Abstract: Recurrent oscillations in biological systems, particularly in models of persistent infections, often indicate underlying slow-fast dynamics. Classical approaches such as Geometric Singular Perturbation Methods (GSPM) rely on the existence of critical manifolds and are limited to systems with small perturbation parameters. However, these conditions are not always met in complex or higher-dimensional models. This thesis introduces a set of refined analytical criteria for detecting slow-fast oscillations beyond the scope of traditional perturbation theory. Applying this framework to a three-dimensional SIR model with secondary transmission and intrinsic population growth, we use bifurcation analysis, center manifold theory, and normal form analysis to identify conditions under which Hopf bifurcations and periodic solutions arise, even without asymptotically small growth rates. Numerical simulations support the theoretical findings, demonstrating the robustness and broader applicability of the proposed method in capturing oscillatory behavior across a wide range of parameter regimes.