The Governor shall appoint the members of the State Board of Health. Each member of the board serving on October 1, 2003, shall hold office until August 1 of the calendar year in which his or her term would otherwise expire. Appointments made for the succeeding members shall be for terms of five years. The term of office of each member of the board shall expire on August 1 of the appropriate year. If a vacancy occurs prior to the expiration of a term, the Governor shall appoint a successor with similar qualifications for the remainder of the unexpired term. No member of the board shall serve more than two consecutive, full terms. If the Legislature is not in session when an appointment is made by the Governor, the member shall take office and act as a recess appointee until the Legislature convenes.