The Ministry of Health has confirmed a fourth positive coronavirus test in New Zealand - the husband of a woman already infected.

The man, in his 30s, returned with his wife from a trip to Italy last week - they have children at two North Shore high schools, which have also been alerted. The family is in isolation, and have been subject to social media bullying.

"Contact tracing is under way and close contacts are already in self-isolation," the Ministry of Health said in a statement.

It said further details would be provided later today as well as any results of other testing around the country.

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Earlier, the wife of the fourth case had already revealed her husband had also tested positive, bringing the numbers of cases in this country to four.

The woman, who had visited northern Italy, has been in isolation at her Auckland home with her husband and two high-school aged children.

The Herald revealed yesterday the woman had been seen at the Westgate Medical Centre, where protocols had been followed - but a staff member told the Herald they were upset they had only found out about the confirmed case in the news.

The woman insisted yesterday she had tried to follow all medical advice, including visits to two medical centres.

"People assume we are these monsters that have all these symptoms out in public infecting people. We've had very, very, very few symptoms," the woman told Stuff.

"Can you imagine my children going back to school at this point? There are already people pulling their kids out of school when they've been told really clearly that there is absolutely zero threat to their children."

Both schools the children attend, Westlake Boys' and Girls' high schools, have been alerted.

Health officials were yesterday also contacting 100 passengers who travelled on two domestic flights between Palmerston North and Auckland with the infected woman before she was diagnosed.

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But the Ministry of Health maintains the chance of widespread community outbreak remains low.

Asked whether parents had kept kids home from the two schools yesterday, Ministry of Education deputy secretary Katrina Casey said: "We are aware that both schools experienced more student absences than is usual for this time of the year but we know they are following our normal procedures for responding to unexplained absences."

On Tuesday night, health officials called for calm and restraint after "sustained bullying" on social media of a family in isolation.

In Thursday's briefing Director-General of Health Dr Ashley Bloomfield called out the "keyboard warriors" who have been critical of those impacted by the disease.

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People affected by the virus should not be criticised, he said.

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"We don't want to be counter-productive and discourage people from coming forward."

The Westgate clinic says it followed Auckland Regional Public Health guidelines and the patient didn't make close contact with anyone but the doctor, but a staff member told the Herald because they were concerned more people could now be at risk of exposure.

The medical clinic was itself busy, and located in a busy West Auckland shopping centre.

The staffer was disappointed the medical centre had failed to inform staff that the woman was a confirmed case.

They were also concerned at her first diagnosis at a different clinic where she was prescribed antibiotics - which won't help cure coronavirus - and then sent on her way.

The person now felt an obligation to let the public know that the infected woman had been through the urgent care clinic on Monday.

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"The person came in as an urgent care walk-in patient. The clinic does have a protocol, an isolation room specifically designated for people and unfortunately the situation happened in which the patient didn't identify themselves as someone that may have been at risk, even though we do have signage asking people to do that.

"The physician that saw her was unaware of anything until he started talking to her and as a result he was exposed.

"The swab for the coronavirus was done and the results would have come back yesterday [Wednesday].

"I assume he would have been notified being the person that submitted the test, so unbeknownst to me and anybody else at the clinic he left, and was asked to self isolate."

It meant the doctor worked all day Tuesday and part of Wednesday after being exposed.

Earlier, the third confirmed case of Covid-19 was an Auckland man in his 40s who had not been travelling overseas.

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Bloomfield told reporters it was believed to be an instance of transmission between family members.

This is the first case of person-to-person transmission inside New Zealand's borders and Bloomfield did not expect it to be the last.

The partner of the now-infected man had returned home from Iran on Qatar Airlines flight QR0920 from Doha to Auckland.

That person did so a week before the Government imposed travel restrictions on Iran.

The Ministry of Health maintains the chance of widespread community outbreak remains low.

Following the third confirmed case, health officials have now notified another two Auckland schools - Auckland Grammar and Ormiston Junior College.

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Two students from the infected man's family are at home, in self-isolation, but none of the family required hospital-level care and Bloomfield confirmed there was no risk to students or staff at either school.

The first confirmed New Zealand case was announced by officials last Friday.

New Zealand is one of 77 countries and territories to report confirmed cases of Covid-19.

More than 95,000 cases of the virus have now been reported globally - more than 51,000 have recovered. More than 3200 people have died.