When the full cost determined to be necessary for the care, support, maintenance, and treatment of any patient is not paid by the patient or his or her relatives within thirty days of receipt of such care, (1) the county in which the patient resides shall pay (a) the first fifteen dollars per day of the unpaid cost for each of the first thirty days at the Hastings Regional Center, the Lincoln Regional Center, the Norfolk Regional Center, or other inpatient treatment facility where the patient is receiving inpatient treatment pursuant to an order of a mental health board under the Nebraska Mental Health Commitment Act or the Sex Offender Commitment Act, (b) the first ten dollars per day of the unpaid cost for each of the first thirty days at the Beatrice State Developmental Center, and (c) the first three dollars per day of the unpaid costs for each day after the first thirty days at any such institution, (2) the balance of the unpaid cost shall be borne by the state, and (3) the county in which the patient resides shall be credited by the department for amounts collected from such patient or his or her relatives in excess of the portion of such costs borne by the state.