The Minister for Health has said that he and the acting chief medical officer are "cautiously optimistic" that the tighter restrictions in Dublin are working.
However, he warned that Cork is one of four counties where there is the possibility of additional restrictions being announced, and said it is being monitored “very carefully”.
Stephen Donnelly said the vast majority of people in Dublin are following the public health restrictions put in place to curb the spread of Covid-19.
Dublin and Donegal are currently subject to Level 3 restrictions under the Government's Covid-19 five-tier response plan, with the rest of the country at Level 2.
The stricter measures in place in the capital include a ban on indoor social gatherings; a requirement for pubs and restaurants to only serve food outdoors, while travel in and out of the county has been limited to work, education and essential purposes.
Mr Donnelly said public health experts tracking the spread of Covid-19 in Ireland need the seven-day rate of the virus to be less than half of the 14-day rate.
Dublin's 14-day incidence rate is 147 positive cases per 100,000 population, while its seven-day rate is 78.
"What we want to start seeing is the seven-day rate becomes less than half of the 14-day rate and that shows that it is plateauing," Mr Donnelly told RTÉ.
"The chief medical officer (Dr Ronan Glynn) and I have been speaking on exactly this point over the weekend, we will be cautiously optimistic as we must always be, but it is very early days.
"We want to see that seven-day rate come down each and every day.
"I have no doubt the people of Dublin have heard this loud and clear, and in the vast majority of cases people are doing what they are supposed to do, limiting their interactions."
He urged the public to halve the number of social contacts they have in a bid to reduce the R number, which is the number of people an infected person will pass the virus on to.
Mr Donnelly said there are no plans for the National Public Health Emergency Team (Nphet) to call an emergency meeting this week, however he warned that could change.
"There are four counties which are being looked at very carefully and they are Cork, Galway, Louth and Wicklow," he added.