Belize is a sub-tropical country so it has a lot of life forms, including mozzies, snakes, bugs, bees, crocs, jaguars, monkeys, and so on.
But like the U.S. or almost any country, a lot depends on where you go. In Belize City and in towns, there are mosquito control efforts. In Cayo district, there are few mosquitos anytime. In the far south, in Toledo district and in low-lying swampy areas, there are quite a few.
Mosquitos tend to be worse after periods of rain, such as in late summer.
On some trips to Belize last a couple of weeks or more, I've hardly even seen a mosquito; other times, I've seen a lot.
DEET helps. So does covering up and wearing light-colored clothes (mosquitos are attracted to dark colors.)
Mosquitos do carry malaria, and there are several thousand cases a year in Belize, mostly among new immigrants from neighboring countries and in remote bush areas. The type of malaria in Belize is mild and the strain is not chloroquine-resistant. Denque fever is also present occasionally; it's carried by a different type of mosquito.
It's very rare for visitors to get either disease.
And Belize is not infected by mosquitos carrying the Nile virus, as is the New York area.
Belize First Magazine http://www.turq.com/belizefirst/