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He said new figures showed that over the past week Covid cases in Edinburgh had increased from 65.9 to 134.3 per 100,000.
And four city high schools – Broughton High, Gracemount High, Holyrood RC High and Wester Hailes Education Centre – have all announced Covid cases in recent days.
Mr Cole-Hamilton said: "What these figures show is that while the vaccine programme is holding back another wave of deaths, the virus is still spreading. The increase in the rate here in Edinburgh is alarming, and speeding up the vaccine roll-out by offering a walk-in centre makes sense.”
He said at the end of last week Health Secretary Humza Yousaf had described Edinburgh to him as “the new Glasgow” in terms of Covid infection.
"If that’s the case, why is it we are still waiting for the door-to-door surge testing and walk-in vaccinations for first and second doses that Glasgow had mobilised within a matter of days of their outbreak?
"Edinburgh’s outbreak has been raging for more than three weeks now and our citizens are as deserving as people in Glasgow of every line of defence against this dangerous new variant.”
He said he had been pressing the Scottish Government for months on how NHS Lothian could be supported to speed up jabs.
"This week we finally saw military personnel deployed but these are few in number and are arriving late in the day.
"In areas like Bolton and Darwen, the government set up walk-in vaccination centres to support those who had missed vaccine invite letters and make it easy for young people to get vaccinated in areas where caseloads were high.
"Meanwhile the Welsh Government say that all over 18s will have had a vaccine invite by next Monday. There are clearly lessons that can be learned from what they have done. Thousands of my constituents are desperate to get vaccinated. They just haven't been given the opportunity."
Mr Cole-Hamilton also posted a video on Twitter urging people to make use of a walk-in testing centre at Blackhall Library. In it he said: “There is going to be a surge of Covid-19 in Silverknowes, Davidson’s Mains and the Blackhall area. That is why Backhall Library has been converted into a lateral flow, walk-in, asymptomatic testing site.”
He said he had just been for a test and urged others to do so too. “It’s available to all the community – you don’t need an appointment, you just walk in. It’s seamless, its safe and it doesn’t take long at all. It’s important we suppress the virus.”
He said the drop-in test centre had not been widely advertised and had seen relatively low numbers of people coming in.
“With home-working, if you went door-to-door – as they are in G41 in Glasgow – you would get lots of people in and lots of people willing to take a test if it’s there in front of them.”