Dr. Albert Ko still remembers the shock he felt when the first babies with smaller-than-average heads were born in Salvador, Brazil, in late 2015.

The Zika virus that emerged in the South American country earlier that year initially didn’t worry the epidemiologist from the Yale School of Public Health as he led a team studying urban health problems amid Brazil’s extreme income inequality.

“We thought this was going to be a mild illness. Everybody got it,” Ko said in an interview.

“But everything flew off the handle in October and November, when we had all these babies being born with...