The deputy state health officers for and under the direction of the Department shall enforce the laws of the State pertaining to the public health, shall enforce all rules, regulations and orders adopted or promulgated by the Department in accordance with law, shall undertake such other duties as may be assigned to them by the Department and shall supervise all public health matters within their respective counties, and in the City of Wilmington, but not in other incorporated cities or towns having duly constituted boards of health. The governing authorities of any incorporated city or town, other than the City of Wilmington, may by resolution duly adopted and approved by the Department designate a deputy state health officer to act as health officer of such incorporated city or town, and the deputy health officer shall then exercise the powers and perform the duties of the local board of health.