It comes after four cases of the new variant were found in Scotland last week, among a total of 103 across the UK.
It has been labelled a “variant under investigation” rather than the more serious “variant of concern”, but Dr Smith said this was not a reason for complacency.
Three of the cases in Scotland were associated with travel, but one other case was a contact of those.
Dr Smith said a further 12 cases of Covid-19 have been identified in the past week among people who have travelled to India so some of these may later be identified as the “India variant” known as B1617.
India has seen a rapid rise in cases recently, with record numbers of cases and deaths reported.
Asked about the new variant, First Minister Nicola Sturgeon told a media briefing on Tuesday: “The most concerning thing about the Indian variant is what's happening in India right now.
“The situation is very, very severe in India, and from what I can see here it looks to be deteriorating.”
She added: “The seriousness of that situation can’t be overstated. But it is indicative of the fact that this is a virus that is mutating, there are new variants appearing in many parts of the world.
“The new variant that has caused us the most problems is the Kent variant that has happened in the UK
“The Indian variant is not yet a variant of concern, it’s a variant under investigation, but it underlines the fact that we can't be complacent as we get this under control here.
“If we allow new variants to come in and spread within Scotland, then our progress will be put at risk, and that's why we continue to to be difficult precautions particularly around international travel.”
Dr Smith confirmed that no new cases of the variant have yet been found in Scotland.
“As of yesterday there are still four cases of the B1617 variant that has been associated with India, which have been discovered in Scotland,” he said.
"Three of those were associated with travel, plus one contact of those three people.
“There have been no further identifications of any other variant associated with 1617. In fact, the team who looks after our genomic sequencing has had a good look back through all the previous data and they have seen no evidence of any previous identification of a similar lineage to this, so the situation remains at four.
“I expect that will increase in time. I think that is very possible. We’ve had another 12 cases associated with travel to India in the last seven days, and it may be that subsequently some of those are identified as being that variant, we just don’t know yet.
“We will continue to monitor all cases.”
India was placed on the “red list” of countries in terms of travel to Scotland on Monday.
This means that non-UK and Irish citizens won’t be allowed to enter Scotland if they have been in India in the previous ten days, and UK and Irish citizens will have to quarantine.
The changes come into force from Friday.