Health officials say they are concerned with rising Covid-19 case numbers in the last two weeks, stressing "we cannot lose the opportunity" to keep under control.
The warning was issued by Acting Chief Medical Officer (CMO) Dr Ronan Glynn as he revealed there were no new deaths recorded on Monday, but a further 11 cases were confirmed to the Health Protection Surveillance Centre.
There have now been 1,746 deaths associated with the virus in Ireland and some 25,638 cases.
Dr Glynn noted in the last 14 days, 165 cases have been recorded, 12% of which are travel-related. Some 20 counties reported new cases in that time. He said there has been an increase in the five-day average number of cases, and also an increase in the last 14 days in comparison to previous fortnights.
"Our research shows that 38% of the population now believe the worst of the pandemic is ahead of us. This does not have to be the case," Dr Glynn said.
"Simple measures like hand washing, physical distancing, face coverings in appropriate settings, cough/sneeze etiquette, and watching out for symptoms are the crucial elements in suppressing Covid-19."
He noted the r-number is "at least one, if not above one", meaning every person with the virus infects at least one other person, and said many people who are requested to get a test as a result of contact tracing are choosing not to do so.
"We have an opportunity to keep it under control and we cannot lose that opportunity," Dr Glynn said, adding it is "a major concern" when anyone doesn't follow public health guidelines.
He said "more work needs to be done" to ensure anyone arriving in Ireland is aware of the requirement to self-isolate for 14 days.
Asked if he had a simple message for those coming from overseas, he said: "Stay in your hotel, stay in your guest house. Now is not the time to be exploring Ireland, unfortunately."
Dr Glynn said, from a public health perspective, a mandatory quarantine "would be a desirable measure" but conceded the Government had to factor in other considerations. But, he said, personal responsibility has to be a factor in containing the virus too. This is the case for travel but also for house parties and other gatherings, he said.
"Of course, there will always be people who don't follow the guidance," he said, noting there "have been clusters connected to house parties and private home gatherings".
"But this is not about blame, it's about protecting people," he said.
Dr Glynn warned "it's all well and good" going to a house party with people that "you think are healthy" but reminded that the virus can spread in asymptomatic people too.
Earlier, it was confirmed the numbers depending on the pandemic unemployment payment (PUP) have dropped 42% since their peak on May 5. PUP recipients dropped by 67,300 in the last week, the largest weekly decrease to date, as the economy reopens.