The disease outbreak became apparent when visitors reported coming down with measles after visiting the park from December 15 to December 20.
This is a common two-word verb that is used in the news when reporting on newsworthy epidemics. It is very common in conversation.
Here's another example from a website:
Each year from October to May, millions of people all across the United States come down with the flu. Kids get the flu most often. But people in every age group — including teens — can catch it.
We use this verb to describe when a person shows symptoms of a disease.
The ebola outbreak resulted in this verb being used frequently:
US health officials are seeking 132 people who flew on a plane with a Texas nurse on the day before she came down with symptoms of Ebola.
Here's to wishing everybody good health!!