When in any county such a proposition, for the levy of a tax for a county tuberculosis sanitarium has been adopted as aforesaid, the chairman or president, as the case may be, of the county board of such county, shall, with the approval of the county board, proceed to appoint a board of 3 directors, one at least of whom shall be a licensed physician, and all of whom shall be chosen with reference to their special fitness for such office. Two additional directors chosen with reference to their special fitness for such office may at the same time be appointed by the county chairman, with the approval of the county board. Whenever a county tuberculosis sanitarium has been established prior to August 2, 1965, 2 additional directors may be appointed by the county chairman, with the approval of the county board, within 60 days from such date.