An 81-year-old man and two of his close contacts, a man and a woman in their 50s, have tested positive for the virus.

The woman is known to be the elderly man's daughter.

More than a dozen other close contact are in quarantine along with hundreds or potentially thousands of people who visited a growing list of exposure sites, including a suburban shopping centre.

Premier Steven Marshall says five of those close contacts have tested negative so far but authorities are very concerned about the current situation.

"We only have one chance to get this right. Our focus from day one has been to go hard and to go early to minimise the level of disruption," he said.

"We've got to respond quickly."

New SA restrictions include a density change to one person to every four square metres in all venues, while all team and contact sport is to be cancelled.

All dining will need to be seated with a ban imposed on dancing and singing.

Masks will be required in all indoor public areas, including shopping malls.

Fitness centres will be closed along with other non-essential retail outlets. Supermarkets and pharmacies and other essential businesses will continue to trade.

Family gatherings will be restricted to 10 people but schools will stay open.

The new rules will apply from midnight on Monday and will be reviewed on Friday.

Police Commissioner Grant Stevens said while SA was not moving to a full lockdown, the restrictions were significant.

"We know there are people who will struggle as a result of these changes. But the consequences of not taking action could be even more severe," he said.

"The intent here is to dramatically suppress all discretionary activity in the community."

Authorities said earlier that the elderly man had returned to Australia through NSW, where he spent 14 days in quarantine.

He spent a further 10 days in hospital in NSW after suffering a fall before returning to SA on July 8, and presented to a suburban hospital over the weekend after developing symptoms.

The man had received one vaccine dose while in Argentina, but it was not known what type.

He has been admitted to hospital with Adelaide's other positive cases transferred to the Tom's Court Hotel which is used to quarantine all those with an active infection.

"We don't draw any conclusions as to where this infection came from. We simply don't know and it is very early days," Mr Marshall said.

"Our primary concern at the moment is putting everyone who may have come into contact with this person into quarantine as soon as possible."

Mr Marshall said the emergency department at Modbury Hospital was closed for a time after the man's presented but was being progressively reopened as it was safe to do so.

Some staff have been placed in hotel quarantine.

The Elizabeth Vale Primary School will also reopen on Tuesday after being closed on Monday as a precaution while checks were made to ensure the elderly man's grandson, who is a teacher, didn't attend over the two-week holiday period.

SA Health named five potential exposure sites on Monday, including an OPSM store at Golden Grove, a food warehouse at Hindmarsh, the Commonwealth Bank at Modbury and an Aldi Supermarket at St Agnes.

There is also some concern for an outdoor reserve at Glenelg.

Chief Public Health Officer Nicola Spurrier said officials were assuming the man had the Delta variant of the disease, though that was yet to be confirmed.

She said the Delta strain could "spread like wildfire".

SA Health expected to have the results of genomic testing on Tuesday.

Australian Associated Press