 2.8 days. Better, but still very dangerous. After five doublings, which would take place in 14 days, the cumulative number of cases would be 32 x 2876 = 92,032. The epidemic is a serious matter.

So far we have used the exponential growth model in thinking about both lilies and the Coronavirus epidemic. Like all models, there are limitations and assumptions that must be respected. Take the lilies. Even in their magical world, exponential growth ceases when the pond is full. Take the Coronavirus epidemic, or, for that matter any other. As the epidemic progresses, the number of susceptible individuals decreases since (a) dead people are not susceptible, nor are (b) those already infected. As these factors increasingly come into play, the rate of growth in number of cases reported declines, and eventually goes to zero. The epidemic is over. This said, exponential increase is a good model for the initial phase of an epidemic.

Another assumption, the general all things being equal assumption, can be violated (and hopefully will be violated) even in the early stages of an epidemic. Active measures such as social distancing can lower the growth rate and so increase the doubling time. It is not a constant of nature, like the speed of light.

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