Meanwhile, HSE chief Tony O'Brien urged caution.
Meanwhile G7 finance ministers meeting in Italy vowed to unite against cyber crime, as it represented a growing threat to their economies and should be tackled as a priority.
But around a fifth of NHS trusts were hit in the attack, forcing them to postpone operations and procedures over the weekend and more could be revealed when people return to work this morning.
At the height of the attack Friday and early Saturday, 48 organizations in the NHS were affected, and hospitals in London, North West England and Central England urged people with non-emergency conditions to stay.
Wallace said the government used to contract for computer services across the entire NHS but that in 2007 that was stopped and left to the individual trusts. "Organisations across Australia have been taking active steps to protect their networks over the weekend", the statement said. "Each health building will have an IT representative to provide assistance in the morning".
"We're obviously working with that business, the Australian Cyber Security Centre is engaging with them".
The anonymous specialist, known only as MalwareTech, issued a warning that hackers could upgrade the virus to remove the kill switch.
There have been calls in Britain for an inquiry into the circumstances surrounding Friday's major incident, with the United Kingdom government and NHS chiefs facing questions over their preparedness and the robustness of vital systems.
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"Thank you to our staff, patients, partners and stakeholders for supporting us over this time".
New versions of the worm are expected, they said, and the extent - and economic cost - of the damage from Friday's attack were unclear.
"We've seen no impact in the health system which is important, we've had no reports of any government agencies impacted by this".
"Since the global coordinated ransomware attack on thousands of private and public sector organisations across dozens of countries on Friday, there have been no sustained new attacks of that kind".
Wainwright said he was anxious that the ransomware attack might spread further once people return to work on Monday and log on to their computers.
'There have also been other reports made to our online cyber security network and the difficulty is of course there are literally hundres of instances of ransomware in Australia each week and so we are now seeking to confirm whether these are examples of the particular ransomware that has cause so much havoc for example in the United Kingdom'.
The virus exploits a vulnerability in Microsoft Windows software, first identified by the US National Security Agency, experts have said.