We can get Dengue here with or without immigrants bringing it. Mosquitos and their larvae get around.
It seems to be spread by the Aedes type, mainly Aedes aegypti - the same one spreading West Nile, Zika and others. Often, it is asymptomatic; but yes, it can be bad.
(My grandmother had malaria as a child in the Tidewater of Virginia; she was still having repeated symptoms for years after.)
Yes, both Aedes albopictus and Aedes egypti can spread dengue. So watch out for any striped mozzies! Both are believed to have been introduced here via import of tires. Distribution of both species is similar in the US.
The mozzies (vector) can only spread it if they have bitten a human or animal carrying the disease, so, actually, immigrants (or returning tourists) carrying the virus are needed.
Aedes albopictus can withstand colder temperatures, perhaps why it is the main vector for dengue in Europe, and it is feared it may the same in US.
My grandfather had malaria in Tennessee — so few people realize we had malaria here before DDT and flood control.
I had multi-drug resistant P. falciparum in SE Asia, so they gave me “toxic doses of quinine” to knock it out. The cure was worse than the disease and I was deaf for two weeks, but it wiped it out, no recurrences. Still not allowed donate blood, though.