Do you think a person who is not the biological father should reimburse the government?

My husband found out after 10 years of lies that he was not the father of a child. He had a hard time convincing her to let him get a DNA without the courts. The mother denied that any other man was the father until after the DNA Test proved he was not the father. She received government istance for years and he owes back pay. Who should foot this bill? I think it human services allows women to lie about the father’s of child without any punishment, then DHS should foot the bill or make her pay them back and write off the back pay. Women who lie should be made to pay the government back after the child is 18.