More than 200,000 vaccines a day are being administered in the UK, Matt Hancock has said.
And he said the UK is on course to deliver 2 million vaccines a week.
The Health Secretary said the Government was on course to reach its target of 13 million people vaccinated by mid-February.
He told Sky News' Sophy Ridge: "Yes we're on course. The rate limiting factor at the moment is supply but that's increasing.
"I'm very glad to say that at the moment we're running at over 200,000 people being vaccinated every day.
"We've now vaccinated around one third of the over-80s in this country so we're making significant progress but there's still further expansion to go.
"This week we're opening mass vaccination centres. Big sites for instance at Epsom racecourse, there's seven going live this week with more to come next week where we will get through very large numbers of people."
Mr Hancock said the public had been following lockdown rules more in the November lockdown.
He did not want to criticise the public as "the majority are following the rules".
He told Sky News' Sophy Ridge On Sunday programme: "We are as you can see, with the police, enforcing stringently against the minority who don't follow the rules.
"But you're right I don't want to criticise the public because the majority of people are following the rules.
"That's what we want to see and we can see that the amount of people who are staying at home has increased, but we've all got to absolutely follow that guidance."
He added: "The data shows more people are following the lockdown than the November lockdown."
But he said "the pressure on the NHS is very, very bad".
He said: "The single biggest thing that anybody can do is to follow the stay at home guidance.
There are limited exemptions. Only if you can't work from home and if you need to go out and get shopping or take some exercise.
"But these are highly-limited for a good reason and that's because the pressure on the NHS is very, very bad and we need to bring the case rate right down.
"So it's on all of us really, it always has been a big team effort."Follow the Official Rokna NEWS Telegram Channel For More and fresh NEWS.