After news broke of U.S. President Donald Trump and First Lady Melania Trump testing positive for COVID-19, a viral tweet contrasted the spate of positive tests in the White House outstripped the rate of positive tests in all of New Zealand:
In an October 2 2020 tweet, a Twitter user wrote:
There’s more cases of Covid in the White House than in all of New Zealand
New Zealand’s Global Standing During the Pandemic
New Zealand’s effective response to and containment of COVID-19 during the global pandemic was a frequent talking point in mid-2020.
A photograph of a crowded, live rugby game went viral in July 2020, as Americans realized that residents of New Zealand enjoyed fewer restrictions on daily living:
At various times, New Zealand went long periods without a single case of community transmission — even as the pandemic raged and deaths increased in other countries. In early August 2020, news circulated that the United States had issued a travel advisory for New Zealand.
That claim was out of context, however, and appeared to reflect the lifting of restrictions rather than imposition of them:
Nevertheless, New Zealand reported intermittent new cases through the summer of 2020 and into the fall.
White House Cases in September 2020
On the night of October 1 2020, news broke that White House advisor Hope Hicks — who is often in close contact with President Trump — tested positive:
Roughly four hours later and about an hour after midnight Eastern time on October 2 2020, reports that Donald and Melania Trump had also tested positive also began to emerge:
Trump himself disclosed that information at 12:54 AM Eastern:
In an October 2 2020 liveblog, CNN reported that Trump’s family members and Vice President Mike Pence initially tested negative — but noted incubation might take longer, and that further, false negatives are fairly common:
Vice President Mike Pence, as well as various Trump family members – including Trump’s son Barron and daughter Ivanka – may have just tested negative for coronavirus, but that doesn’t mean they are in the clear, one pathologist said Friday [October 2 2020].
Dr. Alan Wells, medical director of the University of Pittsburgh Medical Center’s Clinical Labs, points out the rapid tests used to make quick daily diagnoses do not catch all cases, and any test performed too early will give a false negative.
The White House issued two memoranda, one disclosing the President and the First Lady’s diagnoses, the other indicating that Pence had tested negative.
It appeared the number of cases in the White House as of October 2 2020 was confirmed to be three — Hicks, President Trump, and Mrs. Trump.
New Zealand’s Case Counts on October 1 and 2 2020
Google’s pandemic dashboard showed twelve new cases on October 1 2020 in New Zealand, and zero new cases on October 2 2020:

New Zealand’s official pandemic dashboard (archived) provided an up-to-date summary of current cases. In the two screenshots below, it appeared there were between 32 and 43 “active cases” in managed isolation or quarantine as of October 2 2020 in New Zealand:


TL;DR
A viral tweet claiming that there were more cases of COVID-19 in the White House than all of New Zealand, was — although likely intended as to be a bit hyperbolic — was not, mathematically speaking, true. New Zealand reported twelve new cases on October 1 2020 and zero on October 2 2020, and appeared to have a minimum of 32 “active cases” of the virus as of October 2 2020.
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