Isn't it obvious that Frank (the little girl's father) was immune?
With this daughter being sick for a week (he said she started showing symptoms little over a week ago), there's no way he wouldn't be sick also. Yet not only was he without symptoms, but at full strength (unlike his daughter and anyone whom became infected).
If anyone was immune/chosen, it was him, not Brian.
Maybe to address the fact that no disease is 100% fatal in real life. Even the black plague and ebola are not 100% fatal, with some people managing to recover on their own and thus becoming immunized.
For all we know, Frank did become infected (how could he not?) yet managed to recover and thus is now immune. It does happen in real life.
Can't think of another reason, either that or poor writing.