Альфонс Гайдеріх

A rocket engineer that bears a resemblance to Alphonse Elric and befriends Edward in the movie. He is fascinated by Ed's stories about his travels around his original world, but is unsure of whether he can believe in the fact that Ed came from another world where alchemy is real, and his brother looks like him. He is studying under the father of modern rocketry, and wants to prove the scientific worth of his fellow Germans by inventing a rocket that can actually be ridden. He also has his own reasons to do it as fast as possible - he is slowly succumbing to an unspecified disease that eventually causes him to cough up blood.

This is most likely Tuberculosis as it fits all his symptoms and makes him keeping it a secret make much more sense.