The Academy of Light has now been closed for 10 days, while the League One games scheduled at Shrewsbury Town on Saturday, and at home against Blackpool on December 22 and v Hull City on Boxing Day will not take place.
The Black Cats will return to action when they travel to Accrington Stanley on Tuesday, December 29. Lee Johnson's side will return to training on December 27.
Chief executive Jim Rodwell said it was a 'responsible decision'.
"It is with a heavy heart that I have informed the EFL that we are unable to play our next three league fixtures," he said.
"The welfare of our players, staff and opponents remains the number one priority and following detailed conversations with the relevant authorities, we have also taken the responsible decision to close the Academy of Light for a 10-day period.
"On behalf of everyone at the club, I wish all of those affected a swift and complete recovery.”
The club are now likely to face an EFL investigation into the postponements.
The threat of future sanction had led the Black Cats to play against AFC Wimbledon on Tuesday despite having eight players ruled out due to the outbreak.
Head coach Johnson said that the club had felt ‘let down’ by a lack of clarity from the EFL as to what the consequences of a postponement would be.
At that stage, one player had tested positive and two others were displaying symptoms.
The latest testing has confirmed an additional four positive tests, with players and staff affected now self-isolating.
A club statement said: "After one senior player tested positive for coronavirus on Monday, selected players and staff undertook additional testing and a further four cases of coronavirus have now been confirmed.
"The individuals that have tested positive – and those that have come into close contact with them – are now observing a period of self-isolation at home.
"Following extensive dialogue and clear guidance from Public Health England, SAFC has taken the decision to close the Academy of Light with immediate effect for a 10-day period in order to protect its players and staff and to prevent the further spread of the virus."