A county or municipality may establish a hospital or other facility for the care of the sick by: building or contracting for the use of a facility; or joining with another county or municipality to establish a joint facility. A county or municipality may recover the expenses incurred by a hospital or another facility in caring for a patient who is not indigent: from the patient, within 3 years after the patient is discharged from the facility; or from the estate of the patient, if the patient dies in the facility.