The World Health Organisation says countries are correct to put in place local lockdowns - instead of country-wide measures.
Special restrictions are in place in Laois, Offaly and Kildare, while areas of northern England have similar measures.
Last week, it was announced that people in Kildare, Offaly and Laois would not be permitted to travel outside their own county.
The measures came into effect on Saturday and will remain in place for two weeks.
Under the measures, cafés and bars will be closed unless they are doing takeaway or outdoor dining, which is limited to 15 customers.
Cinemas, gyms, cultural venues and other entertainment venues will close. All sport will be cancelled, though non-contact training for up to 15 people can continue. Retail stores can remain open under strict social distancing rules and churches can be open for private prayer.
Dr Mike Ryan, from the World Health Organisation, says countries experiencing local clusters need to be vigilant.
“The trick for them now is to really focus on identifying those clusters of the disease,” he said.
“Identifying any new community transmission and putting in place the kind of localised measures that can contain the virus, suppress the virus and reduce exposure.
“And try to avoid, if possible, having the kind of country-wide lockdowns that did so much economic damage before.”
It comes as the global number of Covid-19 cases has reached 20 million, according to the Johns Hopkins University Coronavirus Resource Centre.
The grim milestone was reached in the early hours of Tuesday morning, Irish time, with 20,011,186 cases worldwide.
The total number of global coronavirus deaths stands at 734,664, according to experts at Johns Hopkins, the American university whose aggregated tally has become the main reference for monitoring the disease.
Health officials believe the real figure of cases worldwide is in fact far higher than the Johns Hopkins tally, due to testing limitations and the fact as many as 40% of people infected have no symptoms.
- additional reporting by PA