Good point, and I agree. However, the mutated variant has to get loose in the relevant animal population for this to happen.
Stamping it out in the affected humans would need to be coupled with measures to keep the mutated form from spreading within the potential animal hosts in the area. Culling the kind of animals the disease was thought to have been transmitted from would have to occur. (I suspect that is the explanation for culling $1 billion of Russian poultry, and the Chinese focus on pigs in an H5N1 infected region.)
Personally, I do not think it can be kept in the box. But we have to try.
Think back to 1997, when this flu first appeared on earth. 18 or so people sick, 6 died. Every single chicken in Hong Kong was killed, in a desperate attempt to stop it.
8 years later, it flies west...