Speaking to the Scottish Parliament’s Covid-19 Recovery Committee on Wednesday, Mr Swinney said the changes to self-isolation rules introduced last week are beginning to ease staffing pressures in health and social care.
The assurance comes as NHS staff absences have doubled since the end of December, to an average of more than 7,000 per day in the week to January 12.
Health boards across the country have said they are under severe pressure, with NHS Lanarkshire suspending all but the most urgent GP services for a month, and NHS Grampian warning it may declare a major incident before the end of the week.
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Asked about existing pressures, Mr Swinney said: “There is a huge amount of pressure on NHS boards around the country. I think it would be fair to say that all boards are under intense pressure.”
He said changes to testing arrangements and isolation periods introduced by the Government “will be beginning to have a welcome effect on easing some of those staffing pressures”.
Mr Swinney said: “We are confident about the sustainability of NHS services at this moment, but I have to say that that is a constantly dynamic position … the individual hospitals will come under greater pressure as a consequence of incidents that take place and demand that presents itself.”
He added: “At this stage I would say that the National Health Service in all parts of the country is coping, but it is coping under enormous pressure, and the headroom that's available to deal with increased demand is very, very limited.”
Mr Swinney also faced questions over limited data on the distinction between hospitalisations “with” and “because of” Covid in Scotland.
According to recent analysis from Public Health Scotland based on a short period in two health boards, around 60 per cent of patients who test positive in hospital were admitted because of the virus.
National clinical director Professor Jason Leitch told the committee he does not know “what the fascination is” with this data, saying the discourse around it “speaks to a slight misunderstanding of how healthcare works”.
"Healthcare is not a single disease,” he said.
"The people in hospital with Covid who are in trouble don't just have Covid.
“They have diabetes, they have leukaemia, they’re 87. There are all kinds of things going on.
“The death certificates often have five reasons for death, not one, and principal diagnosis versus secondary diagnosis is a matter of judgement at the time of death by the junior doctor who perhaps is filling in the death certificate.”
Prof Leitch added: “There are very few people in who are having their leg fixed because they fell over on the ice who get a positive Covid test.
“There will be some, of course, but the vast majority getting care in our hospitals with a Covid test are getting it because they've got Covid, and they've got other things going on as well.”