Sec. 13. The state department may remove a local health officer in the state for any of the following reasons:
(1) Failure to collect vital statistics.
(2) Failure to obey rules.
(3) Failure to keep records.
(4) Failure to make reports.
(5) Failure to answer letters of inquiry of the state department concerning the health of the people.
(6) Neglect of official duty.