The question that needs to be asked here is why a baby of 5 weeks is so chesty in the summer months. In winter when coughs and colds are common it would be easier to understand.In fact she started getting ill at 3 weeks. It could be that she has a problem with reflux and is breathing in tiny amounts of milk into her lungs which can cause the symptoms you describe here. Your baby needs to be examined and treated by a paediatrician so that a proper diagnosis can be made as to the cause of her chest problem and the correct treatment can be given.
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