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Amid reports that no vaccination appointments were available in the Capital before Christmas, he called on the Scottish Government to make available the necessary resources to meet the target set in the face of the fast-spreading Omicron variant.
Mr Murray said: “There are problems across Edinburgh, with people struggling to find appointment slots in locations close to them until the New Year despite the roll-out having to increase.
“Some are having to travel long distances, even outwith the city, the secure a booster appointment.
“The Scottish Government must provide the health service with the necessary resources to meet the ambitious vaccination targets, including widespread drop-in centres.”
His comments came as the UK Government confirmed it has not granted a request from the Scottish Government for extra military personnel to help meet the ambitious vaccination target.
The first person to die with Omicron in the UK was recorded on Monday.
And with First Minister Nicola Sturgeon due to update MSPs on Tuesday, Health Secretary Humza Yousaf said new restrictions were “inevitable”.
Meanwhile, people took to social media to report being unable to book appointments, after the online booking portal opened to those aged 30 to 39.
"Edinburgh has zero appointments”, wrote one user on Twitter, while another added, “Trying to book a booster appointment in Edinburgh and I feel like Schwarzenegger looking for that toy in Jingle All the Way”.
Tory councillor Phil Doggart, who sits on the Edinburgh Integration Joint Board, said he had tried online to book an appointment for a booster vaccination.
"I tried 10 centres in Lothian but none of them had an appointment before Christmas. Quite a few people have got in touch with me and said they've not been able to get anything until the new year.”
He acknowledged health officials had flagged up that finding enough staff to deliver the jags would be a challenge.
“You can't magic people out of thin air and we now have lots of people having to isolate because they or a member of their family has Covid and that's causing all sorts of problems. I don't think there's an easy solution to it.”
And he said bringing in the Army may be necessary. “If we want to meet this target we probably need something more imaginative in terms of solutions and the Army might be it.”
Edinburgh Southern Labour MSP Daniel Johnson said the target meant serious pressures across the system.
"I think Boris Johnson has set a very unrealistic expectation – I don't see how by the end of December we can get every adult vaccinated,
"It's important we have a realistic plan in place, and I'm concerned if you have set that expectation but can't deliver it's going to lead to very high levels of upset and frustration.”
And Lothian Tory MSP Miles Briggs said the target was “hugely challenging.” He said: “I hope not just health board but the government are looking at what additional capacity can be put in as soon as possible.”
Fiona Ireland, NHS Lothian interim nurse director, said: “Since last week we have made over 99,000 booster appointments available across Lothian and are actively working to add more appointments to our booking system.
“We are now entering what will be an exceptionally busy period for our vaccination programme as we work urgently to offer booster vaccinations to all over 18s by the end 2021.”