From the Guardian News Story:

Ultimately, though, where you are sat on a flight can dictate what lurgies you end up with. Howard Weiss, a biomathematician at the Georgia Institute of Technology, modelled behaviour on flights and calculated the risks of becoming ill with airborne diseases. “Direct transmission of diseases such as flu is very unlikely beyond passengers seated about a metre away from the infected passenger,” says Weiss. “For flu, I think the worst case was overall there would be two additional infections in the entire economy cabin.”

Weiss and his team also took swabs from planes. “What we found was the bacteria that you find in an airplane looks a lot like in your home or office. If you are not immune-compromised, then as long as you keep your hands away from your face, you’re probably going to be fine. People have this impression that airplanes are like sewers, but we’ve shown that’s not the case.”