The county board of any county may (1) make an agreement with the Department of Health and Human Services relative to the expenditure of local, state, federal, and other funds or any combination thereof, available for public health in such county; (2) after notice and public hearing, establish and maintain a single full-time local health department for such county and any other counties which combine for that purpose and, pursuant to such combination or agreement, such counties may cooperate with one another and the Department of Health and Human Services and may contribute to a joint fund in carrying out the purpose and intent of sections 71-1626 to 71-1636. The duration and nature of such agreement shall be evidenced by the resolutions of the county boards of such counties, and such agreement shall be submitted to and approved by the Department of Health and Human Services; or (3) cooperate with any city in the establishment and maintenance of a city-county health department as provided in section 71-1630. The duration and nature of such an agreement shall be evidenced by resolutions of the city council of the city and the county board participating, and such agreement shall be submitted to and approved by the Department of Health and Human Services. A city-county health department shall be administered as provided in the agreement between the county and the city and shall be considered a state-approved, local, full-time public health service.