Members of the Board shall hold office until their respective successors have been appointed and qualified. Any member may resign from his office to take effect when his successor has been appointed and has qualified. The Governor and each County Board, respectively, may remove any member of the Board they have appointed in case of incompetency, neglect of duty or malfeasance in office. They shall give such member a copy of the charges against him and an opportunity to be publicly heard in person or by counsel in his own defense upon not less than 10 days’ notice. In case of failure to qualify within the time required, or of abandonment of his office, or in case of death, conviction of a felony or removal from office, the office of such member shall become vacant. Each vacancy shall be filled for the unexpired term by appointment in like manner as in case of expiration of the term of a member of the Board.