The jurisdiction of the local health department shall extend to the entire area represented by the governing body of the county, city, village or town that established the local health department, except that the jurisdiction of a single or multiple county health department or of a city-county health department does not extend to cities, villages and towns that have local health departments. Cities, towns and villages having local health departments may by vote of their local boards of health determine to come under the jurisdiction of the county health department. No part of any expense incurred under this section by a county health department may be levied against any property within any city, village or town that has a local health department and that has not determined to come under the jurisdiction of the county health department.