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Research has shown that the risk of SIDS and fatal sleep accidents increases in the following circumstances.
• If a parent who smokes bedshares with their infant
• If a parent is under the influence of alcohol or drugs that cause sedation
• If a parent is excessively tired
• If other children share the bed with an infant
• If the infant gets caught under bedding e.g. pillows and duvets
• If the bed is a water bed or if the mattress is too soft
• If a parent falls asleep with an infant on a sofa
• If an infant gets caught between the bed and the wall or falls off the bed
• If the infant is rolled on
Research in New Zealand and the UK has shown that sleeping baby in the same room, but not in the same bed, with the parents in the first six to twelve months of life is protective. This is thought to be because parents can see the baby and easily check to see that baby is safe. This protective effect does not work if the baby is in the room with other children probably because the children do not know if an infant is safe or not.
Never sleep baby on a soft mattress, couch beanbag, or waterbed with or without a parent as there is a very high risk of a sleep accident.
No mention of parents being overweight? I would think one of the above factors came into it rather than it being a weight issue if something went wrong.