The future is history.
Why it is hard sci-fi
A 2035 Earth where the surface is uninhabitable after a 1996 pandemic. A prisoner is sent back through an experimental time-displacement program to gather information about the original strain. The film commits to a closed-timeline rule. The protagonist is a witness to his own death as a child at the airport, and the ending closes that loop. The post-collapse society is shown as a working underground bureaucracy with limited resources.
Science inside it
Closed timelike curves with self-consistent loops, the epidemiology of an engineered virus released through commercial aviation, and the operational shape of a research program that must use a partly broken time machine because the budget cannot afford a better one. The films post-pandemic Earth is a working portrayal of an above-ground exclusion zone.
Spoiler alert
The Army of the Twelve Monkeys are a red herring. The actual release is a personal act by a single laboratory assistant flying out of Philadelphia. The protagonist witnesses the release and is shot at the airport in front of his own younger self, which is the moment that sent him forward as a child.