Rapid genomic tests are poised to change the way doctors diagnose and treat infections, but their cost may limit widespread use.

Ryan Springer’s mystery illness began last summer with a dull ache in his chest. Over the next few days, the symptoms grew more alarming: sharp pain with every breath, a rapid heartbeat and a spiking fever.

Emergency room doctors at Carle Foundation Hospital in Urbana, Ill., were stumped. They ordered up a lung biopsy and started Mr. Springer, 47, on a broad-spectrum antibiotic. But his condition worsened, and doctors feared he might not survive the five or more days it would take to get the lab results back…

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