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The sister of an infected man recently returned from Iran and a health worker, both in Sydney, were confirmed on Monday as the first locally-acquired infections.
The 53-year-old male health worker - thought to be a doctor - hadn't travelled for many months and it's unclear how he contracted the virus..
"Our key focus at the moment is to contact staff or patients that may have been close contacts of this gentleman," NSW chief health officer Dr Kerry Chant told reporters.
Australian chief medical officer Professor Brendan Murphy earlier issued a sobering warning for the communit,y stating "it's no longer possible to absolutely prevent new cases coming in".
Efforts are now focused on quickly isolating newly infected people, dissuading Australians from heading to virus hotspots, and in the case of Iran using a travel ban to slow down that route of infection.
Australia has now had 33 confirmed cases: 15 Chinese tourists or residents who had visited China, 10 passengers on the Diamond Princess cruise ship in Japan, six people who recently returned from Iran and the two cases acquired in Sydney.
Fifteen of the cases have been cleared.
Australia recorded its first coronavirus death on the weekend when James Kwan, 78, died in a Perth hospital.
He and his wife were among evacuees from the Diamond Princess, with both falling ill after being taken to Darwin for two weeks in quarantine.
Tasmania reported its first case of coronavirus on Monday, a 40-year-old man recently returned from Iran.
The federal government is looking at strict new powers under biosecurity laws which could be used to detain coronavirus-positive people.
Attorney-General Christian Porter says the government could bar people and large groups from attending public places and authorities would be able to quarantine an entire building similar to the Japanese government's approach to the Diamond Princess.
Prime Minister Scott Morrison on Monday said he'd met with Reserve Bank heads to discuss the impact on the economy.
"This is a health crisis not a financial crisis, but it is a health crisis with very significant economic implications," Mr Morrison told parliament.
Health Minister Greg Hunt said Canberra had imposed a travel ban on arrivals from Iran on the advice of chief medical officers.
Other outbreaks in Italy and South Korea were not considered as risky because they were contained and localised.
Prof Murphy said it was a different situation in Iran where alarm bells had been ringing after more than 50 deaths from about 1000 cases.
"We had a very high index suspicion that the caseload in Iran was much greater than being reported, because of the death rate," he said.
The federal government has upgraded its travel advice for Italy.
Australians intending to travel there have been told to exercise a high degree of caution across the entire country and to reconsider the need to travel to 10 virus-affected towns in Italy's north.
The government has also said any health and aged-care workers returning from Italy and South Korea must not go to work for two weeks because they could infect vulnerable populations.
Globally there have been more than 88,000 infections and almost 3000 deaths spanning 67 countries and regions.
The health emergency has seen stock markets plunge across the world, triggering fears of a global recession.
The Australian market on Monday closed down almost one per cent which was a vast improvement on the more than three pert cent dive it took in earlier trading.
NSW Health Minister Brad Hazzard on Monday urged people to stop shaking hands and instead pat others on the back.
"No hand-shaking," he said. "It's very automatic but don't do it."