If your white blood cell count is high there is something going on.
I thought some mystery tick bite disease, too
I second lyme disease because the cyclical nature suggests a parasite or spirochetes and because antibiotics help
Or another of the tick borne diseases
It can be very hard to diagnose without an expert
If you have it - need to get treated or it progresses
Lyme disease?
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My guess also. Not many doctors have expertise in diagnosis this illness. Seek out a specialist who has an established track record in treating lyme disease. I had a friend who went for years struggling with the illness until he found someone who identified and effectively treated the disease. You never get rid of it, but it can be managed.
I was thinking Lyme also. Difficult to find a Dr who takes it seriously though.