Future Tense

Trouble With the Curve

Why the “cubic model” was such a failure at projecting the coronavirus.

A model of the novel coronavirus SARS-CoV-2.
A model of the novel coronavirus SARS-CoV-2. JENS SCHLUETER/Getty Images

This article is adapted from Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else by Jordan Ellenberg. © Penguin Press 2021.

On May 5, 2020, the White House Council of Economic Advis­ers posted a chart showing deaths from COVID‑19 in the U.S. up to that time, together with several potential “curves” that roughly fit the data so far.

Advertisement
Advertisement
Advertisement

One of those curves, marked “cubic fit” in the chart, represented a stance of extreme optimism; it shows deaths from COVID‑19 dropping to essentially zero in just two weeks’ time. That curve was roundly mocked, especially once it became known that the “cubic fit” was the work of White House adviser Kevin Hassett. Hassett’s biggest previous brush with fame was his coauthorship of the book Dow 36,000, pub­lished in October 1999, which argued that based on past trends the stock market was due for a tremendous near-term rise. We know now what happened to the people who rushed to invest their life savings in Pets.com. The bull market stalled shortly after Hassett’s book came out, then started to drop; it would take the Dow five years just to return to its 1999 high point.

Advertisement

The “cubic fit” curve was a similar overpromise. Deaths from COVID‑19 in the U.S. decreased through May and June, but the disease was far from gone.

What’s mathematically interesting about this story isn’t that Hassett was wrong—it’s how he was wrong. Understanding that is the only way we can learn strategies for avoiding this genre of wrongness in the future, beyond the limited‑in‑application “don’t trust Kevin Hassett.”

Advertisement

To understand what went awry with the cubic fit we have to go back to the great British rinderpest outbreak of 1865-66. Rinderpest is a disease of cattle, or was, until it was finally eradicated from the Earth in 2011, the culmination of a 50-year program. It originated in central Asia, probably before recorded history, and was carried around the world by Huns and Mongols. Sometime in the middle of the Middle Ages, a variant of the disease jumped the species barrier to humans; that spin-off virus is what we now call measles. And like measles, rinderpest is really contagious, which means it can tear through a population extremely fast.

Advertisement

On May 19, 1865, a shipment of infected cattle arrived at the port of Hull in east Yorkshire, England. By the end of October, almost 20,000 cows had fallen ill. In February, Robert Lowe, a Liberal member of parliament and later home secretary, warned the House of Commons in words that would have sounded uncomfortably familiar in 2020:

If we do not get the disease under by the middle of April, prepare yourself for a calamity beyond all calculation. You have seen the thing in its infancy. Wait, and you will see the averages, which have been thousands, grow to tens of thousands, for there is no reason why the same terrible law of increase which has prevailed hitherto should not prevail henceforth.

Advertisement
Advertisement

William Farr disagreed. Farr was a leading British physician of the mid-19th century, the architect of the country’s vital statistics office and an advocate for health reforms in the nation’s crowded cities. He wrote a letter to the London Daily News—his own “cubic fit” tweet—insisting that the rinderpest, far from threatening to burn through the whole bovine population, was about to start guttering out of its own accord. “No one can express a proposition more clearly than Mr. Lowe,” Farr wrote, “but the clearness of a proposition is no evidence of its truth. … It admits of mathematical demonstration that the law of increase which has hitherto prevailed, instead of implying ‘that the averages which have been thousands will grow to tens of thousands,’ implies the reverse; and leads us to expect that the subsidence will begin in the month of March.” Farr went on to make specific numerical predictions, down to the very cow, for the five months to come. By April, he said, the number of cases would be down to 5,226, and by June it would be a mere 16.

Advertisement

Parliament ignored Farr’s claim, and the medical establishment rejected it. The British Medical Journal ran a short, dismissive response:

Advertisement

We will venture to say, that Dr. Farr will not find a single historical fact to back his conclusion that in nine or ten months the disease may quietly die out— may run through its natural curve.

They ventured wrong! Just as Farr had predicted, cases declined through the spring and summer, and the outbreak was done by the end of the year.

Farr’s prediction of a swift end to the pandemic was similar to Hassett’s, and so was his method. He assumed that the graph of the number of rinderpest cases per month would follow a curve of a special type. OK, why hide it, we’re all friends here: He said it would be a curve of the form y = exp(ax3 + bx2 + cx + d) for some constants a, b, c, and d. Then he found the values of a, b, c, and d which best fit the rinderpest case numbers he’d seen so far. And the curve he found, while still growing at the moment he computed it, was about to crest and drop to nothing. That “x3”—i.e., “x cubed” in the exponential—is what makes it a “cubic model,” and this same model (we believe—he never really explained it) is what Hassett meant by the “cubic fit.”

Advertisement
Advertisement
Advertisement

Why should a pandemic follow a curve like that? Is there something about the dynamics of disease transmission that makes the “cubic fit” a reasonable prediction for disease trajectory?

Nope. Farr didn’t even know how the disease was transmitted! It’s a characteristic feature of this kind of analysis, called “curve fitting,” that you can extrapolate from observed data without any knowledge of the mechanism underlying whatever real-life thing you’re studying. And Farr really didn’t have any knowledge. Not fully sold on the germ theory of disease, he thought the slowing of the spread came about because whatever poisonous substance was passing from cow to cow lost some of its noxiousness with each animal it passed through.

Advertisement

Despite his ignorance, curve-fitting turned out all right for Farr due to the specific circumstances of the rinderpest pandemic. A simple curve like the cubic is most likely to work when the disease is spreading under roughly constant conditions. The UK was engaged in a relentless nationwide effort to stamp out rinderpest, and the cows weren’t given a vote as to whether they liked social distancing or not. The effort to halt COVID-19 in the United States was a different story: geographically heterogeneous, and start-and-stop.

Advertisement

The path of a pandemic depends most of all on the complicated iterative dance between disease dynamics and human responses to those dynamics, something a simple curve-fit is highly unlikely to capture. That hasn’t stopped people from invoking Farr’s method to make oracular predictions about pandemics, sometimes going so far as to call it “Farr’s Law” as if it were a part of physics. In 1990, Dennis Bregman and Alexander Langmuir (the latter the creator of the Epidemic Intelligence Service at the CDC) published a paper called “Farr’s Law Applied to AIDS Projections.” Their conclusion was that AIDS had already peaked, and that in 1995 there would be only about 900 cases in the United States. In fact, there were 69,000.

Advertisement

In an epidemic, you make a model based on who’s transmitting to whom and when. Those facts can suddenly change, by mass human action or by government decree. You can also use physics to model the flight of a tennis ball, and tennis players, if they are any good at all, are rapidly and unconsciously computing in that physics model to figure out where a certain shot is going to end up. But you can’t use physics to predict who’s going to win a long tennis match; that depends on how the players react to the physics. Real modeling is always a dance between predictable dynamics and our unpredictable responses. The cubic can’t do that; that’s why it didn’t fit.

Advertisement
Advertisement

What happens now? This spring, as last spring, cases of COVID are dwindling in the United States. But this time there’s an underlying explanation rooted in the biology of the disease—mass vaccination may be creating conditions under which the virus just can’t find enough susceptible hosts to kindle new outbreaks. It’s fair to hope that the pandemic is now less like a tennis ball whacked back and forth with great force, and more like a tennis ball rolling, slowly, to the edge of the court, under nothing but its own steadily diminishing power. If that’s right, simple models (though maybe not quite Kevin Hassett simple) may do better now than they did last May when they tell us to expect current trends to persist. Let’s hope so.

The cover of Jordan Ellenberg's book Shape
Penguin Press

Future Tense is a partnership of Slate, New America, and Arizona State University that examines emerging technologies, public policy, and society.

Advertisement