McElnay and Deputy Prime Minister Grant Robertson provided today's Covid-19 update after the small Taranaki town of Stratford confirmed Delta had been detected in its wastewater.
Both of the self-isolation deaths were under investigation "in order to identify any improvements that may be needed," McElnay said.
"I want to add my condolences to the friends and family of the person who has passed away," Robertson said.
He said the person "was admitted to hospital on the 1st of November and they discharged themselves on the 3rd."
"I am proud of the economic support that we have provided.
"We will continue to support businesses."
Robertson said the Government was committed to allowing Aucklanders to leave the city over Christmas.
"We're doing everything we can to make sure that happens," Robertson said, adding that vaccination certificates would "certainly be part of it" and Covid testing before leaving the city might possibly be another layer of protection.
"It's a big job. We're working through the options."
Officials are committed to letting Aucklanders move about the country by Christmas, he re-iterated.
While allocated time slots for leaving the city aren't completely ruled out, "I don't think that's a particularly practical way to do it", Robertson added.
Robertson said he "continues to keep hope alive" that every DHB will be able to hit the 90 per cent vaccination target.
"We are making good progress," he said.
With millionaire Murray Bolton's landmark win against MIQ today, Robertson was asked when everyday Kiwis will also be able to surpass MIQ.
"We've indicated that that will be early in the new year," he responded, pointing out that the legal argument Bolton took to court included the use of a private jet.
"Mr Bolton's circumstances are quite specific to him.
"I appreciate the fact there is a great desire of people to come back to New Zealand. We're facilitating that the best we can."
But he said the nation will have to "continue to move carefully and cautiously" for the time being given the current outbreak.
Asked why more than 700 people with Covid-19 are self isolating at home but people without the virus are still in MIQ, Robertson said there are "significant logistical issues" to changing the framework.
But the Government is moving in that direction, he said.
"This is all about accumulation of risk. If we've got any risk we can manage or limit, that's what we should do," he said of keeping new arrivals to New Zealand in MIQ for now.
Both Robertson and McElnay said they have been in touch with Auckland's DHBs today and have been assured "the system is able to cope" with the number of current hospitalisations.
"We were notified last night of a strong positive Covid-19 wastewater test result, indicating the detection of the virus in the wastewater system of Stratford," iwi kaiwhakahaere Rachel Rae said.
The sample was taken on November 1, Rae said.
Taranaki has one of the lowest vaccination rates in the country, with 68 per cent of the eligible population fully vaccinated, and 85 per cent with a first dose. In Stratford, the stats are slightly lower, with 63.66 per cent of residents double dosed, and 80.86 single jabbed.
Rae said the Ministry of Health was "urgently scrambling" to see if a recent MIQ returnee travelled to Stratford after being released.
"We've had a wastewater result before...that didn't turn into any cases," Robertson said today.
"Wastewater results are a useful indicator for us" but the next step is testing to see if it holds up, he said.
"There's a number of potential explanations. The best thing we can do is get as many people tested [as possible]."
The ministry reported yesterday the death of a 40-year-old man isolating at his Manukau apartment was not vaccine-related.
The man's death was referred to the coroner, who will determine whether it was coronavirus-related, officials confirmed.
The Northern Region Health Coordination Centre and the ministry will review the public health and clinical oversight of the person, with independent input.
Kia ora, TDHB have confirmed a positive Covid-19 wastewater result in Stratford. Those in the Stratford district and wider Taranaki region with symptoms are encouraged to get a test. Testing and vaccine info >> https://t.co/EvjpME47eNpic.twitter.com/nOu7C6onkR
— Taranaki Civil Defence Emergency Management (@TaranakiCD) November 4, 2021
Across Auckland's three district health boards, 92 per cent of all eligible people have had their first dose and 82 per cent have had their second.
Northland has seen vaccinations rise to 80 per cent of first doses and 66 per cent for second.
Waikato was at 87 per cent for first doses and 74 per cent fully dosed.
Where to get a vaccination in Auckland - without a booking
This map shows large vaccinations centres from the Unite again Covid-19 information page. For more detailed information about your neighbourhood visit Healthpoint.