From learned experience with dengue virus and chikungunya virus in Florida, which are spread by the same mosquito, travelers who come back infected can infect local mosquito populations. With what we know from dengue and chikungunya, it is very unlikely we will have large outbreaks of Zika virus in the United States. One major reason for this is we have better constructed housing with air conditioning and intact screens that protect us from being bitten by mosquitoes in our homes.
When locally acquired mosquito-borne illness is present, the health department works closely with mosquito control to stop further transmission of the virus of concern. Mosquito control and the health department jointly provide public education whenever possible as well since it is very important all residents cooperate and drain containers on their property at least weekly to help successfully control this mosquito.