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The first people in the UK have been hospitalised with Omicron infections, Nadhim Zahawi has said.
The new variant of coronavirus now accounts for a third of cases in London, the education secretary said.
With two doses of a vaccine "not enough" Mr Zahawi encouraged people to get a booster jab - those aged 30 and over are eligible to do so from Monday.
As of Saturday, there have been 1,898 confirmed cases of the Omicron variant in the UK.
But the true number is likely to be far higher.
Scientists from the London School of Hygiene and Tropical Medicine (LSHTM) have warned that the UK will face a substantial wave of Omicron infections without further restrictions beyond the Plan B measures announced last week.
These include recommending people work from home if they can, expanding mask-wearing rules and introducing Covid passes for entry to some venues - with parliament to vote on the changes on Tuesday.
Mr Zahawi described the new variant as a "big bump" in the road on the journey from pandemic to epidemic.
"If we get to one million infections by the end of December, three days later that's two million. If it's half as severe as Delta and 1% of severe infections need hospitalisation that's 10,000 people with the first million and it keeps going that way," he told BBC One's Andrew Marr Show.
Dr Susan Hopkins, head of the UK Health Security Agency, said that she expected to see an increase in the number of people in hospital with Omicron infection.
She said there had not been a report of a death from the variant in the UK yet, although she pointed out that it had only been identified two weeks ago. It is two weeks after infection that you would expect to see people admitted to hospital, with deaths coming after that.
'Big wave'
It is "inevitable" that there was going to be a big wave of infections but what was not clear was the impact that would have on hospitals, she said.
She said the sheer weight of numbers of people being infected with Omicron means it will find the unvaccinated or people who have had a poor immune response to the vaccine.
"This is a big wave coming straight at us - if we see even half the severity that we saw with Delta then we are facing a very large number of hospitalisations and potential deaths," she said.