Boris Johnson stressed the need for caution despite recent falls in the number of coronavirus cases as he warned that the virus “still presents a significant risk”.
But recent data which showed a drop in cases was “encouraging”, the Prime Minister said.
The comments come after a senior minister told the Daily Mail that Covid is “all over bar the shouting”.
But Mike Tildesley, who sits on the scientific advisory body Scientific Pandemic Influenza Group on Modelling (SPI-M), said that it was not yet clear whether the third wave was turning around.
A recent fall in infections could be attributable to people being less willing to “step up” and get tested ahead of their summer holidays, Dr Tildesley, professor of infectious disease modelling at the University of Warwick, said.
Meanwhile testing among school pupils in England will have reduced since the end of term last week, he added.
Official figures on Tuesday showed the number of Covid-19 cases newly reported in the UK dropped for the seventh day in a row.
Asked about the prospect of looking back on the pandemic in October, the Prime Minister told LBC Radio: “We’ve seen some encouraging recent data, there’s no question about that, but it is far, far too early to draw any general conclusions.”
He added: “The most important thing is for people to recognise that the current situation still calls for a lot of caution and for people just to remember that the virus is still out there, that a lot of people have got it, it still presents a significant risk.”
Meanwhile Mr Johnson rejected Cabinet minister Michael Gove’s claim that people who refuse to get a vaccine are “selfish”, adding: “I would put it the other way round and say that if you get one you are doing something massively positive for yourself, for your family.”
Mr Johnson was challenged over his plan to require a vaccine certificate to visit nightclubs from the end of September.
“It’s a very positive thing to do to get a vaccine,” he said.
August 16 is nailed on - there has never been any question of a review date for August 16Boris Johnson
The Prime Minister added: “People can obviously see, when you look at things like travel, like mass events, it’s going to be one of those things that will help you, not hinder you.”
Meanwhile, he said that the number of 18 and 19-year-olds who had come forward to get the vaccine – almost 70% so far – was “stupendous”.
And asked whether it was certain that those fully vaccinated would be able to escape having to self-isolate from next month if they come into contact with a positive coronavirus case, the Conservative Party leader said: “August 16 is nailed on – there has never been any question of a review date for August 16.
“I’m very pleased that this is a country that now has the highest proportion of vaccinated adults of any country in the world – that is enabling us to make the economic progress we are.”
Meanwhile Dr Tildesley said that he hoped that some sort of normality could resume in the autumn.
Asked about the decline in case numbers, he told Times Radio: “Because schools in England closed last week, we haven’t got secondary school pupils doing regular lateral flow testing and so we’re not necessarily detecting as many cases in younger people.
“It’s also been suggested by some that, possibly, because of a high number of cases, because of the summer holidays approaching, people might be less willing to ‘step up’ to testing when they have symptoms.
“What we really need to do is monitor hospital admissions, because at the moment of course they’re still going up – now, of course there is a lag when cases go down, it always takes a couple of weeks before hospital admissions turn around – but if we start to see as we get into August, if we start to see hospital admissions going down as well then I think we would have much stronger evidence to suggest that this third wave is starting to turn around.”
Dr Tildesley added that Covid-19 “isn’t necessarily all over bar the shouting quite yet”, adding: “I think people are aware that Covid isn’t quite over.
“I really hope that this is the turnaround of the third wave and as we get towards the autumn we really are very much getting back to normal.
“But I think, actually, people are doing pretty well at using their own judgment and exercising caution when necessary.
“It’s pretty clear that we are not back to kind of pre-pandemic levels of mixing – people aren’t socialising in the same way they were before the pandemic, hopefully that will come.
“But I think that’s probably partly what we’re seeing in the data – that we’re not seeing a big surge in infections because people are taking a little bit of time to get back to normality.”
Asked about reports that England is expected to open its borders to allow US and EU travellers who are fully vaccinated against coronavirus to enter without the need to quarantine, he added: “If you talk to an epidemiologist, clearly there’s a risk attached to anything like that kind of relaxations but of course if people are double vaccinated, if people are coming from countries that have similar levels of infection to ours, and no real risk of variants of concern that might be introduced into the country, then it’s probably a practical decision that has to be made to try to support the tourist industry.”