SEOUL—As the Covid-19 virus began to spread last month, South Koreans didn’t have to wonder if they may have crossed paths with one of the country’s confirmed cases.

Patient No. 12 had booked seats E13 and E14 for a 5:30 p.m. showing of the South Korean film, “The Man Standing Next.” Before grabbing a 12:40 p.m. train, patient No. 17 dined at a soft-tofu restaurant in Seoul. Patient No. 21 drove her car to attend a weekday evening church service.

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