Ms Fleming had health problems before contracting the flu virus
The family of the first person in Europe to die after being diagnosed with swine flu has suffered a double tragedy with the death of her baby.
Jacqueline Fleming, 38, from Glasgow, died on Sunday at the Royal Alexandra Hospital in Paisley.
Her son Jack, who was born 11 weeks early, died on Monday in a special care baby unit at the same hospital.
The baby was not infected with the Influenza A (H1N1) virus. He was two weeks old when he died.
Ms Fleming had been suffering from underlying health problems since the birth of her son.
In a statement, her partner William McCann said it was an extremely distressing and difficult time for the family.
He said: "My beautiful son was born on the first of June 2009, 11 weeks early.
"He suffered from a number of complications and despite his brave fight he passed away earlier this evening at the Special Care Baby Unit at the Royal Alexandra Hospital in Paisley."
Ms Fleming lived with her two other children - one aged 18 and one of primary school age - and her long-term partner in the family home in Thornliebank, just south of Glasgow.
William Docherty, who knew Ms Fleming, told BBC Scotland he believed she had been admitted to hospital following a stroke.
A female friend of the family said: "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."
Scotland Health Secretary Nicola Sturgeon said: "The death of baby Jack, especially coming so soon after the death of his mother, is a tragedy and I extend my deepest condolences to their family and friends for this unimaginably painful loss."
The Scottish Government said Scotland had no new laboratory confirmed cases of swine flu.
However, GPs in the NHS Greater Glasgow and Clyde area are now, in certain circumstances, diagnosing patients on the basis of clinical symptoms alone - without laboratory confirmation. There had been 71 such diagnoses, the government said.
Scotland therefore currently has 498 confirmed cases, plus 71 which have been clinically diagnosed.
Jacqueline Fleming lived in Crebar Street in the Thornliebank area
In England, the Health Protection Agency said another 59 cases had been confirmed.
Earlier, Ms 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".
She said: "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."