Robert Jenrick said “we all have to be very cautious” after 2,948 new UK cases were recorded on Monday.
The government’s scientific advisers have given stark warnings over the increase in Covid-19 cases.
Among those, England’s deputy chief medical officer, Prof Jonathan Van Tam, said the rise was of “great concern”.
Ministers have singled out young people in particular for not following social-distancing rules.
But one expert cited the government’s “confused messaging” and said it was unfair to blame the young.
The seven-day rate of new cases of Covid-19 in the UK has now risen above 20 cases per 100,000 people.
On Sunday 2,988 new cases were announced, which was the highest figure since 22 May.
At the peak of the virus in spring official figures showed there were 6,000 cases a day, although testing was largely only taking place in hospitals.
Estimates suggest there were around 100,000 cases a day at that point.
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Mr Jenrick told BBC Breakfast: “The virus is still very much with us, it’s still concerning.”
He said if people followed the government’s guidance “we should be able to continue to control the virus but we’re going to have to be especially cautious as we go into the autumn and winter”.
“If we all play our own part then we should be able to maintain our daily lives in this sort of new normal but we’ve got to be very cautious because, as you’ve seen, the number of cases is rising.”
Mr Jenrick added there was a particular responsibility on younger people to follow government guidelines on Covid-19, so that infection rates would not spike again.
“We have to keep hammering the message home. Of course the people in those age categories are unlikely to become extremely unwell as a result of having the virus.
“But they are able to pass it on to others,” he said.
“There’s a responsibility on younger people to not just stay at home, obviously to go out and go to work and to enjoy pubs and restaurants, but to do so in accordance with the guidelines.”
His comments follow Health Secretary Matt Hancock’s warning that warned young people that the UK could see a second spike in coronavirus cases if young people do not follow the rules.
“Don’t kill your gran by catching coronavirus and then passing it on,” he told BBC Radio 1’s Newsbeat on Monday.
However, Prof Susan Michie, a behavioural expert on the government’s Scientific Advisory Group for Emergencies (Sage) was critical of the government’s “confused messaging” over the coronavirus restrictions.
She told the BBC News Channel that the constant changing of the guidance and variations between nations had left young people “very confused about what it is they are and aren’t meant to be doing”.
She said ministers had almost signalled “go out and about as usual” to young people by lifting restrictions and added it was unfair to then say “actually you are the problem” rather than take any responsibility as a government for the messaging.
Another Sage member, Prof Andrew Hayward, told Radio 4’s Today programme he was worried that the rise in virus cases over the last few days might “get out of hand” if control measures were not taken seriously.
His Sage colleague, Prof John Edmunds, said although cases were rising from a “relatively low level”, they were now increasing “exponentially”.
He told ITV’s Robert Peston on Monday that the R number – which rates Covid-19’s ability to spread – had risen above 1.0 and the UK was in a “risky period”.
“We can see the epidemic is taking off again… I don’t think we’ve hit that sweet spot where we’ve been able to control the epidemic and allow the economy to return to some sort of normality,” he said.
Earlier Prof Van Tam described the latest change in coronavirus infections across the UK as a “great concern”, adding: “People have relaxed too much. Now is the time for us to re-engage, and to realise that this a continuing threat to us.”
Prof Van Tam added that hospital admissions and deaths were “at a very low level” in the UK and the rise in cases was most prominent among those aged between 17 and 21 – but the country risks following the trajectory of some EU countries.
Wake-up call for the public
Prof Jonathan Van Tam was known for his plain speaking at Downing Street press conferences earlier in the pandemic.
Now England’s deputy chief medical officer has once again come out with a blunt warning about the spread of the virus.
Put simply, he believes there’s been a significant and concerning change in recent days. No longer is this all about local outbreaks – instead, in his view, there is now a general spread of the disease in many parts of the country.
He suggests social distancing has broken down for some parts of society and points to rising case numbers amongst 17 to 21-year-olds.
That does not mean hospital admissions, currently relatively low, will definitely rise. But Prof Van Tam thinks that is very possible.
He implied the next week would be critical as officials and ministers studied the emerging data.
This is a wake-up call for the public to get real about social distancing, from a medical leader who is clearly worried.
The Sage scientists’ comments come as more parts of the UK face tougher restrictions following a rise in the number of cases.
In Wales, the county borough of Caerphilly is to be placed under a local lockdown from 18:00 BST on Tuesday.
Welsh Health Minister Vaughan Gething said there was evidence of “community transmission” in the area which had largely been caused by a failure to follow social distancing guidelines.
He told Radio 4’s Today programme: “It is that breakdown in social distancing, that breakdown in respecting the rules around extended households, that is driving infection rates.”
Stricter rules on visiting other people’s homes were also extended to two more areas in the west of Scotland from midnight.
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Meanwhile, council leaders in north-east England have said the average number of new cases in the region has doubled to about 80 per day, in just over a week.
A joint statement from seven council leaders said: “We have seen cases where individuals with symptoms have had a test, then gone out and infected others before getting their results – reckless and selfish behaviour.”
The statement also warned businesses that they would be shut down if they did not record customers’ details to aid the NHS Test and Trace contact tracing programme.
“We all have to do our bit and to play our part if we are to prevent a potential second wave,” the leaders said.
The warnings come after Transport Secretary Grant Shapps announced a change in England’s quarantine policy, adopting an approach which allows islands to have separate rules from a country’s mainland.
He said travellers arriving in England from seven Greek islands will have to self-isolate for 14 days from 04:00 BST on Wednesday.
They are Crete, Lesvos, Mykonos, Santorini, Serifos, Tinos, and Zakynthos (also known as Zante).
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