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Coronavirus vaccines work. But this statistical illusion makes people think they don’t.

Is the vaccine wearing off? It’s an exhausting thought for those of us who believed the battle against covid-19 would be won once enough needles plunged into enough arms. But outbreaks of the delta variant have blossomed even in places with high levels of vaccination, including Israel, Britain and my own home of Madison, Wis. Recent reports from Israel that nearly 60 percent of people hospitalized with severe covid-19 were fully vaccinated raised particular alarms about the limits of the protection that vaccines provide.

–Jordan Ellenberg, a math professor at the University of Wisconsin, is the author of “Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else.”