One snowy afternoon in October 1985, eight children from the tiny farming community of Lyndonville, N.Y., went sledding together. Within a few weeks, they all got sick.

David Bell, the local doctor who treated the children, recalls that their symptoms were similar to the flu: sore throats, fevers, muscle aches and severe fatigue. After three days, they hadn't recovered. Then a week. A month. Ninety days.

Six...