Mich. Comp. Laws § 46.413

Sec. 13. (1) If the county board of commissioners does not fill a vacancy in the office of commissioner as provided in section 12 within 30 days, that vacancy shall be filled by a special election. The special election shall be called by the county board of commissioners. (2) The person elected at the special election to fill the vacancy in the office of commissioner shall serve for the remainder of the unexpired term.