Minister for Health Simon Harris has said that this will be a very challenging and difficult week for the health services following last week's snow storm.
He has appealed for the public to be patient and said many of the staff at hospitals are exhausted after a very stressful and intense few days.
Mr Harris said that the best assistance that the public can give them is to only use HSE services where absolutely necessary and in cases of emergencies.
The minister said that there is a serious backlog in hospital discharges with upwards of 700 discharges delayed as a result of the disruption caused by the snow.
He said that there is also a steady rise in demand for hospitals services as a result of slips, trips and falls and he appealed for people to visit their GP and injury clinics rather than the HSE where possible.
The HSE has cancelled all non-urgent surgeries planned for tomorrow, Tuesday 6 March.
It says there is no need for patients to contact their hospital as hospitals will be in touch over the coming weeks to reschedule appointments.
The HSE says outpatient appointments have recommenced across all hospitals with day surgery recommencing in some acute hospitals from today.
The areas of Kildare, Wexford and Wicklow are still experiencing difficulties with access following the storm and the HSE has asked people to bear in mind that the recovery of services in these locations will be slower.
Weather conditions have contributed to a big rise in hospital overcrowding today, with 664 patients on trolleys or on wards, waiting for admission to a bed.
The hospital worst affected is Cork University with 63 patients waiting, according to figures from the Irish Nurses and Midwives Organisation.
There are 50 patients waiting at University Hospital Galway; 47 at University Hospital Limerick; 45 at the Midland Regional Hospital, Tullamore and Connolly Hospital Blanchardstown in Dublin has 43 patients waiting.
The HSE has said that due to the weather, it has not been possible to discharge patients who are ready to leave hospital.
As a result, fewer beds are free for emergency department admissions.
HSE/TrolleyGar figures put the number of patients on trolleys today at 518.
Unlike the INMO figures, the TrolleyGar figures do not give the number of patients placed on wards, who are waiting for admission for a bad, having already been admitted from an emergency department.
The HSE figures represent a 50% increase on the same day last year.
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In a statement, the HSE said that because of restricted access to hospitals for patients due to the heavy snow fall, there was a 41% reduction in attendances at emergency departments.
It said that as a result of this and with regional variations, there was a sharp increase in ED attendances this morning with 507 people awaiting admission to hospital.
It also said that the National Ambulance Services continues to be under pressure and is priortising emergency calls.
HSE National Director for Emergency Response Damien McCallion has asked the public to be patient as services attempt to return to normal after Storm Emma.
He says it will take at least ten days for things to return to normal.
Speaking on RTÉ's Morning Ireland, he said that the re-scheduling of cancelled outpatients appointments and elective procedures will not begin until the week after next.
Anyone who is deemed urgent will be contacted and brought in for their procedure.
Mr McCallion said it is anticipated that there will be an increased demand on emergency services over the coming weeks and urged people to only attend hospitals in the case of an emergency.