Milder winters, burgeoning mice and deer populations and a bumper acorn crop from two years ago mean this year’s tick season is expected to be bad and more widespread, experts say. With that comes the threat of more tick-borne diseases, including the most common, Lyme disease.

States like Connecticut—home to the town of Old Lyme where the disease was first diagnosed—are already reporting a higher number of ticks infected with the Borrelia burgdorferi bacterium, which causes Lyme disease, as well as other tick-borne pathogens....