Exponential growth is so powerful not because it’s necessarily fast, but because it’s relentless — Dr. Ethan Siegel, Senior Forbes contributor
Let me preface this by saying that just like everyone else commenting about the coronavirus, I don’t know what I’m talking about. So take this all with a grain of salt. Now consider that quote above. It’s interesting because its partly true and partly false: exponential growth is relentless…until it isn’t.
To better understand the growth of the coronavirus I want to recalibrate our intuition about time and numbers. Instead of talking about days I am going to talk about doubling periods (which for now, probably corresponds to something like 3 days) and for patient counts, I am going to talk in terms of powers of 2. So, every doubling period the number of cases is going to go from 2^n to 2^(n+1).
Now let’s ground this in some actual numbers to get us started. When you only have 1 patient, you have 2⁰ cases. The maximum number of cases in the US would be something like 2²⁸ = about 268 million. So the space we are dealing with is about 28 periods max. But if we reach n = 28, things are getting pretty apocalyptic. So exponentials are “relentless” only for a little while. They never last.
Where are we now? Last I checked (this is the site I have been using…not sure if it is better than any other) the US has about 50,000 known cases. This obviously isn’t the total number, but it represents a little under 2¹⁶. I am going to estimate that due to our lack of testing, we are currently about 3 periods behind…so we probably have something like 2¹⁹ cases.
In other words, we are talking about a pretty narrow range of interest here. We are already at period 19, and period 28 is catastrophe. This is not to say I think we are going to get there…it just means that we are mainly interested in how the next 9 periods play out. 9 periods is a lot of doubling, but if each period is three days, it can take place in less than a month.
So my first conclusion is that whether or not the US experiences an apocalyptic event will be determined in the next month. Let me also point out that I think it is pretty much inevitable that we are going to experience a few more doublings. Interestingly, China, who has way more people than the US (about 2² times as much) only got to around period 17. Since they have more population, their apocalyptic maximum would have been something like 30. But they stopped it with about 13 periods to go.
So back to the US. Since we don’t have quite the same command of our population as China, I think it is inevitable that we get to at least period 22. We will reach this number in 3 periods, but it could take as many as 6 periods before we realize that we have reached this level. So in my view, we will almost certainly continue to see exponential growth in the number of reported cases for another 6 periods (or about Easter). This is assuming that actual cases cease doubling in about 3 periods.
I am assuming that at some point the number of reported cases is going to start to catch up to the number of actual cases. However, it is very possible that this will happen after the pandemic starts to break (i.e., cease to double). One result of this is that the pandemic will probably break before we know about it (because it is already much worse than the official numbers).
What’s really interesting about this way of looking at things is that in exponential terms, there is a really fine line between a nothing burger and total apocalypse. Period 22 is not that bad and period 28 is the collapse of our medical system…and the difference represents a few weeks time.
An interesting comparison is Moore’s Law, the prediction that the number of transistors on an integrated circuit would double every 2 years. Although that trend (like all exponential trends) is bound to end sometime, the number of transistors is now approaching 2⁴⁰ in some chips, resulting in the explosion of computing power that we are all now very familiar with.
Ok, so let’s use this basic framework to analyze a recent comment by President Trump:
“Our country wasn’t built to be shut down. This is not a country that was built for this…America will again and soon be open for business. Very soon. A lot sooner than three or four months that somebody was suggesting.”
He goes on to suggest we could be up and running in 3 or 4 weeks. Where will we be in 4 weeks? 4 months? 4 weeks is about 9 periods (assuming the continue to last three days). If we are in period 19 now (i.e., 2¹⁹ actual cases), then in four weeks we will be pretty much at the maximum number of cases given our population. If we reach that point, we might as well open our economy because everyone will already be infected and there will be no reason to stay isolated.
So that is one scenario where Trump is right. We zip on up to level 28 (2²⁸ actual cases), all the baby boomers die, and we open up the economy from there.
Another scenario where we could open up our economy in 4 weeks is that our social distancing strategy is really super effective and the pandemic has already broken. Then we will level off at between 2¹⁹ and 2²⁰ cases (between 500,000–1,000,000), the disease will run its course in those who are already infected, and we will be fine to open up our economy.
But the more challenging scenarios occur in the middle. Perhaps the most likely scenarios are that we will reach somewhere between level 23 and 27 within 4 weeks and then start to level off. This is not a super bold prediction. Level 23 represents about 8 million cases, and level 27 represents almost 150 million cases. That is a pretty wide range. At a 1% death rate (chosen arbitrarily because it is easy to do the math and we don’t really know what it will be) that would represent a range of between 80,000 deaths (a bit worse than a typical flu season) and 1.5 million deaths (proportional to the 1918 Flu, which lasted a few years, all rolled up into a few months).
At a doubling rate of three days, these two scenarios are only separated by 15 days. So what do we do if 3 weeks from now (7 periods) we appear to be at level 23? With millions of cases, and the potential for hundreds of millions more, there is no way we open up the economy. By the same token, within 6 weeks we will know for darn sure whether the doubling rate has leveled off (and probably where it leveled off to).
Given this state of affairs, let me put on my dictator hat for a moment and describe what I would do. I would establish several different clear levels of response, something like:
Response Level 0 — business as usual
Response Level 1 — wash your hands, don’t touch your face, don’t go out if you’re over 65, and quarantine known cases
Response Level 2 — social distancing, work from home if possible (but not enforced by govt), close major events (i.e, anything with hundreds of people)
Response Level 3 — close schools, airports, no gatherings over 10
Response Level 4 — going outside without a pass is a crime, even to the grocery store, everything is delivered
Response Level 5 — military goes out and tests every single person, the infected are taken and placed into internment camps (since there isn’t room in hospitals).
The whole nation needs to be on the same page about this. No more decisions made at local (or even state levels). The national government applies a threat level to every state and that is it. We reevaluate every day. If cases are falling for a week or so we can relax the threat level by one. (Note that there is probably a two week lag, so current cases give us feedback about whether what we were trying two weeks ago was working). If cases are still rising, we need to ratchet up the threat level again.
Right now I see the US as somewhere between Threat Level 2 and 3, with a few places bordering on level 4. Once we hit contagion level 23 we need to be ready to go to level 4 all around. If we reach contagion level 26 or 27, we need to start thinking about entering Response Level 5. I am optimistic we won’t get there, but we need to be ready for the worst case scenario.
In any case, I actually think Trump is right that we will know where we stand in about 3 or four weeks. Of course, by then it might be too late if we have chosen the wrong response level. If we are able to successfully flatten the curve, we should be ready to raise and lower the response level multiple times based on how the virus responds. In that scenario, it could take 3 months, or even 3 years of tinkering with the response level.
Plagues of the past lasted years, not months. But despite the ineptitude of our federal government, we actually have much better data than we ever did back then and we can tailor our response accordingly.
Update: During an epidemic, it is kind of fun to have a granular metric so you can watch as the nation goes from level 23 to 24, etc. in real time. However, for future posterity, we need a catchy way of expressing how bad an epidemic is in the final tally, and I am not aware of anything that has really caught on (like the Ricther scale for earthquakes). So here is my proposal:
First, future generations care about deaths, not infections. Second, we want everything to be on a scale of 0–10, so we are going to take our population to the power of 1/10. For the world that would give us 7,800,000,000^.1 = 9.75. For the US, it gives us a base of 350,000,000^.1 = 7.15. Then to determine the level of the epidemic for a given population you find the number of cases and take the logarithm using your population-adjusted base.
The US is currently a little under level 3.3 in terms of deaths — log_7.15(650). I think we can get to a level 6 before we start really stressing our system. The worst case for the coronavirus is probably around level 7.5 (about 2.5 million deaths in the US). For comparison, the annual flu (all sources) is something like a 5.5, the Black Plague was about a level 9.5, and absolute annihilation of the human race is exactly 10 if we base our scale on deaths.
