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Hancock said he had been told “there should be a public execution” and these type of threats had been in the “high single figures” in the last six months.
He told the Telegraph’s Chopper’s Politics Podcast that the abuse had ramped up after his resignation.
Hancock said: It’s both the quantity but also the unpleasantness of vitriol.
“And then also there there are direct threats and it is not acceptable in a democracy.”
He said the greatest amount of abuse came from anti-vaxxers, adding: “They are very aggressive.”
He said: “I hope he doesn’t mind my saying this, but I got advice from Tony Blair [who said]: ‘Just don’t look at Twitter. What on earth are you looking at that for?”
He said there needed to be recourse to tackle anonymous accounts online but said they didn’t need to be banned.
The Tory also called on other politicians not to be drawn into abuse.
He said: “It’s also on us as politicians to rise above this, not to get tempted by the Twitter mob and to make sure that we act, talk and behave - and what we say is in a reasonable way, a reasonable tone and content.”
Mr Hancock faced an online backlash recently when he revealed he had been offered the position of unpaid “UN special representative on financial innovation and climate change” for the United Nations Economic Commission for Africa.
The job offer was announced on the same day that a highly critical joint probe by the Commons health and science committees criticised the Government for “one of the most important public health failures” in UK history at the start of the pandemic, when Mr Hancock was health secretary.