Mich. Comp. Laws § 46.412

Sec. 12. (1) If a vacancy occurs in the office of commissioner by death, resignation, removal from the district, or removal from office, the vacancy must be filled by appointment within 30 days by the county board of commissioners of a resident and registered voter of that district. Except as otherwise provided in subsection (2), the individual appointed to fill a vacancy shall serve for the remainder of the unexpired term. (2) If the vacancy occurs more than 7 days before the nominating petition deadline as provided in section 11 for the general November election that is not the general November election at which a successor in office would be elected if there were no vacancy, the individual appointed shall hold office only until a successor is elected at the next general November election in the manner provided by law and qualifies for office. The successor shall hold the office for the remainder of the unexpired term.