Sec. 10. All charges made under this chapter must be:
(1) signed by the superintendent; and
(2) forwarded for payment to the auditor of the county from which the patient was admitted according to the following schedule:
(A) All charges accruing between January 1 and June 30 of each year shall be forwarded to the county auditor on or before October 1 of that year.
(B) All charges accruing between July 1 and December 31 of each year shall be forwarded to the county auditor before April 1 of the following year.