This comes after a female patient at the Mersey tested positive for COVID-19 on Thursday.

Coronavirus: All the latest updates on COVID-19 for Tasmania

Public Health deputy director Scott McKeown said a decision to test asymptomatic staff would need to be made by the Tasmanian Health Service.

A spokesperson for the THS said all Mersey staff who request a test are able to have one carried out.

Dr McKeown said close contacts of the COVID-positive case had been quarantined and strong infection control measures were in place.

"There's a very high level of vigilance for symptoms in any staff or in any residents in that ward ... to ensure we pick up early anyone developing the disease," he said.

But Labor leader Rebecca White said the government needed to start testing for asymptomatic cases of COVID-19.

"Not only should we be making sure all of our health workers are tested, and all our frontline workers have priority testing, but we should be doing random testing at places like supermarkets," Ms White said.

"We're encouraged by the fact Tasmania has the capacity to test up to 2000 people now but it's worrying only 600 people were tested [on Saturday]."

Ms White said random testing would provide an understanding as to if the virus was in the community and would give confidence to lifting restrictions.

Meanwhile, Health Minister Sarah Courtney announced maternity services, radiation, oncology, the ICU and the Spencer Clinic at the North West Regional Hospital would be up and running this week.

"Maternity and ICU will be coming on, we are aiming, for the first half of the week. As each service opens we will continue to update the community," Ms Courtney said.

Ms Courtney would not confirm if a deep-clean of the Mersey was being considered after the two Burnie hospitals were reopened.

"We are working through what, if any, further measures are needed there," she said.

When asked if the government was considering providing compensation to people or the families of people who contracted coronavirus in the state's public hospitals, Ms Courtney said this was not something the government was contemplating at the moment.

"The most important thing that we do at the moment is ensuing we are monitoring coronavirus [and] we take all the measures we can in our health facilities," she said.

Speaking as Minister for Women, Ms Courtney said a new partnership with the Federal government would invest an additional $1.4 million into family violence prevention services in Tasmania.

"We are working with many non-government organisations to help ensure we have the right programs in place," she said.

"It's ensuring we have those reach out services. It's ensuring we have accommodation for women and their children who cannot remain at home."

She said the government was also working with GPs and pharmacists to ensure women could access the help they needed.

As of Sunday morning, there were 50 active cases of COVID-19 in Tasmania.

Eight cases were being managed in hospital with one person in intensive care.

Of a total 221 cases, 158 people have recovered and sadly 13 people have died.

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