It’s the math, stupid
Linear thinking during an exponential crisis
Never in my life has this math nerd seen a basic mathematical concept become so central to a critical public discussion. That concept is exponential growth, and as the global COVID-19 pandemic unfolds we are seeing a clear split between those who understand — or at least accept — exponential growth, and those who either do not understand it or attempt to disregard it. The latter camp unfortunately includes the President of the United States, who two days ago made a problematic comparison of traffic deaths (which are tragically high but fairly stable) and COVID-19 deaths (which are growing at a breakneck pace) as part of a rationale to ease restrictions and “have the country opened up” by Easter.¹
Infectious disease lends itself very well to straightforward mathematical analysis. I’m not a pandemic specialist, but wanting to better understand the public discussion, I’ve been tracking data on COVID-19 cases in a few countries around the world over the last few weeks.² Even a nonspecialist like me, using basic Microsoft Excel functions, can calculate the rate of increase in reported COVID-19 cases and make simple projections. Yes, understanding where the numbers come from is important. Are the case numbers complete? Are the sources unbiased? How much has the unavailability of test kits suppressed case numbers in the U.S.? But the math itself is pretty basic, and this math reveals that the growth of COVID-19 in the weeks before strong controls are put in place is quite consistent across countries and regions.
Warning: math ahead
As of this writing, current case numbers are fairly small in the U.S. as a fraction of the total population, but the number 33% is vastly more important than the current levels. 33% is the rate at which U.S. cases have been growing PER DAY from March 2 (when the 100th case was reported) to March 24. Anything growing by 33% per day will increase by almost a factor of 10 every 8 days. So 100 cases on March 2 becomes 1,000 on March 10, 10,000 on March 18, and nearly 1 million on April 3. This daily growth rate is similar to that experienced by the U.K., Italy, and South Korea in equivalent stages of the outbreak.
This is just simple, apolitical math. But it still feels shocking. Based on the experience of other countries, what is happening today was easily predicted based on the number of cases three weeks ago. It shouldn’t be a surprise, and yet it is, at least to most of us. The recent precipitous decline in the stock market shows that even investment professionals — whose success depends in part on being very good at math — were surprised.
Every day the pandemic seems to get worse, with more new cases and deaths than the day before. But this should not be surprising: the more people who have the virus, the more new people will catch the virus each day, and so on. That is how pandemics — and exponential growth — work. This pattern will persist until something is done to disrupt it, such as putting strong containment measures in place to separate people from one another.
We — journalists and non-journalists alike — can’t seem to wrap our heads around this rate of growth. Our brains seem geared toward linear growth: a certain number of new cases per day, and a similar number of new cases the next day, and the day after that. We think we understand exponential growth, but most of our experience is based on low growth rates, for example population growth or U.S. inflation, both of which also follow exponential growth patterns.³
Changing the math
Looking ahead, what will be happening just two weeks from today — 3.8 MILLION cases in the U.S. if the 33% daily growth rate continues — is equally foreseeable based on the cold truth of exponential growth. But again using simple calculations, we can easily see the huge impact that a decrease in the daily growth rate would have. If that 33% daily growth rate were to magically decrease overnight to 14% — still a scary number! — we would have 380,000 cases three weeks from now; not a 19% reduction but a 90% reduction! Acting quickly and decisively is crucial; if the daily growth rate continued at 33% for 7 more days and then dropped to 14%, we’d have 1.1 million cases two weeks from today. Much better than 3.8 million, but way more than 380,000.⁴
14% is a made-up number, but these calculations illustrate that if quick and aggressive measures successfully reduce the rate of spread of COVID-19, this will have a big impact on how severe this pandemic will be in just two weeks, let alone two months. Timing is everything, as San José Mayor Sam Liccardo alluded to last week when Bay Area county health officers issued shelter in place orders: “History will not forgive us for waiting.”
And yet, at this moment when aggressive, UNIVERSAL action is needed, we see a big divide. On one side of the divide are people who are able to do the math and understand what it means, or who at least believe the infectious disease specialists who are doing the math. On the other side are those who discard or disbelieve the numbers because they don’t fit their linear mental model — the projections don’t “feel right” — AND do not believe public health professionals. This divide seems particularly acute in the U.S.
Let’s repurpose James Carville’s famous 1992 utterance: it’s “the math, stupid.” With the stakes so high, we need to set aside our natural desire to have numbers that “feel right” to our linear brains; we need to understand and accept the exponential math. If we don’t want to (or can’t) believe the math, we at least need to trust the public health experts who are doing the math, and the public officials who are putting in place aggressive measures to CHANGE the math, provided those public officials have credibility. And speaking from my perspective as a San Franciscan, officials at the state and local levels do have that credibility.
Math nerds, unite!
All you math nerds who throw your hands up when reluctant math learners exclaim “I’ll never use this!” — now is your time to shine. Let your nerd flag fly! Help everyone you know understand the simple but powerful math behind the COVID-19 pandemic. Help others move from a linear world to an exponential one, because that’s the world we live in. When they’ve made that move, gently remind them to act accordingly by washing their hands, practicing social distancing, and following the guidance of public health professionals and local officials.
[1] Philip Bump provides an excellent critique of this dangerous analogy in the Washington Post.
[2] There are numerous data sources; I’ve been primarily using WorldOMeter.
[3] Even there we are surprised by how this plays out in the long run, like when we see someone in an old movie buy a newspaper or a soda for a nickel. That’s just exponential growth run backwards over several decades.
[4] Grant Sanderson of the YouTube channel 3Blue1Brown walked through this mathematical logic with similar calculations in a much more graphical and entertaining way here. I can’t help noting that I learned last week that 3Blue1Brown is a spinoff from the amazing Khan Academy.
