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More than half a million people booked their Covid booster jabs on Monday – despite the NHS website crashing as thousands sought top-up doses.
Boris Johnson revealed the figure a day after pledging that boosters would be offered to all over-18s by 31 December.
Long queues formed at walk-in vaccination sites on Monday, with some people waiting five hours to get jabs.
The PM and the NHS are now appealing for volunteers to help get jabs into arms and run vaccination centres.
No 10 said hundreds more sites, mobile units and pop-ups would be set up across the country including at football stadiums, shopping centres and racecourses in the next week, as the UK bids to slow the spread of the Omicron variant.
Deputy Prime Minister Dominic Raab told BBC Breakfast he accepted there had been "teething problems" in the bid to offer a jab to all adults before the new year, but he hoped they would diminish over the course of the week.
"It does take a few days just to make sure we get to a steady state," he said, adding: "We'll keep straining every sinew to make sure we can reach that target."
One doctor told BBC Radio 4's Today programme there had been an "incredibly rapid reaction" from GPs to assist with the booster rollout at "incredibly short notice".
But Dr Jess Harvey, from Shropshire, urged people frustrated with long queues both at clinics and on booking websites to be patient as "there is increased capacity that's going to be created" in the coming days.
More than half a million people booked their booster jab today – an incredible feat.
We're doing everything we can to deliver millions more boosters in the coming weeks.
Thank you to everyone who has answered the call – please tell your friends and family to Get Boosted Now.
Meanwhile, fully-vaccinated people in England who come into contact with any Covid case should now take daily lateral flow tests for seven days.
Until now, contacts of Omicron cases had to isolate for 10 days. Most unvaccinated adults will still have to isolate if they are contacts of any Covid cases, Omicron or not.
On Monday, ordering lateral flow kits from the government website was temporarily suspended amid high demand.
By the early hours of Tuesday morning, it appeared the tests could once more be ordered from the government website.
Monday saw online booster jab bookings open for people aged 30 and over – a day after the prime minister announced an expansion to the rollout amid the "Omicron emergency".
People aged 18 to 29 will be able to book their booster jabs online from Wednesday.
Senior health sources have said the end-of-year target is highly ambitious and delivery may take longer, BBC health editor Hugh Pym said.
The prime minister and NHS England chief executive Amanda Pritchard are appealing for "tens of thousands" of volunteers – from trained vaccinators to stewards – to join the "national mission" and help achieve the accelerated booster rollout.
Mr Johnson said: "Many thousands have already given their time but we need you to come forward again, to work alongside our brilliant GPs, doctors, nurses and pharmacists, to deliver jabs and save lives.
"So please come forward if you can."
At the moment there are nearly 3,000 vaccine sites across the country run by more than 90,000 volunteers.
After last week's NHS recruitment drive for 10,000 paid vaccinators, 4,500 people have registered their interest while 13,000 have come forward as volunteer stewards.
Modelling by the UK Health Security Agency (UKHSA) estimates the current number of daily Omicron infections is around 200,000, according to Health and Social Care Secretary Sajid Javid.
Omicron has risen to more than 44% of cases in London and is expected to become the dominant variant in the city in the next 48 hours, he said.
There are 4,713 confirmed cases of the Omicron variant, with 10 people, aged between 18 and 85, hospitalised with the variant in England, according to UKHSA.
Mr Raab, however, caused confusion on Tuesday when he told Sky News there were 250 Omicron cases in hospital, then immediately afterwards told the BBC there were nine cases. When contacted by the BBC, the UKHSA said neither answer was right, and the actual number was still 10.
The UKHSA added all were diagnosed on or before admission to hospital and the majority had received two doses of a vaccine.
At least one person in the UK has died with Omicron, Mr Johnson confirmed on Monday.
The UK confirmed 54,661 new coronavirus cases on Monday, as well as 38 deaths within 28 days of a positive test.
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