I performed a literature review. The articles that I found dated back to the 1970s and 1980s. The presence of cats in the home varied among these studies. A Swedish study in which there were no cats or furry pets in the home found a decrease in bronchial reactivity. Other studies in which there were a mix of cats in the home did not find this reduction to occur. Unfortunately, I could not find any data that addresses your question of acquiring a pet after a course allergen immunotherapy.
Jacqueline A. Pongracic, MD, FAAAAI