Well i dont have astma, but my younger cusin has.
I believe it depends on the astma.
I was very upset last summer when i found out that a divemaster had taken my cusin (12 years old) for an introduction dive. My cusin was aware of that he should not dive with his astma as it is very bad, if he tries to run 100m he will need his inhalor, and he told this tp the DM and he said that it is ok.
Let me tell you this... he is no DM anymore
If you are old enough, you can take the decission yourself. And if you are alergic to cats and nothing else, well i dont belive you are going to see any cats underwater, perhaps a catfish
...
But if you case is severe, just like my cusin, then you have nothing to do underwater...
/Jonny