PM Jacinda Ardern defended the government's testing regime, saying it was down to medical professionals to make the choice to test.
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A Hamilton City Council worker is in self-isolation while being tested for Covid-19.
While the council would not comment on the specific case, citing privacy, the Herald understands the staff member had been overseas before showing flu-like symptoms.
The council employs more than 1000 staff across its 27 business units.
Hamilton City Council people, safety and wellness manager Dan Finn said his team had been communicating regularly with staff and management around its response to Covid-19, including advising staff with concerns or queries to contact dedicated support people.
Some other staff members were on leave as a precautionary measure as people they shared a home with returned to New Zealand after the travel rules changed on Monday morning.
The council was also working closely with the Ministry of Health and other agencies to get the best advice.
It has introduced other precautionary measures including additional cleaning, asking staff to minimise face-to-face meetings and to stay at home if they are feeling unwell.
All work-related international travel has been stopped and any domestic work-related air travel must be signed off by the chief executive.
The council has also postponed a park opening, an open day at Waikato FMG Stadium and a citizenship ceremony and scaled back other events to adhere to the Government's advice of avoiding gatherings of more than 500 people.
