Great finding! This is very non-intuitive knowing the way inactivated vaccine should work to enhance our immunity. Strains of the flu are generally grown in fertilized chicken egg and they are killed or inactivated before their antigen is used for vaccination. Seems like problem is with how our body is reacting to those virus antigens and developing immunity against it after vaccination. One hypothesis could be the strain of virus antigen people are getting rather amplifies the immune response in a bad way by messing up with the inflammation signals, particularly of our adaptive immune system that is suppose to remember the antigen it encounters for future defense.
Or simply there are so many strains of flu given to us these days through vaccine that our immune system get friendly and used to with the virus antigens. As a result, immune cells do not not attack the real virus at all (do not even release antibody) when its time to defend us from influenza. The real influenza then does its job of getting us sick by replicating quickly in our body. I will wait till someone contest or backup the PNAS paper you have highlighted.
I found this paper in the Lancet journal [1] saying influenza vaccine is not that effective (only roughly works about 35% of the time) and other paper saying 43% people believes flu vaccine can give you the flu [2]. Influenza vaccination is indeed a confusing thing for many people and it’s worth highlighting and talking about it in the platform like Steemit. Looking forward for more post like this.