From Longman Dictionary of Contemporary Englishcome down with something phrasal verbGETto get an illness I think I’m coming down with a cold. → come→ See Verb table
Examples from the Corpus
come down with • Several monitors have come down with heat rash and heat stroke this summer.• She started coming down with it when we were on our business trip, arranging this weekend.• She came down with this summer flu everyone's getting and went home before I got in. 