Between mid-May to the end of June, 35% of those identified as a close contact of a confirmed case did not take up the offer of a test.
Dr Glynn said: “Every case has the potential to turn into a cluster, which in turn has the potential to spread through a community.
“If you are identified as a close contact, please take up the offer of a test without delay.”
He said the international impact of COVID-19 was changing rapidly and it was right that the Government waited until later this month to publish green lists of countries which it was safe to travel to.
“The situation globally could get worse over the next week or two in which case we would have to think again about who is or is not on the list,” he added.
For the first time since March, there have been two consecutive days with no reported deaths while four new cases have been confirmed.
Acting Chief Medical Officer, Dr Ronan Glynn, says that is a positive development.
"The most difficult aspect of this, obviously, throughout the process has been reporting the deaths and I am aware of the effect this has had on families across the country," said Dr Glynn.
"So it is brilliant that we have had two days but we always caution there is a weekend effect and sometimes cases are not notified as quickly at the weekend in terms of deaths as they would be during the week."
Dr Glynn, said while the figures are positive, they will be keeping a close eye on hospital admissions.
"If we see those moving in the wrong direction and moving quickly in the wrong direction that is a clear sign that we need to act quickly.
"For the last number of weeks now, we have had hospitalisation and new admissions to hospital in single figures.
"If we were to look at data today and see there were 20 additional hospital admissions that would be a very significant cause for concern."
