But "second wave" was also disputed, thanks to its roots in describing influenza outbreaks, said Dr Tom Frieden, who served as director of the US Centres for Disease Control and Prevention under Barack Obama's presidency.
He said that the phrase "implies that this [virus] will act as the flu acts", with seasonal peaks and troughs and relatively rapid mutations.
"We don't know that - it might, but it might not."
The World Health Organisation has warned against "thinking about seasons".
Margaret Harris, a spokesman at the United Nations agency, told a virtual briefing this week: "What we all need to get our heads around is that this is a new virus and this one is behaving differently."
She added that the virus was instead unfolding in "one big wave" - a trend that was evident from a global perspective.
It is now six months since the World Health Organisation declared an international health emergency and the pandemic is accelerating. Globally, cases doubled in the last six weeks.
So it was not surprising that countries were seeing an upturn in cases as they attempted to return to some semblance of normality, said Hans Kluge, WHO's regional director for Europe.
He said that the first wave had never really stopped.
In Europe, this trend was pronounced, with 36 countries seeing an increase in infections, based on a seven day rolling average compared with the week prior.
Many were creeping up only marginally from a very low base, including Greece and Norway, where the rate remained below three per million people.
But among the 10 countries with rates of daily new cases above 40 per million, nine were experiencing rises, including Luxembourg and Romania.
Analysis by the Telegraph of the Government Response Stringency Index, published by the Blavatnik School at Oxford University, suggested those countries that had the most relaxed lockdown restrictions were seeing more pronounced surges.
It also showed that spikes in Europe occurred approximately two months after governments first began significantly easing lockdown measures.
Spain's surge began around 50 days after the country's first major drop in its lockdown stringency score, while Belgium's began just more than 60 days later.
Professor Jose Vazquez-Boland, chairman of Infectious Diseases at the University of Edinburgh, said: "What we are facing is a comeback of community transmission after removing the lockdown measures. We have to be clear that the efficacy of lockdowns is only temporary.
"There will be a resurgence of new cases every time social restriction measures are lifted as long as the virus remains in circulation."
While headlines focused on a resurgence of cases in Spain amid the UK's decision to introduce a quarantine for holidaymakers returning from the country, other parts of Europe were also on a worrying trajectory.
In Luxembourg and Croatia, where the average stringency index over the past fortnight had been under 30 out of 100, there were significant increases in Covid infections.
In the UK and Portugal, where the government response over the last two weeks averaged 64 and 70 respectively, there had not yet been a significant rise in the infection rate, although cases were beginning to climb again across Britain - from 8.6 to 10.2 in the space of two weeks.
