The streets are narrow, the houses small, the population vast. The whole neighbourhood covers no more than a square mile, but it has eaten up the space relentlessly.

Corrugated iron stretches out in every direction, a patchwork quilt of metal covering an impoverished society.

Peel back the roofs, and they’d reveal a community living like sardines. Imagine the entire population of York, forced to live within the walled city centre, and you can begin to conceive of the density.

This is Korogocho, the third largest slum in Nairobi. It is home to more than 150,000 people including, in this dimly lit, sparsely furnished, 12ft x 12ft room, three young brothers who know all too much about the harsh realities of life.

Vincent Walong is like any 13 year old boy. He plays football, he jokes about with his brothers Athanas and Gilcrest, ten and seven, and he enjoys listening to music.

But daily life is difficult and lonely. Twelve months ago, the boys’ mother Rosemary died after contracting HIV and – having been fathered by different men – the boys have been abandoned by their myriad relatives.

They are seen as outcasts, explains Stephen Ngugi from Christian Aid's Nairobi office – a “bad omen” for the rest of the family.

Another brother, Mesheck, 14, provided a humble but valuable income for a while, carrying goods for a local radio salesman. But he has been in hospital for the past few months, laid low by a spinal condition that his affected him to varying degrees since birth. At the age of 13, and with no family support, Vincent is now the head of his household.

“When we went to bury my mother, the relatives said they did not want to see the children in their household, or to stay with us,” he says.

“We want to live with them – but the relatives did not want us in their homestead.”

Until their mother’s death, the boys lived elsewhere, but they could not afford the rent on their own and were forced to move.

A neighbour now pays the rent for their simple new home, a basic structure with a small window, a threadbare rug on the floor, a couple of old chairs and a single lightbulb hanging from a timber frame.

This one room is the bedroom, the living room and the kitchen, the boys cooking over a naked flame just a few inches from the battered foam mattress that they share each night.

Behind their new home, the Dandora rubbish tip sprawls over the horizon. The air is putrid, but every morning, the boys head there, or to nearby alleyways, to scavenge for plastic containers that they can sell to local market traders.

On a typical day, they collect three kilogrammes, which earns them 45 Kenyan shillings (the equivalent of 30 pence). They use the money to buy charcoal and basic food – small fried donuts from a nearby stall, porridge and ugali, a maize-based doughy substance that resembles mashed potato.

There is usually enough plastic to pay for breakfast, but the work does not end there.

“To cook lunch, we need a larger amount,” says Vincent, speaking through a translator.

“After we have taken breakfast, we start collecting containers again. To get money for lunch, we have to collect more. When we have collected enough, and been given the money, we wash the utensils and – if the charcoal has remained lit – we cook ugali.”

When they cannot collect enough plastic, says Vincent, they forgo lunch – “and then we make sure we get enough for supper”.

The boys are keen to better themselves, but face challenges at every turn. Vincent wants to learn to read; Athanas wants to become a mechanic, but invariably when they go to school, they are sent home.

The fees are 250KSh (£1.65) a month each, which is more than the boys can hope to afford. Although their teachers sometimes allow them to stay, giving them sought-after teaching and lunch, it is hit and miss at best. As often as not, they are sent home.

The support the boys do receive comes from the Kenya Network of Women with AIDS (Kenwa).

Rosemary was one of their clients, and Kenwa provided food when needed. They will be at the forefront, also, of trying to pay Mesheck's hospital bill, which currently stands at 38,000KSh (£250).

The boys’ case, sadly, is all too familiar for Kenwa, which has worked with the victims of HIV and their families for nearly 20 years now, but which has finite finances and infinite demand.

Vincent, Athanas and Gilcrest will go to school for the next six months, their fees having been paid by one kind-hearted individual. But in an area where poverty and struggles skulk on every corner, their story is, tragically, far from a one-off. And their future, as they look beyond the rubbish dump, is a long way from being rosy. They, like Kenwa, need all the help they can get.

• Kenwa is supported by the Anglican Church of Kenya’s development services arm, which is part-funded by Christian Aid. This year, Christian Aid’s Christmas appeal is focused on HIV, marking the 30th anniversary of the discovery of the virus.

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• If you would like to donate to Christian Aid’s Christmas Appeal, or would like to find out more about its work on health and HIV, visit christianaid.org.uk/christmas or call 0845 7000 300.