There have been ten detected cases of Covid-19 since mandatory hotel and quarantine testing began in the quarantine facilities on 26 March, according to Niamh O'Beirne, the HSE's National lead for Covid-19 Testing and Tracing.
She said 413 people have been swabbed.
Ms O'Beirne said since 31 March, contact tracers have been asking additional questions to index cases, who were noted to have contracted the virus through community transmission.
She said of the 3,476 cases in the last week, it was indicated to do close investigation on 787 of those cases.
She said about 60% of those cases resided in Dublin.
Ms O'Beirne said when people were asked where they had been over the previous seven days "a high number" (over 55.8%) said they had been in supermarkets and petrol stations.
"In second place was workplace at about 15%, and then we have school, transport, travel and social gatherings outdoors.
"There were other areas men can point to a lot much lower volume, birthday parties, weddings, funerals, personal care appointments, and cafes indoors and outdoors."
She explained that this information is passed on to the local departments of public health where it is determined whether further investigation needs to be going on any particular setting.
Earlier, Minister for Foreign Affairs Simon Coveney said more EU countries will be added to the list of countries on the mandatory hotel quarantine list.
Mr Coveney said the Government is looking at issues that are undoubtedly going to flow from that, such as capacity and potential legal issues.
Speaking on RTÉ's Morning Ireland, he said various Government departments have been working to address the necessary capacity and systems issues before it "dramatically increases" the list.
He said such discussions would conclude this week.
Any decision would then be a matter for Cabinet.
Mr Coveney said: "Make no mistake, the Government is at one about the need to respond to public health advice that we need to keep variants of concern out of Ireland and to use mandatory quarantine to the fullest extent that we can."
Sinn Féin's Health spokesperson David Cullinane has said that it was clear there was "push back" in government on public health advice in relation to mandatory hotel quarantining and he said that was not just risky, it was reckless.
He said all public health advice must be fully implemented.
Plans to add those countries to the list were shelved but 26 others were included.
People from 59 designated countries and territories arriving into Ireland must pre-book accommodation for 12 nights of mandatory hotel quarantine on arrival and pay for the stay.
The length of stay can be reduced if a person receives a negative test for the virus on day ten of quarantine.
Countries on the list include Albania, Argentina, Austria, Bahrain, Brazil, Chile, Israel, Jordan, Monaco, Mauritius, Rwanda, South Africa, the United Arab Emirates and Zimbabwe.
Yesterday, Minister for Transport Eamon Ryan said more EU countries will be added to the list and said he had "very good conversations" with the Taoiseach, Tánaiste and Minister for Health on the situation.
However, Minister for Health Stephen Donnelly said that adding a "lot of countries where we have a lot of Irish citizens" to the list needs to be "thought through carefully" and the Government needs to make sure it gets this right.
There have now been one million doses of the Covid-19 vaccines administered in Ireland, HSE Chief Executive Paul Reid has told a briefing this afternoon | Follow live updates: https://t.co/3WgoocIrVu pic.twitter.com/C99UwJ6wtf— RTÉ News (@rtenews) April 8, 2021