Ohio Rev. Code § 339.11
The board of county commissioners may enter into an agreement with one or more corporations or associations organized for charitable purposes or for the purpose of maintaining and operating a hospital in any county in which such hospital has been established, for the care of indigent persons who are sick or have disabilities, including indigent persons receiving the tuberculosis treatment specified in section 339.73 of the Revised Code. The document used to verify the agreement shall specify the terms that have been agreed upon by the board and such corporations or associations. Such board shall provide for the payment of the amount agreed upon in one payment, or installments, or so much from year to year as the parties stipulate. This section does not authorize the payment of public funds to a sectarian institution, except when the payment is made pursuant to sections 339.71 to 339.89 of the Revised Code. The board may employ the necessary and properly qualified employees to assist it in carrying out all responsibilities devolving upon such board by reason of any agreement entered into in accordance with this section.
R.C. § 339.11