There is only one way to get malaria.  Malaria is a disease that is only transmitted to people by mosquitoes of a certain type.  Those mosquitoes can carry the germs that cause malaria.

Jeff and the rest of his unit run into a bunch of mosquitoes at one point while they are riding along.  They ride through a swamp with tall grass and stagnant water -- perfect for mosquitoes.  The men ahead of Jeff start yelling about "gallinippers" and Jeff soon finds out what those are.

This is presumably where he got the malaria.