The county department shall enter into an agreement signed by the patient or a legally responsible relative or guardian for reimbursement of the net cost to the county in furnishing such hospitalization: Provided, That such an agreement between the patient and the county department shall be deemed to be in existence in respect to an emergency hospitalization. The spouse, parent and adult child or any such patient being of sufficient ability shall be jointly and severally liable to the county department for the reimbursement of the expenses incurred by the county in furnishing such hospitalization to the extent that such expenses are not reimbursed from another source. Such liability may be enforced in an action at law.