Actually, the tropical disease experts say that the advent of widespread air travel in the 60s was the cause of the spreading of Dengue, so it's the tourists you have to watch out for,not illegal immigrants "guests".
We had a few cases of Dengue here on Nevis
last year. The government
department employees around to every home on the island, handing out a flier explaining where mosquitoes breed and how to eliminate standing water
. They walked around our property checking to see if there was anything holding water
Most of our mosquitoes are a small variety that doesn't seem to be a disease vector. I watch out for the larger Aedes aegypti, which is the vector for Dengue and Yellow Fever (which we don't have here). I've had a few of them land on me over the years, but so far have managed to squash them before they bit me.