Sorry to hear that you son has this problem.
Looking at it and taking his age into account, it does sounds like impetigo. So then it would differ from the problem what the people on this forum have. (But of course, this problem can still be discussed here.)
From which sites (nose, throat, active lesions) were the swabs taken? And how often?
Impetigo can be caused by staph and strep bacteria. Doctors often look for staph, but strep can also cause this. If correct, detecting strep bacteria requires different tests, so they should also perform those. Did they test for strep (streptococcus) (Maybe they did all at once.)
Does it only appear at his face?
How did the lesions respond to topical antibiotics? Was it also applied inside the nose? (For example with Bacitracin ?)
The recurring of impetigo can be caused when the bacteria is in the environment and/or when your son has a weak skin. Allergies and conditions like eczema can weaken the skin, but also scratching and overuse of antiseptics on the skin. Also note that even asymptomatic carriers may transfer it to him.
Things you could try:
- Wash his clothing at at least 60 degrees Celsius, preferably higher, like 75, because not all washing machines reach the set temperature level.
- Add an antibacterial laundry cleanser when washing, like: http://www.dettol.co.uk/products/laundry-sanitiser/dettol-antibacterial-laundry-cleanser/
- Fabric softeners may cause allergic reactions to some people. You could try without it, or add a household cleaning vinegar instead.
- Preferably use a dryer to dry the clothes. This can also kill bacteria. Although clothes wear out quicker when using a dryer.
- If your son is allergic to substances, try to avoid it. Note that food may also cause allergies.
- Avoid rubbing the face when the face is washed and wet. Try to pad it dry with a clean towel, or maybe preferably a clean paper towel.
- Wash your hands often, although too often may dry/damage the skin, which may cause an opposite effect.
A difficulty can be that you want to get the skin clean from staph bacteria by using anti sceptics when cleaning, but at the same time, application of it damages/dries the skin, which may cause an opposite effect. So you have to find a balance in this.
Heavier approaches that some may use are bleach baths: a warm water bath with bleach added to it. (Be careful with the concentration, because it can be aggressive on the skin and eyes. If the skin dries out too much, it may cause an opposite effect.)
A solution that works for our skin problem here on this forum, is the application of Soolantra (topical ivermectine, available on prescription), it has anti-demodex and anti-inflammatory properties, but it isn’t meant to treat impetigo, and may be completely useful for impetigo. But if it’s not impetigo, then it may be interesting to try it.