Another North West Regional Hospital worker and a patient at the hospital have tested positive for coronavirus.
They are included in the four new cases of COVID-19 diagnosed on Sunday, bringing the state's total number of confirmed cases to 86.
Of the other two cases, one was related to the Ruby Princess cruise ship and one was a contact of a known case.
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Premier Peter Gutwein on Monday confirmed two of the newly diagnosed cases were part of the North West Regional Hospital "cluster".
He said an investigation was underway to identify the close contacts of the two cases in the community and the hospital, and the risk to people at the hospital.
"We are throwing additional resources at this," Mr Gutwein said.
Mr Gutwein said locking down the state's regions was not being considered at this time but he would take this step if necessary.
"I'd urge you, unless you need to travel outside of the North-West region for work or another reason, don't," he said.
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Chief Medical Officer Tony Lawler said the patient who tested positive, an adult male, had been transferred from the North West Regional Hospital to the Mersey Community Hospital on April 2.
"At the time of transfer the patient had not developed any symptoms," Professor Lawler said.
He said six staff at the North West Regional Hospital and 21 staff at the Mersey were now self-isolating.
He said further transfers between the two hospitals would not occur unless specifically approved and a process was underway to work out the details of how the North West Private Hospital would be utilised in the COVID-19 response.
Public Health director Mark Veitch said there were a total of five cases associated with the North West Regional Hospital.
"Four of those case are health care workers at the hospital. One of those cases was a patient," Dr Veitch said.
"We have no evidence of further spread in the community in the current circumstances."
He said if there was evidence of further spread, enhanced testing would be conducted.
Of Tasmania's 86 confirmed cases, Dr Veitch said 40 cases were in the South, 24 were in the North, 19 were in North-West and three were people from interstate who were diagnosed with COVID-19 in Tasmania.
Three people are being treated in hospital and 29 people have recovered.
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