Santana said she and her partner were in a state of "desperation and pain", and her doctor had given her something to help her sleep at night.
"We love him, we miss him so much. At night, for us, is really, really hard for us to go to sleep."
The family, who also have a 7-year-old daughter, moved to New Zealand from Brazil nine years ago, and don't have other family here for support.
Santana said her son had never gone missing before. The morning she last saw him, she said he was "very quiet" and told her he was going for a walk.
He was last seen on Badham Road near the intersection of Rangitata Island Road at 11.40am.
He left his home on Rangitata Huts Road in Clandeboye about 10am, wearing light blue shorts and a light grey zip-up jacket with half a zip and light green stripes on the sleeves.
"We think that Yuri is probably hiding somewhere," Santana said. It is believed Yuri may not want to come back so he can avoid going to hospital or taking medication for his depression, she said.
The night before he went missing, she and Yuri had been "playing together and having a good time".
Santana said her days were mainly spent "driving around", but she and her family had put their trust in God.
"I believe that God is in control of that," she said.
She begged Yuri to return home.
"Our heart is bleeding for him . . . please Yuri."
Police ask anyone who sees Yuri or has any information as to his whereabouts to call 111 or contact Crimestoppers on 0800 555 111.
