Ohio Rev. Code § 749.05
The term of office of the appointed members of the board of hospital commissioners shall be four years, but the members first appointed shall hold their offices, respectively, as determined by lot at the first meeting of the board, for the periods of one, two, three, and four years, and thereafter one member shall be appointed each year for the full term of four years. The mayor with the consent of the legislative authority shall fill any vacancy on the board of hospital commissioners not later than sixty days after the vacancy occurs. If the vacancy remains unfilled on that date, the legislative authority shall have thirty additional days to fill the vacancy. If the vacancy remains unfilled for ninety days after the vacancy occurs, the remaining members of the board, by majority vote, shall appoint an individual to fill the vacancy. Any member appointed to fill a vacancy occurring prior to the expiration date of the term for which the member's predecessor was appointed shall hold office as a member for the remainder of that term.
A majority of the board shall constitute a quorum.
R.C. § 749.05