The Covid-19 cluster at an Auckland high school has worsened today, with the number of students and staff testing positive for coronavirus now rising to 16.

The decile 7 Catholic girls' school Marist College, with 759 students in Mt Albert, has been closed all week after a teacher tested positive for the virus at the weekend.

This afternoon, board of trustees chairman Stephen Dallow confirmed that 16 students and staff have now tested positive for the virus – rising by one from yesterday.

Ten of the cases are teachers, including principal Raechelle Taulu, and six were students.

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One student is in Year 9 and the others in Year 12 and Year 13.

There are now 368 confirmed or probable cases of coronavirus in New Zealand.

In a video update to parents and caregivers this morning, Dallow said many of the school's cases are now "feeling much better" and some are back to full health. To his knowledge, none are in hospital or requiring serious care but some are "experiencing the full-on flu".

"We do ask our families to keep us updated," he said.

He urged students to keep in contact with each other but also revealed that yesterday they learned of some online bullying by students from other schools.

The school principals have been spoken to this morning and the bullying was being "dealt with swiftly", Dallow said.

Students from two Auckland schools have ridden the same school bus as Marist College students with coronavirus while they were infectious, it's also emerged.

A "handful of students" from Lynfield College and Mount Albert Grammar were now considered casual contacts, parents were told yesterday.

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Mount Albert Grammar headmaster Patrick Drumm told parents students aboard nine Kiwi Coach bus rides last week are now at risk of contracting coronavirus.

Covid19.govt.nz: The Government's official Covid-19 advisory website

"While this will be concerning for you, Auckland Regional Public Health Service considers most of the students on the buses to be at low risk of catching Covid-19," Drumm said.

"Between four and six students who sat near the infected student on each bus are at slightly higher risk, but it is impossible to identify who these students are."

New Zealand has another 85 new coronavirus cases today and one patient is now being treated in a hospital intensive care unit.

As alert level 4 began at 11.59pm on Wednesday, these students are now in lockdown with the rest of the country.

"We ask that they take this seriously and remain inside, with no visitors," Drumm said.

"They should not be socialising with others - even at a distance, and they should not go out for essential supplies to the supermarket or pharmacy for 14 days."

There was a small possibility that students aboard these buses will develop symptoms, such as a fever, cough or difficulty breathing.

"In this case, please ring Healthline on 0800 358 5453 and tell them that your child is a contact of a Covid case," Drumm said.

"They will advise you what to do."

Mount Albert Grammar students were asked to respect the privacy of the Marist students and not to speculate about or confirm who they were through text messages or on social media.