But Bloomfield has indicated he won't ask the police to take action over the Covid breaches.
Bloomfield says there were no new community cases overnight.
Bloomfield told Newstalk ZB's Mike Hosking he was looking for zero days of community cases to determine when lockdown should come to an end and the alert level lowered.
He said one of the cases which sparked Auckland's lockdown had been about for a week while infectious and that was a key reason why the alert level was raised.
He said he wasn't getting the police involved so far this lockdown because the families needed support to navigate isolation.
On not getting police involved at the outset, Bloomfield said: "In the first instance we get our health providers to go out because often what it is the families just need support to stay isolated safely. If we have any issues or challenges or if we can't people then we don't hesitate to involve the police."
He said he spoke yesterday to the Police Commissioner who told him they are "willing and ready and standing by as they need to".
He said in this outbreak close contacts were being called daily and if nobody was at home, health officials would then involve the police.
Bloomfield imagined the MIT student that sparked this lockdown "wasn't having a great time" at the moment.
"I imagine he's extremely remorseful at the moment. It's a pretty heavy burden to bear," he said.
Bloomfield said with the current cluster with 15 cases and high infection rates in families, the positive tests from casual exposure was a concern in this outbreak.
Bloomfield said it was difficult to cordon off a community in this outbreak, unlike what happened in Sydney during a recent outbreak.
He revealed officials discovered the connection between the two families on Sunday morning
"It really helped our investigation to enlist that information early. It was volunteered early and that's actually really helpful with what next steps we take."
Bloomfield said there would be an update tomorrow on how many people had been vaccinated, saying they were ahead of schedule.
It comes as Prime Minister Jacinda Ardern delivers her most scathing rebuke of rulebreakers so far, saying their actions were not tolerable.
However, a member of a Papatoetoe family who went to work at KFC Botany and later turned out to test positive for Covid-19 says she was never told to self-isolate.
The woman, known as Case L, told Discovery the Prime Minister's comments - that she should have been self-isolating - were "upsetting", given she claimed she had received information saying the opposite. She now wants an apology from the government.
Case L is the sister of a Papatoetoe High student, Case I, who had earlier been considered a casual-plus contact. She says her sister had received a text saying she needed to isolate but family members did not. She went to work at KFC on Tuesday February 23 and tested positive on Friday February 26.
"It's not fair on our end that we're getting all this backlash for something that we haven't actually done," she told Discovery.
Ardern's office said letters from public health were sent to the family on February 17 and 19 saying they needed to be tested. Case L's family said such advice was never received.
"If they tried to contact us multiple times and send us letters and stuff, where is this evidence?" she told Discovery.
Ardern denied the claims the woman had not been told. In an interview with The Hui, Ardern said the government had reviewed the situation and information sent to Case L's household.
Most family members in the Auckland cluster, she said, were asked to be tested and remain in isolation until they received their results. "They had two school members who were not meant to leave their home until they were tested," Ardern told The Hui.
Ardern said dealing with those who broke the rules was a "matter for police, not politicians".
The police, however, can only launch an investigation if they are asked to by the Ministry of Health - and at this stage say they have received no such referrals from Bloomfield's ministry.
A spokesman for the Ministry of Health said: "Decisions on referring matters to the police for prosecution may be made at a later date.
"At this stage, the full focus of the Ministry of Health remains on managing the Covid-19 response."
Pressure will now be mounting on Bloomfield after it was yesterday revealed that the mother of a now Covid-infected family contracted the virus after going for a walk with another Covid-19 positive mother during the first February lockdown.
That will be compounded by the fact Ardern yesterday revealed another infected person – the MIT student who went to the gym after their test – had gone through a managed isolation facility late last year.
During his stay, he experienced a "sustained period of propaganda" around the need for self-isolation, Ardern said.
Microbiologist Siouxsie Wiles said the last thing needed at the moment was people not disclosing things because they're frightened of being prosecuted.
"I can totally understand why people would want it [prosecution] and think it would be a deterrent but I worry it would be counterproductive."
"But how you communicate those consequences, it may be better to talk about the health of their families and their community, appealing to that more than to the threat of prosecution and legal consequences."
Ardern was not mincing her words when talking about the rulebreakers at her post-Cabinet media conference yesterday.
"It is not appropriate and it is not okay, the members of a team of five million to let the rest of us down," Ardern said.
Previously, Ardern would only go as far as saying she was "frustrated" with the rule-breaking gymgoer.
But yesterday, her clear frustration was directed at members of the Covid-infected families flouting level 3 rules.
"No one – no minister, nor politician – none of us that I have spoken to think that this is tolerable … What has happened here has been a clear breach, and everyone is frustrated by it."
She would not, however, go into detail as to what sort of punishment rulebreakers should receive as that was a "matter for police".
A police spokesperson said that officers can only investigate a breach, where someone failed to comply with health orders, if there is a referral from the Ministry of Health.
"At this stage, we have not received referrals and we continue to focus on working with individuals, whānau and communities to support them to isolate and be tested."
In September, the Ministry of Justice revealed that until the end of June last year, roughly 1000 charges were filed relating to breaches of the Covid-19 act.
Some 159 of those 1000 were convicted; 24 per cent were jailed and 26 per cent were given community service.
In terms of using the police to prevent lockdown breaches, Bloomfield said he has been in talks with Police Commissioner Andy Coster about this very issue.
"If they [health workers] need support from police because they can't find people, then we absolutely involve the police."
Ardern has called on people to "continue to support and back each other".
"If that means calling a family member or a work colleague out for not following the rules, we should do that."
But, she said it should be done "with kindness".
Meanwhile, Ardern rejected criticism from the likes of Act's David Seymour that the Government has not been doing enough to ensure people would not break lockdown restrictions.
She said the Government's system is one with "many layers" of protection.
This includes symptom and isolation tests – including in-person checks – to ensure compliance.
"But we also ask people to follow the rules and to play their part."
"Quite simply, we cannot do this alone."
University of Canterbury law professor John Hopkins said while certain breaches of Covid-19 rules can be prosecuted under the Covid-19 Public Health Response Act but thinks it's unlikely there would be prosecutions relating to the latest outbreak.
"It's subject to what exactly happened on the ground. We don't know officially what's occurred. My suspicion would be [they won't be prosecuted] at this stage, but it's a police decision.
"In this case, you'd have to ask yourself what prosecution would gain. You're trying to get people to adhere to these rules. It's interesting that there has been talk of prosecution now because in the past there's been very little."
"What you're wanting is compliance. You've got to question whether a prosecution in this instance would deliver what you want to the public good, which is to ensure people comply for the future."
There were no new community Covid-19 cases yesterday.
But Bloomfield revealed that there are 21 close contacts at the Manukau Institute of Technology – these are classmates and teachers – and a further 21 casual plus contacts.
He also said there were 154 casual plus contacts of the gym.
All causal and close contacts have been contacted and have been asked to follow health advice.
