Six months after the outbreak of a lethal pathogen, a man must travel across the country to a potential safe zone with his infected father and a young girl who possibly holds a cure.

The problem with that logline is that it sounds so much like the SyFy movie of the week. The logline needs to be jazzed up. Maybe there is a twist in the characters or the environment?
Maybe the environment is medieval Europe during the time of the great plague?
Maybe the character is the one who cause the plague and is now trying to reverse the damage he caused.
Maybe the protagonist has a cured, but is losing his memory and the girl is the main driver of the periple?
Your script may be the best script, but the logline does not attract me. The movie from the script may be a masterpiece, but the logline sounds like a cheap Direct To Video.

Maybe you can make the audience think the little girl is the cure when in reality she is the carrier of the pathogen? Your idea has been done so many times. Thinking up a unique twist or unexpected direction will help.

What nicholasandrewhalls said. I just took the logline for "The Children of Men" and swapped out words. What's the unique hook in your story? What is about that hasn't already been done?