Speaking to BBC Breakfast, Hancock said that for those aged 70 and above, take-up was 90%, rising to as high as more than 97% among those aged 75-79, while in care homes it was more than 90%. "It's important for you, it's important for your patients and of course it's important for the whole of society".
"Right now, as of today, at the latest count there are still over 23,000 people in hospital with Covid - that's more than in the April peak - so we've still got some way to go, but we are looking to set out that road map on Monday".
It means there are a significant number at risk in the 50 to 70 age group as well as younger adults with health conditions.
The Government is aiming to get an offer of a vaccine to the estimated 17 million people in the next five groups by the end of April.
Achieving that will be no easy task.
The government has prioritised giving as many people as possible the first dose of vaccine with the second dose due to be delivered within 12 weeks of the first.
As always, everything depends on supply.
AstraZeneca supply lines remain strong and by Easter the first deliveries of a third vaccine made by Moderna should arrive. There can be no let up.
Hancock said the British government was speaking to other countries across the world about giving British people certificates showing they had been vaccinated so that they could travel overseas in the future to countries that require them. Johnson wants children back at school on March 8 and has promised to announce a "road map" plotting a route out of the restrictions on February 22.
Vaccine Minister Nadhim Zahawi last week said the country will not issue "vaccine passports".
Adult carers of disabled people with these conditions, and younger adults in care homes, are also part of this group.
The government says it has hit its target of vaccinating the top four priority groups across the United Kingdom and more than 1.5 million people in the East of England have now been given their first jab.
Prime Minister Boris Johnson urged those eligible to receive the vaccine to come forward.
Mr Hancock said that 90% of the over-70s had accepted the vaccine - a "much higher uptake than we could possibly have hoped for".
The Department of Health said 88% of the UK's coronavirus deaths occurred in the priority groups who have now been offered the vaccine.
The government has said it will take "cautious" steps, rejecting calls from Conservative MPs to commit to abolishing all Covid-19 restrictions by the end of April.
He said that meant further reducing the spread of the virus by offering "decent" financial support for self-isolation, new guidance on mask-wearing, and help for workplaces to be Covid secure.
"Our current virus, which is the dominant one in the country, only requires one or two more mutations to partially escape immunity - and that means immunity naturally gained or immunity gained by being vaccinated - and so it is imperative that we monitor the situation as closely as we can", he told ITV's Good Morning Britain.
With the expectation that increasing numbers of people will be protected from serious illness, ministers will this week finalise plans on how England's coronavirus measures can be eased, ahead of the prime minister setting out a "road map" out of lockdown next week.
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