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A few times each year, a particular chain letter pops up in my inbox. “We’re starting a collective, constructive, and hopefully uplifting exchange,” it starts, exhorting me to send a “favorite text / verse / meditation” to a previous participant in the chain, and to forward the message to another 20 friends. In my personal social circles, chain emails (“FW: FW: RE: funny!”) largely died out some time in the 2000s. But I first received this message in February 2014—the oldest example I can find online dates from just the same time—and it has gatecrashed my inbox at fairly regular intervals in the three years since.

“Seldom does anyone drop out because we all need new ideas and inspiration,” it proclaims. Though perhaps consoling, this assertion is a blatant, implausible lie. And I can prove it.

The email instructs us to pass it along to 20 people within five days. Suppose that everyone who receives the message really does continue the chain. On the first day, the email would be sent out (by its original creator) to 20 people. By the fifth day, those 20 people would have sent it out to another 20 people each—that’s 400 new people receiving the message. By the tenth day, those 400 people have sent it to yet another 20 people each—suddenly we’re up to 8,000 new recipients. From there, things start to get a little out of control.

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Around 3 billion people have Internet access (only about one-quarter of those are primarily English speakers, but we’ll ignore that for now). With 25.6 billion new emails to be shared between 3 billion people in the immediate run up to Day 35, every Internet user on Earth would have to receive over 8 copies of the email during a few-day stretch.

You might think that this incredible growth is unfair; if we assumed, say, five or 10 forwarders instead, we might not reach such absurd conclusions. But we’d reach absurd conclusions even if we assumed just two forwarders. (There’s an old fable that illustrates this exponential point, about the sage who invented the game of chess.) With 20 forwarders each time, we reached everyone on the Internet by Day 35; with 2 forwarders, we only delay that outcome until Day 155—about five months instead of one.

The great thing about chain emails, as opposed to infections, is that we, and our friends, aren’t just passive and involuntary vectors.

While this simple model is enough to show that, in the long run, fewer than 2 people on average must be continuing the chain each time, there’s clearly more to the story than that. Based on my inbox, this email chain is neither dying out nor taking over. What sort of behavior could possibly explain that?

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The spread of infection, of course: “Epidemiologists have been breaking their skulls trying to think about how to model infections,” says Marta Andrés-Terré, an immunology PhD candidate at Stanford University. “You’re trying to model hundreds of variables, many of them being stochastic. You basically have to figure out what’s going on without being able to see what’s going on. It’s like the Sherlock Holmes of medicine, you have to just follow the trace.”

Luckily for us, chain emails are much simpler; still, several key concepts from epidemiology can help. A major one is the “basic reproduction number,” known as R0, which measures how many new cases of infection will be caused by one infected individual. So long as R0 is larger than 1—that is, so long as each new case leads to more than one additional new case—the infection will continue to spread and grow. By contrast, if R0 is below 1—if each new case leads to less than one additional case—the infection will peter out. When our chain email author claimed, “Seldom does anyone drop out,” she was saying, in epidemiological terms, “R0 of this email is close to 20.”

“I’d say that chain mails are more like endemic diseases where the R0 hovers around 1 with the occasional outbreak where the virus hits a person with lots of vulnerable social contacts,” says Tali Cassidy, an HIV epidemiologist at Médecins Sans Frontières in Khayelitsha, South Africa. “You could also liken it to a disease like measles where most people are immune, by vaccine or herd immunity, but occasionally we let our guard down and an unvaccinated person wanders into a vulnerable population, and suddenly there’s an outbreak.” Since most people in the broader population are already immune they are not infected, even when they are exposed, “and then the R0 drops below 1 again,” says Cassidy. It’s possible, she speculates, that my (anecdotal) data about the spread of the email—seven emails in three years, neither speeding up over time nor dying out—is “just the midpoint between exponential spread, tempered by increasing immunity.”

Incidentally, a form of natural selection can act on the chain email when its recipients tailor the chain message to suit the people they forward it to. For example, another version of the same chain message I’ve been receiving (still starting with the words “Collective, Constructive and Uplifting Exchange”) asks for a Bible Verse instead of a general favorite text. I don’t know which version came first but, somewhere along the line, a writer “mutated” the message she received. If the new version is better suited to a particular social circle, it will spread faster and become more prevalent within that group.

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The great thing about chain emails, as opposed to infections, is that we, and our friends, aren’t just passive and involuntary vectors. As a result, those of us in the anti-chain message brigade can be part of the solution. “We’re the therapy,” says Andrés-Terré. “We go to others and say, ‘This is so dumb, we shouldn’t reply.’ We’re not just endpoints for the pathogen, we’re actually working to stop this thing. It’s a public health campaign.”

You could join by forwarding this article to 20 close friends. Most other readers are doing it….

Uri Bram is the author of Thinking Statistically, an informal primer on the big ideas from statistics. Follow him on Twitter @UriBram.

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