In current times in America, certain mosquitos have the west nile virus which is potentially deadly. I have no idea of the percentages of how many mosquito’s actually have the virus, but would any mosquito be permissible to kill on shabbos due to some of them having a potential deadly virus?
If people are generally afraid of mosquitoes on regular weekdays (because of the danger of infection), and quickly kill them to avert danger, it would be permitted to kill them even on Shabbos. The reason for this is that for matters of pikuach nefesh (danger of death) we do not follow the “majority.”
However, if the custom of people is not to fear the mosquitoes during the week, and people do not hurry to kill them (because of the extreme unlikelihood of their posing a danger), it follows that one may not kill them on Shabbos, either.
It stands to reason that if people are not concerned for the mosquitoes on weekdays, they cannot be suddenly concerned for them on Shabbos, to the degree that it would be permitted to kill them.