The Deputy First Minister said the spread of the Omicron variant had been “really aggressive” and urged people to take lateral flow tests before attending any large events.
Nicola Sturgeon has asked people to limit socialising as much as possible around Christmas and work from home where they can.
New guidance for retail and hospitality has also come into force, requiring businesses to take “reasonable measures” to reduce the spread of coronavirus.
Appearing on BBC Breakfast on Sunday, Mr Swinney said: “I think it’s pretty clear that people in Scotland are changing their patterns of activity as a consequence of the seriousness of the threat that we face from Omicron.
“The regulations came into place on Friday morning and it’s very clear and noticeable that people are following many of these restrictions.”
Good progress was being made with the vaccine booster programme, he said.
Asked what it would take for more measures to be introduced, he said the Government was continuing to monitor the “really aggressive” spread of Omicron.
He said: “We will, of course, be updating parliament in the course of this week, on the further views that we have about any other measures that we need to take.”
The Deputy First Minister was also asked about large events such as Edinburgh’s Hogmanay celebrations.
Mr Swinney said the Government was not asking people to change their plans for Christmas Eve, Christmas Day or Boxing Day, but to “dial down” their interactions around these days.
He said: “Obviously there’s some larger events that take place, we’re saying to people before they go to them they should test themselves with lateral flow tests to make sure that they are negative.
“They should be vaccinated, they should get the booster jag.
“We’re saying to people in a whole variety of different ways to take steps to avoid being carriers of the virus.”