SKALA, Greece—Manolis Giannakakos doesn't remember how he got to the hospital. What he does recall is a searing pain that felt like someone was driving screws into his skull and then violent chills and delirium.

Doctors told the 56-year-old math teacher in this small town near Sparta he had a dangerous disease mostly eradicated from Greece in 1974: Malaria.

"When...