A high school student is among four new people who have tested positive for coronavirus in South Australia.

Key points: A Sacred Heart College student has tested positive for coronavirus

A Sacred Heart College student has tested positive for coronavirus Students who shared classrooms with him will be contacted and the classrooms will be cleaned

Students who shared classrooms with him will be contacted and the classrooms will be cleaned A woman who has no overseas travel history has also tested positive in South Australia

Sacred Heart College's senior campus in Somerton Park will be cleaned over the weekend and students who shared two classrooms with the boy will be put into quarantine.

The boy contracted the virus from one of his parents, who tested positive on Thursday, despite not travelling overseas himself.

He is in a stable condition.

Chief public health officer Nicola Spurrier said there was no need to close the Catholic school for "a prolonged time" because the weekend would be enough time to clean the rooms and contact students' families.

Principal Shana Bennett said all precautions would be taken to ensure the health and safety of students and staff.

"Our thoughts are with the Sacred Heart College student and their family," she said.

Sacred Heart sports teams will not compete this weekend.

Another woman in her 40s who received a positive test on Thursday also contracted the virus within Australia — the first such case in South Australia.

She had no recent overseas travel history, but had previously been in Melbourne.

The other positive tests recorded on Friday include a man in his 60s and woman in her 50s who travelled to the US, and a man in his 50s who travelled to Europe.

There are now 16 confirmed cases of COVID-19 in South Australia.

Dr Spurrier said South Australia was now moving out of the "containment phase" of coronavirus, with cases of person-to-person transmission beginning to take place.

She said South Australians had a responsibility to take precautions not to spread the virus, particularly by staying home if sick.

"It is so very important that all of us start to think about social distancing," she said.

SA Health has banned work-related international and interstate travel for its own staff.

Fringe and Adelaide Festival to go ahead

Prime Minister Scott Morrison and state and territory leaders want all non-essential gatherings of more than 500 people to be suspended from Monday amid fears about the spread of coronavirus.

Chief public health officer Nicola Spurrier (left) says SA is out of the "containment phase". ( ABC News: Isabel Dayman )

Adelaide Fringe director Heather Croall said all shows would go ahead as planned until the event's planned end on Sunday, since the advice was only for events after Monday.

"At all stages we've followed the Government advice, which is why this morning we were all standing by and waiting to hear what the advice was going to be," Ms Croall said.

She said no refunds would be available for ticketholders who choose not to attend.

"If people stay away at their own choice, that's their own choice," Ms Croall said.

Adelaide Festival executive director Rob Brookman said extra hand sanitiser and paper towels were among the measures being implemented by venues.

"We expected that we might start to see some empty seats where people simply decide even though they bought tickets not to turn up," he said.

"This hasn't happened.

"Indeed, the rate of ticket sales for the festival has actually accelerated during this last week rather than dropped off."

Organisers of Tasting Australia, a food and drink festival held in Adelaide and regional South Australia, have announced they will postpone the event following the Government's announcement.

It was due to be held at the end of March.

Education Minister John Gardner said schools would not be counted as mass gatherings, but school assemblies would be cancelled until further notice.

He said if anyone in a school tested positive, it would be closed for 24 hours.

"The situation essentially is in the event of a positive test, a school will be closed for at least 24 hours to allow public health officials to identify who that student or indeed staff member has been in contact with … to allow appropriate settings to be in place for those people if they need to be isolated or tested and enabling time for the school to be cleaned," Mr Gardner said.

Earlier on Friday, Adelaide submarine maintenance facility ASC closed part of its operations after one of its workers tested positive for coronavirus.

The Federal Government-owned company said all employees who had direct contact with the worker were undertaking self-isolation/