Tasmania's Premier Peter Gutwein says he cannot rule out that two coronavirus infections in the state's north-west are the result of community transmission.

Key points: Tasmania's coronavirus cases now stand at 66 after four more were confirmed late Sunday

Tasmania's coronavirus cases now stand at 66 after four more were confirmed late Sunday Investigations into two of the state's cases have so far failed to find the source of transmission

Investigations into two of the state's cases have so far failed to find the source of transmission The Public Health Service says people are likely to die from the virus in Tasmania

It came as he also warned "there will be deaths" in the state.

"Tasmania is not immune from this; there will be at some stage community transmission. That is going to occur," the Premier said on Sunday.

"I expect also unfortunately that there will be deaths. People will die and we need to ready ourselves for that."

The state's tally of confirmed cases stood at 66 after four new cases were revealed on Sunday night.

Three cases were associated with the cruise ship Ovation of the Seas and one with Voyager of the Seas.

All cases were already in quarantine.

Three new cases were announced on Saturday: one in the north-west, one in the north and another in the state's south.

"Two of those cases we're continuing to aggressively track and trace," the Premier said.

Deputy Director of Public Health Scott McKeown said Public Health Services was continuing to investigate the two cases in the Devonport area and its surrounds, one a health worker from the Mersey Community Hospital.

"Both of these cases have not travelled overseas and have not been on cruise ships recently," he said.

Late on Sunday night the State Government confirmed the two cases were continuing to be investigated.

Dr McKeown urged all Devonport residents to adhere to social-distancing rules and said there had been an increase in testing in the area.

"We can't rule out community transmission in the Devonport area. We don't have evidence of it but we want to investigate it further," he said.

It has been revealed the healthcare worker spent 30 minutes working at the Mersey Community Hospital while infectious.

Health Minister Sarah Courtney said the employee had no close contact with patients while infected, but a handful of employees were being monitored and quarantined.

Future of food markets uncertain

Early on Sunday, the Premier asked Tasmania Police to investigate social distancing at Hobart's Farm Gate market amid concerns crowds were not adhering to guidelines.

The market in Hobart has been operating as an essential service. ( ABC News: Lucy MacDonald )

"If you are going to have a crowd that looks like a football crowd at a food market then that's not working and we will need to look at the model going forward," he said.

The Sunday market has been allowed to continue operating because it has been deemed an essential service.

Stallholder with the Tasmanian Juice Press, Bentley Deegan, said vendors had to look at new ways of doing business to remain viable.

"As far as surviving goes, I guess we do really need to adapt," he said.

"We're moving to an online shopfront.

"Our customers have adapted really, really well to this.

"A lot more people are being health conscious, going for our healthier cleanse of vegetable juices.

"[We're trying] to move to that old-school delivery format where people can get what they need and not have to worry about leaving the house, still self distance and act responsibly."

Tougher measures imminent

Mr Gutwein also reiterated plans to turn the island state into "a fortress", flagging a further clampdown on quarantine measures after Sunday night's national cabinet meeting.

The National Cabinet has decided to limit the number of people present at gatherings to two people.

Previously the maximum gathering size was 10 people.

The rule does not apply to people who live together.

In addition, outside gyms and skate parks will be closed from Monday.

Boot camps will also be reduced to two people.

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Mr Gutwein said before the Prime Minister's announcement on Sunday evening that while Tasmania was not in the same position as the larger states, his intention was to take steps in line with those states.

"My reasoning is that around the world, as the impacts of the spread of the virus have gotten worse, harder actions have then been taken," he said.

"I'm of the view that we should take those harder actions now to stop things from getting worse."

"I'll have more to say about the steps we will take and the timing of those steps in coming days."

People urged to comply with quarantine

More than 1,800 tests have been completed to date across the state.

Dr McKeown said 314 people have been in close contact with infected cases across the state, and of those 257 were linked to cruise ships or overseas travel, placing significant demand on the state's health system.

About 5,000 people are in isolation.

Dr McKeown said people were likely to die from the virus in the state and Public Health Services was "buying time to get the health system ready".

More than 1,000 compliance checks have been completed across the state, 98 per cent of which have found people are correctly self-isolating.

The Premier said while no-one had been charged for flouting isolation rules, a small number of people were being interviewed and could face fines of up to $16,800.

"If found to have broken the rules … they will be summonsed and they will be prosecuted," he said.

"Ask yourself: is this essential what I'm doing? If it's not, don't do it," the Premier said.