Health Minister Chris Hipkins said today the source of the outbreak remained a mystery.
"One of the things we are testing is whether it came in on frozen goods," Hipkins told Newstalk ZB's Mike Hosking, referring to the Americold coolstore in Mt Wellington.
That's where one of the positive cases - one of the coolstore workers - showed the earliest symptoms, on July 31.
Experts have cast doubt on whether Covid could have been picked up from imported goods but Hipkins said it needed to be investigated.
"The fact this all seems to have originated in a coolstore means we need to pursue that and look at that. We don't yet know how these people got this virus in the first place.
"The epidemiology suggests it has come from Australia or the UK because that's the epidemiological DNA of it but none of our cases in managed isolation match the same sort of fingerprint as the outbreak we have in Auckland. It's a mystery. Everyone can speculate but they are not based on facts."
Hipkins' comments came as the total number of cases in the country has reached 1271, after 13 new cases were confirmed yesterday - 12 in the community and one in managed isolation.
Other young people who have tested positive include a baby under the age of 1, two preschoolers, a boy aged between 5-9 and three students aged between 10-14.
Hipkins was confident all port, border and isolation/quarantine workers would be tested by the end of the week.
"There are a lot of them and while we have testing running at full bore we are prioritising those most at risk. Those who who are more likely to be people coming off the ships, or working on the ships," Hipkins told Hosking.
He said people working airside at Auckland Airport, dealing with passengers, were more at risk than, say, a coffee shop worker on the other side of the airport.
Hipkins insisted contact-tracing was operating at a gold standard level. "It's ultimately a human system. We are rapidly identifying people within 48 hours of them being identified as a contact and getting them tested when appropriate."
Hipkins said New Zealand was in the same consortiums as the UK and the US to help fund research and get access to a vaccine as quickly as possible. "We are doing everything we can to make sure we are at the front of the queue for that."
Director general of health Dr Ashley Bloomfield said all 12 community cases are Auckland-based and none had travelled outside of the region. All have a connection to the existing cluster.
In the AUT case it appears the alert level 3 lockdown came into force just at the right time to help reduce any potential further spread.
The student only started experiencing symptoms on Saturday, but authorities have determined their infection period began on Thursday.
In an email to all staff vice-chancellor Derek McCormack said the university understood the student's movements and physical interactions were limited to the MBA programme and specific areas of the city campus.
AUT campuses have been closed since Auckland moved up alert levels on Wednesday and were thoroughly cleaned that day.
The Ministry of Health also reported yesterday it was aware that one of the positive cases previously confirmed in Tokoroa was present at the Toi Ohomai Institute of Technology Tokoroa campus on August 10 and 11.
In a statement to the Herald, the Ministry said public health services have several measures when a potentially infectious person has attended or visited institutions.
They included temporary full school closures, deep cleaning, close contact tracing and testing, casual contact tracing, and giving advice.
Schools with links to cases have provided information directly to affected people.
When the exposure risk is regarded as significant, the ministry has publicly issued information primarily through the 1pm case update press conference.
When asked to confirm the total number of cases linked to education providers, the ministry said it was not giving additional detail at this stage, while investigations continued.
A student at the University of Auckland tested positive for Covid-19 last week, but the risk to staff and other students is believed to be low as the student hadn't been on campus since the end of July and doesn't live in student accommodation.
Manukau Institute of Technology students have also been told a student studying in TechPark's general engineering area based at South Campus, Ōtara, has tested positive, but was not on campus while they were infectious.
Meanwhile, Mount Albert Grammar School headmaster Pat Drumm told parents in the school's weekly newsletter that a student who tested positive for the virus was "doing very well".
About 100 close contacts made up of staff and students are self-isolating.
Confirmed cases have also been linked to Glamorgan School at Auckland's North Shore and South Auckland's Southern Cross Campus.
All pupils and their families have been asked to be vigilant for any symptoms.
A pre-schooler at Taeaofou/Puaseisei preschool on Winthrop Way, in Māngere East, has also tested positive.
