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I hope he is not allergic to the dog.
My child got sick yrs and yrs ago...similar situation. I got rid of all my toxic cleaners, floor mops w/ spray, toxic candles and laundry detergents. I used dry Swiffer pads. vacuuming, vinegar/water and Seventh Generation products...Even switched to organic bar soaps/soy candles.
My child never went back to the dr.
Even if a dog does not shed, there is still dander. Bathe the dog once a week....if it's not too cold/weather. When you say new house...Do you mean new construction? Outgassing? I know that I cannot sit in a new car or house for very long. Could be the house---If walking pneumonia is ruled out,