It’s mango season in Kenya. Evidence of this fact is heaped two stories high outside a greenhouse on the outskirts of Nairobi: a 900-ton mound of mango detritus reeking sweetly in the February sun.

The New World screwworm is a flesh-eating parasite that infests open wounds on live animals and humans. While eradicated in the U.S. decades ago, the parasite is spreading north through Central ...