This thesis investigates the stochastic dynamics of the SIR epidemic model through Large Deviation Theory and the Wentzel-Kramers-Brillouin (WKB) approximation. While standard mean-field equations effectively capture macroscopic epidemic trends, they fail to account for finite-size fluctuations and rare anomalous trajectories. By mapping the exact stochastic Master Equation onto a deterministic Hamilton-Jacobi formalism, the epidemic evolution is redefined as a classical mechanics problem, where optimal fluctuation paths minimize a physical action. The core objective of this work is to provide a quantitative methodology capable of determining whether an observed anomalous outbreak is a rare stochastic fluctuation of a standard epidemic, or if it is driven by fundamentally different underlying kinetic rates (beta and mu).