The latest figures show that in the week to 5 June, a total of 137 children aged 14 and under received a positive Covid test every day, compared to 143 a day in the week to 4 January, as the second wave was at its peak.
Dr Gurdasani said mitigation methods such as masks for children of all ages, as well as smaller class sizes and ventilation, were necessary to help prevent the virus spreading further.
Her comments come as Devi Sridhar, professor of public health at the University of Edinburgh, said she believed Scottish teenagers should be vaccinated as soon as possible, in a bid to prevent in-school transmission.
Dr Gurdasani also warned that while hospitalisations may be slightly lower due to vulnerable groups being vaccinated, children and their parents, who may not yet be fully vaccinated, could be at risk of serious health outcomes such as long Covid and even, potentially, diabetes.
She said: “Infection rates in under 14-year-olds has risen rapidly, and reported infection rates are the same as the peak of the second wave.
"This is unlikely to be due to increased testing, as testing appears to have remained flat during this period. Recent increases over the past several weeks are likely to be real.
“All this again points to rapidly rising infection in school-age children, similar to what we've seen with the Delta variant in England, where it appears as if schools have been driving transmission and spread back into the community. It argues for robust mitigatory measures in schools.”
She added: “Both in England and Scotland, sufficient attention hasn't been paid to school mitigations, and we're behind most other countries, and CDC (US guidance) recommendations on this.
"We urgently need masks in primary and secondary schools, much more attention to ventilation, smaller bubbles and class sizes, outdoor classes and PE [protective equipment], where possible. The more mitigations, the better.
"I certainly think there should be much more talk about mitigations in schools. There can be a lot done without school closures, but our governments haven't even attempted to do this, because of a continuing false narrative that transmission isn't happening in schools.”
She said that while the link between cases and hospitalisations will have been weakened, it is “certainly far from broken”.
Dr Gurdasani warned while few children require acute hospital treatment for Covid infections, the long-term health impact could be more significant – as well as for their parents, who may not yet have been able to have a vaccine, or have only had one dose, which is only partially effective against the current strain predominantly circulating in Scotland – the Delta variant.
A recent study showed that one dose of the AstraZeneca or Pfizer vaccine is only 33 per cent effective against the new variant.
She said: “There is a real impact on children. Seven to 8 per cent of children who get infected go on to have symptoms for 12 weeks or more.
"We don't know the long-term consequences of this virus on children. There are now links being suggested between onset of type one diabetes among young people who get infection.
“It is also worth pointing out that although cases as reported by Public Health Scotland among children seem high, these are almost certainly underestimates, as children are more likely to be asymptomatic, and not get identified as cases.
"We have seen this in comparisons of data from Public Health England with the ONS [random surveillance data].”
Dr Gurdasani said she believed the vaccination programme should include high school children in a bid to reduce infections. The Pfizer vaccine has been authorised in the UK for children aged 12 and over.
“I think vaccinating adolescents should also be a priority, as it's clear that a substantial proportion of transmission is happening in these age groups,” Dr Gurdasani said.
"It is very unlikely we will be able to achieve any level of control of the pandemic through vaccination unless we control transmission in children and offer them vaccines too.”
Prof Sridhar, told Good Morning Britain on Monday: “Children can still get long Covid and can still be chronically ill from this.
"Given that we know children can transmit, where we are going to see problems going forward is not going to be in care homes, it’s not going to be in hospitals, it’s going to be in schools, because this is where you’re going to see large groups of unvaccinated kids together, and we are going to have outbreaks.”
She added: “If we want schools to continue without disruption in the autumn and lift restrictions so children can have a normal experience, we need to vaccinate them, and if we wait and watch for the evidence it will be too late in the next few weeks.”
A Scottish Government spokesman said that Covid cases in schools “remain low”.
He said: “The current rate of absence for Covid-related reasons amongst pupils is around 2 per cent. The vast majority of these – about 85 per cent - are pupils who are isolating, rather than being off because they have a Covid-related sickness.”
“At this stage, our expectation is that the total number of children and young people in Scotland experiencing Covid-19 is small and those developing long Covid is a small percentage of these.
"At present, our NHS in Scotland is already delivering care for children and young people with long Covid, ensuring care is tailored to the individual needs of the child through existing pathways such as for chronic fatigue syndrome.”
The spokesman added: “We continue to closely monitor the situation in schools and work with partners to ensure guidance reflects the latest scientific evidence and advice. We have confidence on the decisions made on managing local outbreaks by local public health teams in partnership with schools and local authorities.”