Ashling Murphy's death has sparked promises of a 'zero tolerance' response to gender violence.
The popular 23-year-old school teacher was attacked as she went for an afternoon run along a canal in Tullamore, Ireland.
Irish police are still hunting for her killer, after she was found dead on the banks of the Grand Canal, Co Offaly.
The murder has caused widespread anger and shock in Ireland and beyond, with tens of thousands of people attending vigils in recent days to remember Ms Murphy.
Ireland's Justice Minister has said that a “zero-tolerance” approach will be central to a new Government strategy tackling gender-based violence, as the investigation into the murder of Ashling Murphy continues.
Ms Murphy's funeral will take place on Tuesday at St Brigid’s Church, Mountbolus, in Co Offaly.
The Garda said it had made “significant progress” in its investigation, but was not releasing details for operational reasons.
It is understood that gardai have identified a new person of interest, who is believed to be in hospital in the Dublin region receiving medical care, and are waiting to speak to him, according to the PA News Agency.
UTV News has approached gardai for comment on the claims.
Searches are being carried out in properties in Tullamore and Dublin as part of the investigation, it is understood.
A complaint from another woman, who has said she was followed on the same canal path in the hours before the murder, remains something gardai are investigating.
On Sunday, Helen McEntee said that a new Government strategy to tackle domestic, sexual and gender-based violence will be published by the beginning of March.
It would, the Justice Minister told Newstalk, take a “zero-tolerance” approach to violence against women.
She told Newstalk: “I think what we’ve seen this week really is an outpouring of grief right across the country from women, men, children, all of whom have come together in solidarity with Ashling’s family and her community.
“But in particular, have come together to demand that there is zero tolerance for this.
“I myself have often decided, ‘well, I’ll go out for a walk at this time of the day or I’ll go to this area because it could be safer’. That shouldn’t be the case.
“And what we’ve seen now is everybody in society coming together to say this should not be the case. We should not tolerate this.”
On Sunday, opposition parties stressed the need for urgent Government action to prevent violence against women.
Sinn Fein TD Kathleen Funchion told RTE’s The Week In Politics her party believes there needs to be a centralised approach to tackling the issue.
She said that too often responsibilities and roles are split between departments and agencies.
On the same programme, Social Democrats TD Jennifer Whitmore said: “We need the Government to have a very, very clear focus to work to resolve these issues.
“A society where 50% of the population are scared to go out walking by themselves at night is a dysfunctional society.”
She said that a “cultural shift” was required in Ireland.
As vigils and memorials to Ms Murphy continue to be held, prayers were said at masses across the country on Sunday for the young teacher.
At a vigil in north London on Saturday, people held candles and stood in silent tribute outside the London Irish Centre.
Traditional music was played in honour of Ms Murphy, a talented fiddle player, while some of the crowd quietly sang or hummed along.
Anna Johnston, cultural officer at the London Irish Centre, said people had come together in solidarity with those who knew and loved Ms Murphy “and all the women of Ireland and further afield who are angry, distressed and heartbroken”.