Lifting lockdown is 'long, long, long way off,' says Health Secretary Matt Hancock
Mr Hancock said there was evidence that restrictions in place were having an impact while the vaccination programme was making "brilliant progress".
Three quarters of all those over 80 in the UK had now been vaccinated, with a similar number of those in care homes, he said.
However, Mr Hancock said that case numbers were "incredibly high" and the NHS remained under intense pressure.
"There is early evidence that the lockdown is starting to bring cases down but we are a long, long, long way from being low enough because the case rate was incredibly high," he told Sky News's Sophy Ridge on Sunday programme.
"You can see the pressure on the NHS - you can see it every day."
Mr Hancock said that while he hoped schools in England could reopen by Easter, it would depend on the levels of infection in the community at that time.
"We have got to look at the data, we have got to look at the impact of the vaccination programme," he said.
"The Education Secretary (Gavin Williamson) has said that we will ensure schools get two weeks' notice of return. I don't know whether it will be then or before then. We have got to watch the data."
Following the emergence of new variants of the virus in Brazil and South Africa which may be less susceptible to the vaccines, Mr Hancock said the Government would adopt a "precautionary" approach to protecting the UK's border.
Ministers are expected to meet this week to discuss a proposal to require people arriving in the UK to pay to quarantine in a designated hotel to ensure they are following the rules on self-isolating.
Mr Hancock said that so far there were 77 known cases of the South African variant in the UK and nine of the Brazilian. He said that all the cases of the South African variant were linked to travel.
"There is not what we call community transmission where you find a case that you can't find the link back to travel. At the moment it is all linked to travel," he told BBC One's The Andrew Marr Show.
He said the new variants had been identified because both Brazil and South Africa had "decent-sized" genomic sequencing programmes but other countries were less well covered.
"The new variant I really worry about is the one that is out there that hasn't been spotted," he told Sky News.
"There's probably those elsewhere that simply haven't been picked up because the country doesn't have that genomic sequencing service."
Meanwhile, the deputy chief medical officer for England, Professor Jonathan Van-Tam, has called on those people who have been vaccinated to keep following the rules on social distancing.
The latest Government figures showed the number receiving the first dose of the vaccine across the UK has passed 5.8 million, with a record 478,248 getting the jab in a single day.
However, writing in The Sunday Telegraph, Prof Van-Tam said it was still not known if people who had received the jab could still pass on the virus to others, even though they were protected from falling ill themselves.
"If you change your behaviour you could still be spreading the virus, keeping the number of cases high and putting others at risk who also need their vaccine but are further down the queue," he said.