While mosquitoes in the Virgin Islands [1] do not spread malaria or yellow fever, there have been cases of dengue and dengue hemorrhagic fever. Dengue is a virus characterized by fever, headaches, and joint and muscle pain, sometimes accompanied by nausea, vomiting, and rash. Symptoms usually appear 4–7 days after the person is bitten by a dengue-carrying mosquito. There is no cure or vaccine for dengue. Treatment is available for the symptoms. In rare cases, dengue can be fatal, especially in cases of small children or older adults.
The only way to avoid dengue is to avoid mosquito bites. Use a mosquito repellent containing DEET, and stay within screened-in or air-conditioned rooms. Avoid being outside during the early morning and late afternoon hours favored by mosquitoes.
Links:
[1] http://www.moon.com/destinations/virgin-islands/discover-the-virgin-islands