MSPs and their staff are to be given personal safety training by Police Scotland amid security concerns following the brutal killing of MP Jo Cox on Thursday.

An investigation has also been launched after one newly elected parliamentarian complained to the force about a constituent who has repeatedly confronted her and verbally abused her.

The Sunday Herald understands several MSPs contacted Holyrood’s security chief for advice in the hours after Cox was shot and stabbed by Thomas Mair.

Shortly before the MP’s death was announced, the head of security sent an email to every MSP offering “sessions on personal safety”.

It stated: “You may have heard of the sad news this afternoon regarding ... Jo Cox MP ... the Security Office and Police Scotland will be running sessions on personal safety on Tuesday 28 June 2016.”

The security head also said the team is on standby until then and offered a telephone number for worried MSPs to call. The following day police contacted MSPs and offered to have their home and constituency office addresses “flagged” for a quick response.

They also suggested officers should be alerted if their movements are likely to attract the attention of “opposing factions or individuals”.

At the personal safety sessions in Holyrood on Tuesday, politicians and their staff will be encouraged to carry a personal attack alarm and regularly change their route to and from parliament.

The Herald:

They will also be advised to provide relatives and staff with regular updates about their whereabouts and, when staying in Edinburgh, use a variety of hotels and vary where they stay.

Some MSPs are believed to be reviewing regular drop-in surgeries which leave them vulnerable to attack.

Labour MSP Monica Lennon, 35, has stopped publicising surgeries on social media after she was singled out for abuse.

She was twice targeted by a man in his 50s during the independent referendum campaign. On one occasion he followed her from street stalls and shouted abuse.

He resurfaced after an event in South Lanarkshire recently when Lennon was accosted as she left.

Police are hunting for the man who is yet to be charged in connection with Lennon’s allegations.

She said: “On Monday night I held my regular advice surgeries in Hamilton, where I’m nearing the end of my five-year term as local councillor.

“Usually I post where I’ll be on social media so that the opportunity reaches as many constituents as possible. But not this time.

“A matter which is still being investigated hung over me. After consulting the police and being advised to ‘use my common sense’ I decided to say nothing. A decision that left me feeling conflicted. Being approachable and accessible has meant everything to me.”

Lennon was quick to reject comparisons between her experience and a three-month campaign of harassment against Jo Cox in the run up to her death following a surgery in her constituency.

However, she admits to being “haunted” by the abuse she has received “both online and face-to-face”.

Lennon said: “Tributes have rightly focused on Jo and the person she was. But with the daily business of politics grinding on, I’ve been thinking about the 50 people from across Scotland who also found themselves elected as new MSPs last month.

“We’re all in the middle of hiring staff and setting up offices, as we endeavour to give our constituents the best service possible. Only 13 months ago Jo Cox was going through the very same motions.

“Past experiences of harassment while doing my job – both online and face-to-face – have been haunting me this week. When the news of Jo’s death reached me as I left the Scottish Parliament chamber on Thursday I’ve never felt so far away from home.

“Political leaders in Scotland and the UK rightly celebrate our democracy and always go to great lengths to stress the haters who pollute our social media and our streets are a tiny minority.

“Thankfully acts of violence are rare. But I am concerned that politicians rarely report incidents of concern.”

SNP MSP Humza Yousaf has been vocal about online abuse he has received from right-wing extremists.

He revealed some of the messages he has received to the Sunday Herald which included threats to his life.

“I have had racist abuse shouted at me in public as well as online,” he said. “Some of the threats I receive online have a clear physical threat.”

A survey of MPs found that four out of five had been victims of an attack or attempted attack.

The ground-breaking study by psychiatrists working with Home Office found that 192 of the 239 who responded had been targeted.

Comments by politicians included: 'Pulled a knife on me in the surgery'; 'repeatedly punched me in the face'; 'came at me with a hammer'; 'hit with a brick'; 'shot with air rifle'.

Conservative MP David Mundell revealed on Friday that he received a death threat described by police as “credible” after he voted for airstrikes in Syria in December 2015.

“It’s very concerning, not just for yourself but for your family, your staff in particular,” Mundell said.

“The police do offer very good guidance and support in relation to how to carry out your duties in the safest possible way.”

However Monica Lennon suggested more could be done to support politicians under threat.

“From my own experience, I believe there is significant scope for improvement in how Police Scotland and political parties themselves deal with threats to politicians,” she added.

“Those inside politics rarely ask for support to avoid a backlash. But if we truly value freedom and democracy, we must do more to stamp out intimidation and harassment, and that means ensuring politicians and their staff can serve the public without fear.

“Jo was one of those people who made the world better and made the rest of us believe it could be better still. We owe it to Jo to make it so.”

A police spokeswoman said: “I can confirm that police in Hamilton have received a complaint of verbal abuse towards a 35-year-old woman. No-one has been arrested at this time and police enquiries are continuing.”