The man, 43, sat in the dock in the County Court on Wednesday as details of how his family home was full of junk, rubbish, food scraps, mould and faeces for years leading up to the death of his younger son, on August 1, 2012.

The boy died days after cutting his big toe on an open can of cat food that was discarded in the house, in Melbourne's north-western suburbs.

He was never taken to hospital and the cause of death remains unknown although, the court heard, it could have been from tetanus, botulism, food poisoning or from inhaling toxic gases released by rotting food.

The boy's parents both pleaded guilty to two charges of reckless conduct endangering serious injury – one count for each of their two sons -– and one charge of failing to register the birth of a child.

The boys' mother died this month, which means the charges against her discontinue.

The father, whose name is suppressed so not to identify his surviving son, will be sentenced next month, after chief judge Michael Rozenes has assessed the level of culpability.

The maximum penalties for each of the major charges the man faces are five years in jail.

Prosecutor Susan Borg said the man, who held a full-time job while his wife stayed home with their sons, never questioned "what was going on, but turned a blind eye".

"If Your Honour just looks at the photographs, how could you walk in this household and think everything was OK?" Ms Borg said.

The prosecutor said it was not the Crown's insistence for a jail term, but that a message needed to be sent "that those who have children have a duty of care to ensure they're looked after".

Defence counsel Hilary Bonney called for a suspended jail term, as the man had already been punished through the loss of his son and wife, who died from organ failure related to severe alcoholism.

Ms Bonney said his remorse, lack of prior convictions and good prospects for rehabilitation were reasons he should be spared jail, as were his dealing with his own circumstances and the mental-health problems experienced by his wife.

Throughout the hearing, the court was told the man was incapable of standing up to his wife or solving major problems himself.

Ms Bonney said her client tried to have his wife clean the house, but feared she would flee with their sons if he sought outside help.

"He would encourage her to clean up. He even got a mini skip three times and encouraged her to clean up and when she did, say, 'See, isn't this a better way to live?' "He hoped if he kept pushing her she would see this was a better way to live ... but he said he couldn't combat her mental state and feared the thought of losing his family."

Psychologist Elizabeth Warren told the court the man had anxiety and dependency disorders, which led to a "paralysis of decision making".

Pressed by Judge Rozenes and Ms Borg while in the witness box, Ms Warren acknowledged the man should have known his children were unsafe living like they did.

"Even in the last days he thought the [boy's] toe was addressed adequately. I agree it's incomprehensible, but that is what he thought," she said.

The surviving boy now lives with guardians and has no contact with his father.

Ms Warren said the man wanted one day to "regain some sort of meaningful communication with his older son if he could".

The court heard the man considered himself the breadwinner of the family, while his wife handled the running of the house.

The court was told the boys never went to school, but that their mother went to great lengths to have her husband believe they did, including having the oldest boy recite stories about school and buying medals to claim they were won in athletics races.

Judge Rozenes said he found it difficult to comprehend how someone who functioned with others at work could return home and find such appalling living conditions acceptable.

Ms Warren replied: "It's hard to explain but people do adjust to incredible things."

Once the family became caught in a downward spiral, the court heard, they became isolated from their families and community.

The court heard neighbours complained about the state of the home, but council staff and the police could not get anyone to answer the door and no action was taken.

The Department of Human Services became involved only once the boy died, Ms Bonney said.

In documents previously tendered to Melbourne Magistrates Court, a woman who lived next door reported concerns to the department in 2009, but the DHS child protection service did not act because an "immediacy of risk had not been established".

Witness statements reveal another neighbour said someone they believed to be a DHS worker visited the family's home, up to two years before the boy died, and yelled: "You're not fit to be parents, I'm going to do everything I can to take the kids away".

The father, whom the court heard felt like a failure and believed that he had let his sons down, had his bail extended.

Ms Bonney said he had cleaned the family home since his youngest son died and that it was now immaculate.

"He sits in it in silence," she said. "He says he has lost everything."