Working with a simple mathematical model in which chance plays a key role, researchers calculated how long it would take a bacterial infection or cancer cell to take over a network of healthy cells. The distribution of incubation times in most cases, they contend, is close to ‘lognormal’ — meaning that the logarithms of the incubation periods, rather than the incubation periods themselves, are normally distributed.

It’s been known for more than 60 years that the incubation periods of numerous diseases follow a certain pattern: relatively quick appearance of symptoms in most cases, but longer — sometimes much longer — periods for others. It’s known as Sartwell’s law, named for Philip E. Sartwell, the epidemiologist who identified it in the 1950s, but why it holds true has never been explained.

“For some reason, [biologists don’t] see it as a mystery,” said Steve Strogatz, the Jacob Gould Schurman Professor of Applied Mathematics. “They just see it as a fact. But we see it as, ‘Why? Why does this keep coming up?'”

Through mathematical modeling and application of two classic problems in probability theory — the “coupon collector” and the “random walk” — Strogatz and docto to generalize too broadly, this theory holds up following countless simulations and analytical calculations performed by Ottino-Löffler. And this could be helpful in explaining not only disease proliferation, but also other examples of “contagion” — including computer viruses and bank failures, the researchers say.

“In a very stripped down, simplified picture of reality, you’d expect to see this right-skewed mechanism in many situations,” Strogatz said. “And it seems that you do — it’s sort of a basic vocabulary of invasion. It’s a powerful underlying current that’s always there.”

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