COLUMBIA, S.C. (AP) - Police have searched several states for a South Carolina boy and they have repeatedly questioned his mother, who is accused of lying to authorities, but there is still no sign of the toddler who was reported missing nearly a month ago. Authorities said they have two theories on the fate of Amir Jennings, an 18-month-old boy from Columbia whose grandmother says she hasn't seen him since Thanksgiving. I'm really just focused on making sure that Amir is OK, or has some harm come to him? Columbia Police Chief Randy Scott said. That's what we don't know. That's what we cannot find out. Police have chased down multiple leads from the boy's mother, who has told them her son was with friends and family in the Carolinas and also Georgia. She has been arrested and is in custody. Amir's grandmother contacted police in early December, saying she was worried about the boy and her daughter, a former Winthrop University student who struggled with depression and had begun acting erratically since her son's birth. Police say they began looking for Zinah Jennings, 22, but also considered that relatives said she had repeatedly left town for days at a time, taking her son to visit friends in neighboring states. "The grandmother told me specifically that, when she was in school, she was a very good person, a very good student," said Scott, adding that relatives had filed several previous missing persons reports on the mother. Several weeks later, on Christmas Eve, police investigating a one-car wreck just blocks from Jennings' home were surprised to find the driver was the young mother they'd been looking for. Interviewed at a hospital, police say Jennings immediately began giving conflicting statements about where the boy was. Police spoke with Amir's father, who told them he had seen the boy during Thanksgiving but generally has had little contact with him. After several dead ends, the mother was arrested Dec. 29 and charged with lying to police about her son's whereabouts. Authorities say they have stepped up their efforts to find the boy and are hoping a tip line will yield some information. Police did not know if Jennings had an attorney. On Thursday, the woman's mother said the family was planning to speak out. "Our family will be making some statements today," Jocelyn Jennings Nelson said.