Ticks are sometimes more annoying and way less ecological useful. (without mosquitos for example there would be way less birds, bats, ...)
So that's good for the birds, and bad for the humans that want to get rid of all the pesky annoying mosquitoes, not just get rid of mosquito born disease.
In that case, not even need to exterminate any species.
That's typically done by introducing some Wolbachia in their gut.