I have a brother who has a girl friend and they had a child together.they both moved from place to place till they wound up at my moms place for about seven months. While living at moms she found out there were both addicted to pills and were stealing from her.one day out of the blue they picked up and left to Arizona.(WITHOUT THERE CHILD)not telling anyone mom found out when they called and asked for money for gas.now a year later they have called and said they want there child toi be sent to them.they are both jobless and are still hooked on pills.mom has been careinf for the child and has created a stable home for him with love and attention that a child should get.so my question is what rights does mom have? Knowing that the child would be in harms way if she were to send him out to Arizona.and was this child abandoned?I know the law says if they leave the child in a safe place it's not but they left the state and knowbody knew for a week where they were to me that's abandoment but please correct me if I'm wrong in my thinking

Abandonment is leaving a child unsupervised and uncared for. It would appear that the child was left with your mother so abandonment does not apply.

Your mother needs to go Court and request formal custody of the child - and she needs to prove her charges that the parents are unfit.