An application for county general assistance or for county health services shall give a right of subrogation to the county furnishing such aid. Subject to sections 68-921 to 68-925, subrogation shall include every claim or right which the applicant may have against a third party when such right or claim involves money for medical care. The third party shall be liable to make payments directly to the county as soon as he or she is notified in writing of the valid claim for subrogation under this section.