Mosquitoes - those winged creatures that suck the blood and depart with a malarial or filarial blessing. We don't have mosquitoes here.
Nobody needs to take a malaria shot before coming to Dubai.
Filaria, water-borne diseases or any others that most travellers worry about are unknown here. If you stay here long enough, the rich food sans exercise is going to hit you right there in the belly. Dubai is relatively disease-free. The roads are clean, the food stores regularly inspected and the water is no bother. If you have a delicate stomach though, stick to mineral water.