However, Mr Anderson would not go into details about Leon's death.
"We understand the public are interested in why a 'normal' day ended so tragically for Leon.
"Unfortunately, Canterbury police are unable to provide any further information or specific details of this ongoing homicide investigation into how Leon sustained his injuries."
Mr Anderson said police were focusing their investigation around Leon's activities throughout Wednesday before he was admitted to hospital.
"Police have already spoken to members of his family and staff from the school he attends who have all been co-operating with police. We will finish speaking to the remaining people who came into contact with him over the next few days.
"Leon was in the presence of his family and teachers throughout the day on Wednesday."
Friends of Leon's parents, James and Emma Roberts, have paid tribute today to "a beautiful wee soul".
Last night, Amy Lloyd wrote on Facebook: "Sadly lost my favorite wee friend today, I loved him so much. Rest in peace Leon xx."
Julian Llewellyn Jayet said Leon was born just before the Canterbury earthquakes.
"Now his life has gone it hurts a lot of people and it's hard to comprehend the little fellow's life being so short."
Alexis Lyford added: "An angel only loaned to us. Goodbye darling boy. My heart has broken."
It is understood that Mr Roberts, who married Leon's heavily pregnant mother Emma earlier this year, was by Leon's side in the ambulance when he was rushed to hospital.
He leaves behind a 7-year-old brother, Lucas, who is also autistic, and 10-year-old sister, Nadia.
Neighbours of the Papanui property are distraught at Leon's death.
They said the "normal, noisy" family has lived in the Housing New Zealand property for about 12 months.
The mum is said to be heavily pregnant.
Neighbour Gary Lee-Cooper, a father of young children, said he brought his family to Christchurch from Auckland two years ago to escape violence after friends were murdered.
"I can't escape it. I'm shocked, devastated," he said today, looking across the road to the house surrounded by police tape, a tent up in the front yard where specialist officers are conducting a scene examination.
When Mr Lee-Cooper was told the boy had died, he "broke down and cried".
"A 5-year old boy? This is a child. I've got kids myself... It's heartbreaking."
Another neighbour, who did not wish to be named, said they were a "a normal, noisy family".
"I was out the back most of Wednesday, hanging washing out, cleaning up, and there was nothing unusual. And then it all happened. It was all go from there. It was weird, very weird," she said.
"It's extremely sad. My heart goes out to them."
The high front fence was put up recently to keep the boys on the property, the neighbour said.
"They're a bit of a handful," she said.
The child had recently started at a local primary school. The principal was not available for comment this morning.
A local woman, who said she did not know the family, laid flowers at the house this morning.
"I live in the area, and my son is nearly 5, so it's a bit close to home," she said.
"My thoughts are there with the family. It can't be easy."
Police and Victim Support are working with his immediate family who are "extremely upset to lose their child in such tragic circumstances", Detective Senior Sergeant Scott Anderson said this morning.
"Canterbury Police will continue to speak to those who came into contact with the child on Wednesday and the scene examination of the Papanui address will continue over the weekend," he said.
"The investigation aims to determine the circumstances and events which led to the child being seriously injured."
Mr Anderson earlier said the boy attended school on Wednesday morning before returning home, as normal, about noon.
Detectives have already spoken to the boy's family, school staff and people who were in contact with him after he arrived home.