The current recommendations for a child with what is considered "Early localized" infection - a rash with no signs of any other symptoms is 6+ weeks of abx. You can see this either on Dr. Jones' site (link above in previous post in this thread) which is being quoted from Dr. Joseph Burrascano's "Advanced Topics in Lyme Disease" - which can be found in our 'New to Lyme?' thread.
If it was my child that showed a rash with no other symptoms, personally I would go after 8 weeks of abx at the proper dosage. BTW - what is the dosage of Doxy? You need to be sure the abx are being given at the proper dosage otherwise the only thing that's being done is to keep the bacteria from replicating, not actually killing them. This would allow a relapse - which of course, no one wants to see!!
The problem with Doxy and why most doc's won't prescribe it for children is that it needs to be in high doses - 400 - 600 mgs each day. The "staining" of his teeth could be permanent, just so that you know. Most knowledgeable docs will prescribe Amoxicillin at 50 mgs per each Kilogram of weight of the child divided into 3 doses a day, according to Joseph Burrascano's Advanced Topics in Lyme Disease.
If necessary, get to a LLMD -Lyme Literate Medical Doctor or a Holistic practitioner. One thing is for sure - the infection was caught early, so don't take a chance that it's not all killed off so that it will have the opportunity to come back at a later time.
Have I confused you yet? Giggle!