To be quite honest, any answer to this question is purely speculative. The illness itself is purely informational in origin so there is no question whatsoever of anything like a vaccine ever existing.
Realistically speaking, a complete change in the way in which society is structured is required before this illness will disappear. A social form in which it is possible for anyone to feel an outsider must go before asthma can cease to exist. A new social form in which co-responsibility is the norm would eliminate this problem. As this is the direction in which mankind is currently moving, I would expect asthma to finally disappear over the next 100 years or so.