We are not given details of how Bud contracted the infection, but it's significant that a member of the family home had already contracted the disease: it means the pathogen was already in the home.
My theory: the dog is the source of the infection. Throughout the movie we see the dog being allowed to roam free inside or outside. This was the crucial mistake that doomed everybody within home, despite the extreme precautions taken.
In fact, Bud's original infection may have been brought in by the dog originally, but we aren't given enough information to know.
Either way: Travis got sick before everyone else, and the only behaviour he had over the rest of his family was sleeping closely with the dog every night. If Bud's leftover pathogen had spread to the dog or its fur, then Travis was clearly likely to be the next victim.
When the dog began barking erratically and ran away from the home (likely just chasing an animal, but being disobedient in manner), it was merely the dog showing symptoms of being infected similar to the strange dreams and thoughts shown by Travis. As the infection flourished within both, we can explain the unlocked door as Travis acting erratically. His "dream" of going outside at night was real, and when he returned he did not leave the door locked as he should have, leading to the sick dying dog being able to return to the red door.