British Health Secretary Matt Hancock said on Sunday it was still a "long way" before COVID-19 cases would be low enough to lift the lockdown.
Government data on Sunday showed that there were another 30,004 infections and 610 deaths, bringing the totals since the start of the pandemic to 3,647,463 cases and 97,939 fatalities.
Over 6.3 million people have now received a dose of COVID-19 vaccine.
Hancock made his comments in an interview with Sky News, saying: "We should be worried enough, all of us, about this pandemic to follow the rules and it is just so important that people do."
He added that they "don't know the degree" to which new variants from South Africa and Brazil "might have a lower response to the vaccine".
"The new variant I really worry about is the one that is out there that hasn't been spotted," Hancock said.
He later told the BBC in a later interview that the government had identified 77 cases of the South Africa variant in the UK.
"That's why we have got such stringent border measures in place against movement from South Africa," he told the BBC.
