Dengue fever is present in Belize, but it is fairly uncommon compared with most of the rest of Central America. Last year, El Salvador, Honduras, Guatemala and Costa Rica had serious epidemics, with tens of thousands of cases, including a few deaths from the rarer hemorhagic varities. The Yucatan also had many cases. I think it's also present in some areas of the U.S. It's an emerging disease that is becoming more common.
I don't know the number of cases in Belize last year, but it was probably in the hundreds.
Usually dengue occurs in and after the rainy season. Last year the Belize government sprayed for it in a lot of areas.
It is carried by the Aedes aegypt mosquito, which does prefer more urban areas and often is found inside houses, unlike the mosquito that carries malaria. I know quite a few people on the mainland who have had dengue over the years but haven't heard of many cases on the cayes.
I had what I think was a mild case a couple of years ago, but I never confirmed it with my doc. Symptoms usually are similar to the flu, with joint aches.
There's no vaccine for it, and unless you have the more serious variety, no treatment except rest and Tylenol (don't take aspirin as bleeding can result.)
I personally wouldn't be much concerned about it in Belize unless things change.