If antibiotics are not working then there is less likelihood of an actual bacterial infection and it is most likely environmental. This being the spring season there are probably a lot of new vegetation popping up and there may be some irritant or intrusive growth in your yard. Have a walk around and look for any thorny, prickly, sticky, or intrusive growth on the ground. Could also be some form of parasitic creatures on the ground because of the change in season.
Can you tell if this happened on a seasonal basis? Has it happened before at the same time period? Have you eliminated allergies as a cause?
Try spreading a broad spectrum yard treatment that will kill off yard bugs to eliminate the possibility of fire ants and other such critters.
I have read the Epsom Salt and warm water used to bathe (salve) the paws may help treat the infection but will not prevent it.
Check for these plants in your yard - Common ragwort (Senecio jacobaea)