Syndrome, or MERS, outbreak, which has Korea on edge.
What's the meaning of a phrase "have something on edge"?
I know that 'on edge' means being worried or anxious.
Does it mean that MERS makes Korea worried?
The Subj HAVE Obj Compl is a sort of "stative" version of causative Subj MAKE Obj Compl.
The MAKE construction names an event whose result is that Obj enters the state expressed by Compl. If MAKE is in the simple present it implies repeated events and consequences:
George made Mary nervous. ... That is, on one occasion Mary became nervous as a result of something George said or did.
George makes Mary nervous. ... That is, whenever George is present Mary becomes nervous.
The HAVE construction is understood not so much as an event as a situation which causes Obj to be continuously in the state expressed by Compl.
George has Mary nervous. ... That is, Mary is always nervous as a result of something George did or does.
The sense of your sentence, then, is that Korea is currently and continuously on edge because of the MERS outbreak.
Compl may also be an infinitive or gerund clause; the two have very different senses.
An infinitive clause designates a single perfective (completed) event performed by Obj at Subj's explicit request or order.
George had Mary check her math. ... That is, George told Mary to check her math on a particular occasion.
If HAVE is in the simple present, Obj performs this action repeatedly or habitually.
George has Mary check her math. ... That is, George has established that Mary is to check her math on each occasion.
A gerund clause designates an imperfective (not completed) activity—in effect, a state of activity—which Obj is continuously performing at the time in question. This activity may have been expressly ordered or requested by Subj, or it may have been undertaken in response to something entirely different done by Subj.
George has Mary checking her math. ... This may merely report that Mary is currently engaged in the check George ordered; or it may report that something George said or did has Mary so uncertain that she is constantly checking her math to make sure she hasn't made mistakes.