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An Australian man has been charged with murdering a woman who was last seen leaving her home a month ago.

Samantha Murphy has not been seen since she went out from her home in Ballarat, Victoria, for a run on 4 February.

Her disappearance led to a huge search, but police have still found no trace of the 51-year-old mother of three.

A 22-year-old man was arrested on Wednesday at his house in Ballarat, about 115km (70 miles) northwest of Melbourne. He is yet to enter a plea.

On Thursday police said he had deliberately attacked Ms Murphy on the day of her disappearance in a bushland area where she was known to exercise, near her East Ballarat home.

Victoria Police Chief Commissioner Shane Patton told reporters that the suspect had not disclosed the location of Ms Murphy's body, and that detectives did not believe he was previously known to her.

Mr Patton described it as a "tragic outcome" for her family.

Just before she went missing, Ms Murphy was seen on security cameras in her driveway wearing a maroon running shirt and black leggings.

After she failed to return that afternoon, authorities began scouring nearby areas and examining thousands of hours of CCTV footage.

Ms Murphy's phone, Apple watch and headphones - which she was carrying at the time - have still not been located.

Police thanked the local community on Thursday for their help with the case.

"I know that Samantha's disappearance has had a profound impact on the Ballarat community," Det Acting Supt Mark Hatt said.

In a plea to the public last month, Ms Murphy's family described her as a strong woman who would be determined to return home.

"Mum we love you so much and we miss you. We need you at home with us. Please come home soon," her daughter Jess Murphy said.

Police had previously warned it was unlikely that Ms Murphy was still alive.