Ms Fleming had given birth to an eleven week premature baby a fortnight ago
The first person in Europe to die after being diagnosed with swine flu has been named as 38-year-old Jacqueline Fleming from Glasgow.
Ms Fleming died on Sunday at the Royal Alexandra Hospital in Paisley. She had been suffering from underlying health problems and had recently given birth.
Her family said it had been "absolutely devastated" by her death.
In a statement, Ms Fleming's family said it was doing everything to support her two sons and partner.
The statement, issued by NHS Greater Glasgow and Clyde, said: "We can confirm that our beloved Jacqueline Fleming passed away yesterday afternoon.
She was a really nice lady, really kind, a quiet woman, just a family person really
"Our whole family is absolutely devastated and we are doing everything we can to support Jacqueline's two sons and her partner.
"Jacqueline has been ill in hospital for a number of weeks but nothing can prepare you for such shattering news and to that end we would ask the media to respect the privacy of our entire family so that we can grieve in peace."
William Docherty, who knew Ms Fleming, from the Thornliebank area of Glasgow's southside, told BBC Scotland he believed she had been admitted to hospital following a stroke.
A female friend of the family said: "It is a tragedy. She had an 18-year-old son and a boy of around nine or 10 and a new baby who will grow up without a mother now.
Jacqueline Fleming lives in Crebar Street in the Thornliebank area
"I think they are taking it really badly. She was in hospital for a couple of weeks and there were days she was getting better and days she was taking a turn for the worse.
"They hoped she was going to pull through and it was a shock when she died. The family are really devastated."
She added: "She was a really nice lady, really kind, a quiet woman, just a family person really."
It is understood Ms Fleming's baby does not have swine flu.
The Scottish Government said on Monday there had been a further 71 new cases of swine flu confirmed by GPs, bringing the total to 569. All of the new cases were in the Greater Glasgow and Clyde NHS area.
The Health Protection Agency said another 59 cases had been confirmed in England.
Earlier, Scotland's Health Secretary Nicola Sturgeon said that while Ms Fleming's death was tragic the public should take some reassurance from the fact that most cases of the virus were "relatively mild".
Speaking to the BBC News Channel, Ms Sturgeon said: "Obviously this death is a tragedy and I would express my deepest condolences to the family of the patient concerned.
"However, I do also think it is important to stress that in any flu outbreak, unfortunately, we will see a small number of deaths and that doesn't change the fact that for the vast majority of people contracting this virus the symptoms are relatively mild."
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