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21 October 2014 Last updated at 09:51 BST
Eleven-month-old Harmonie-Rose Allen became ill in September and was rushed into Bristol Children's Hospital where she was diagnosed with a rare form of meningococcal septicaemia.
Harmonie-Rose's family want to raise money to help with her care and have stuck pink ribbons all around the estate where she lives.
BBC reporter Ali Vowles went to meet the family.

The Colin Bloomfield Melanoma Appeal was launched in February to raise awareness of skin cancer and educate children in schools about sun safety.
The target was reached on Monday after the Freemasons in Derby donated £10,000 with a new target set at £75,000.
Radio Derby presenter Colin Bloomfield - who has stage four melanoma - said the donations were "hugely generous".
BBC Radio Derby is working with the Derby Telegraph and charity Skcin, which specialises in skin cancer prevention.
The station launched the appeal after Colin's diagnosis in 2013, and his openness and optimism on the disease touched thousands of listeners.
The 33-year-old had a malignant melanoma removed from his leg 10 years ago but in 2013 he was told the cancer had returned and in November he was given months to live when it spread to his brain.
"Every single pound has made such a big difference," he said,
"As a child I never understood the seriousness of getting burnt but I'm living through it now.
"[Melanoma] is a truly horrible disease that a lot of people don't understand the seriousness of.
"We are not used to seeing the sun in this country and when we go out and enjoy it, it is about doing it safely and responsibly.
"If we can hammer home that message to kids in schools that can only be a good thing."
Earlier this month Derby County Football Club raised more than £14,000 after donating £10 for every away ticket sold at Norwich.
The appeal aims to take the sun safe scheme to 200 schools in Derbyshire and East Staffordshire.

The 38-year-old slow left-armer took 5-3 against New Zealand during Sri Lanka's 2014 World T20-winning run.
He claimed 18 wickets in 17 matches in the shortest format and also took 74 wickets in 71 one-day internationals.
Herath will continue at Test level and needs only three more wickets to become the third Sri Lankan to reach 300 after Muttiah Muralitharan and Chaminda Vaas.
Some of his best performances have come in the five-day arena and he was the world's leading Test wicket-taker in the calendar years 2012 and 2014.
He is the only left-arm bowler in the history of the game to take nine wickets in a Test innings, capturing 9-127 against Pakistan in Colombo in August 2014 - one of only two Test nine-wicket hauls by any bowler in the 21st century.
While more successful on the subcontinent than abroad, he had brief spells in county cricket with Surrey and Hampshire.
Sri Lanka play three Tests against England, who beat them in the World T20 quarter-finals last month, in the summer.
The series begins at Yorkshire's Headingley ground on 19 May and there are also matches at Chester-le-Street, Durham on 27 May and Lord's on 9 June.

The 19-year-old will now be eligible to play against Barnsley in the Football League Trophy final on Sunday.
Kenny, who joined up with the England Under-20 squad over Easter, has so far made 13 appearances for the U's.
"He has done really well for us and is enjoying his football," said Oxford head coach Michael Appleton.
"He is highly thought of at Goodison Park and has a big future."
A fundraising campaign is under way to help a baby girl from Bath who has to have all of her limbs amputated after contracting meningitis.
[NEXT_CONCEPT]
A skin cancer appeal backed by a BBC presenter has reached its £45,000 target after just six weeks.
[NEXT_CONCEPT]
Sri Lanka spinner Rangana Herath has announced his retirement from limited-overs internationals.
[NEXT_CONCEPT]
Oxford United have extended the loan of right-back Jonjoe Kenny from Premier League side Everton until the end of the season.