The family of a South Australian grandfather who passed away on Monday after battling coronavirus has implored South Australians to abide by health advice and directions from the government and police.

Key points: An Adelaide man has died after contracting coronavirus

An Adelaide man has died after contracting coronavirus He acquired the virus interstate

He acquired the virus interstate South Australians are being encouraged to stick to local parks for walks

The plea comes as SA Health confirmed an additional four COVID-19 cases have been diagnosed in the state.

The family of Francesco 'Frank' Ferraro, 75, released a statement via SA Health on Tuesday afternoon, saying Mr Ferraro was "a loving husband and father to three children and grandfather to eight".

"Soon after returning from an interstate family event in March 2020 he felt unwell and learnt he had contracted the COVID-19 virus," the family's statement read.

"He was soon admitted to the Royal Adelaide Hospital where his condition deteriorated.

"Despite the best efforts of the medical professionals and staff and the love and support of his family he passed away on Monday, April 6 2020.

"The family would like to take this opportunity to implore all South Australians to abide by and comply with all the advice and directions given by the government and police."

Mr Ferraro was from metropolitan Adelaide and acquired coronavirus while interstate.

He was being treated in the Royal Adelaide Hospital's intensive care unit.

Eight in ICU as four more cases confirmed

On Tuesday afternoon, SA Health confirmed a further four positive cases of coronavirus in the state in the past 24 hours, taking the total to 415.

Deputy chief public health officer Mike Cusack said there had been one additional case linked to the Ruby Princess.

There has also been another additional case linked to the cluster at Adelaide Airport, with another baggage handler a confirmed case.

He said there was also one more suspected case of community transmission that was currently being investigated.

The number of recoveries has increased significantly to 89, as was expected with more data available from GPs.

Dr Cusack said eight patients remained in the ICU, with six of those critical.

SA Pathology has now undertaken almost 34,000 tests on South Australians.

The national death toll is approaching 50, after fatalities in New South Wales, Victoria and Tasmania.

The state recorded just two new positive cases both yesterday and the day before.

Premier Steven Marshall said it was a "very sad day for our state".

"My heart goes out to the family, the friends of the man who suffered from this disease," he said.

"I learnt overnight of his death and my condolences go to his family and to his friends.

"I think in many ways this just underscores for the people of South Australia how devastating this virus is."

Chief public health officer Nicola Spurrier said Mr Ferraro had underlying health problems.

SA COVID-19 snapshot: Current confirmed cases: 421

Current confirmed cases: 421 Deaths: 3

Deaths: 3 SA Pathology tests: 34,593 Latest information from SA Health

"This is not just a statistic — it is a person that has been a real integral part of a family," she said.

"So it's very important for us to all sit back and say 'it is hard to do this social distancing, it's hard not to be able to do the things we love doing, it's hard to change your Easter plans', but these are such important things to be doing for all of us so we are able to protect the most vulnerable in our community."

SA Health urging people to stay home for Easter

Dr Cusack also urged people not to travel over the Easter long weekend to prevent the spread of the virus.

"There clearly is virus in the community which we;ve seen with the baggage handlers and I think it;s important that we do relate vigilant," he said.

"Particularly over the long weekend where ... people would want to travel and get together, I think thats' why it's crucial.

"As they did in China when they recognised they had infection and they needed to put holidays on hold to an extent.

"That's why it's important that people don't travel and give the virus a chance to break out."

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Police enforcing coronavirus rules

Police Commissioner and state coordinator of the coronavirus response, Grant Stevens, said nine fines and 13 formal cautions had been given in a week to people breaking coronavirus rules.

They include people and businesses flouting self-quarantine and gathering rules.

Mr Stevens said almost 3,000 self-quarantine compliance checks had been done.

"Our observance is that most people are genuinely trying to do the right thing, so where we can see that there is an effort to comply, we're providing that education and further instructions to people so they fully understand their obligations," he said.

"But anybody who receives a fine I would suggest would be blatantly disregarding [the rules]."

The South Australian Government says people can go to local parks for walks, so long as they are only five to 10 minutes' drive from their homes.

Environment Minister David Speirs said campgrounds, the Mount Lofty Botanic Garden and the Warren and South Para reservoirs close to the Barossa Valley — where there is a coronavirus cluster — were closed.

A sign encouraging social distancing at the Waterfall Gully–Mount Lofty walking track in the Adelaide Hills. ( ABC Radio Adelaide: Spence Denny )

Mr Speirs said the Nationals Parks and Wildlife Service was not patrolling walking tracks around the state, instead relying on the public to do the right thing.

"We're saying it's OK to get into them, but keep that social distancing, be sensible and don't travel all across the state to do this - stay local," he said.

He said people who not live near Adelaide's beaches should also stay away.

"If you get there and don't feel comfortable — if you think there are too many people there — please go to some other area of open space," he said.