Head lice are no respectors of persons. They prefer short hair, and hygenic conditions, otherwise other parasites tend to wipe them out. i.e. dirty kids are not the main source. With my kids, I have always found that a daily combing with a fine comb sterilised in raw alcohol has done the trick. The technique is that you take layers of hair, comb right out from the scalp, and pull right out to the end in one stroke, and after each stroke, wipe the comb on a piece of WATER WET TISSUE. If there are Nits, they will show red rather than a normal grey.
Please make sure that the comb is dry after sterilisation, you just need to blow on it, otherwise you will strip essential oils from the hair.
In UK we can buy "Surgical Spirit", which is ok, but in most other regions, it is best to buy good strong vodka. All you have to do is to saturate a bit of (cotton) cloth with vodka and put it in a plastic bag so that it doesn't dry out. Use that to sterilise your comb, make sure the wad doesn't dry out. Renew the wad each week. Don't share combs or brushes.
Look at the condition of the hair to see if it needs feeding.
If this doesn't work, I'll eat my hat, but at least then you can resort to chemical treatments.