  We investigate SIR models with vital dynamics, reinfection, and randomness at
the transmission coefficient and recruitment rate. Initially, we conduct an
extensive analysis of the autonomous scenario, covering aspects such as local
and global well-posedness, the existence and internal structure of attractors,
and the presence of gradient dynamics. Subsequently, we explore the
implications of small nonautonomous random perturbations, establishing the
continuity of attractors and ensuring their topological structural stability.
Additionally, we study scenarios in which both the transmission coefficient and
the recruitment rate exhibit time-dependent or random behavior. For each
scenario, we establish the existence of attractors and delineate conditions
that determine whether the disease is eradicated or reaches an endemic state.
Finally, we depict numerical simulations to illustrate the theoretical results.
