This study develops and analyzes an SIRS epidemic model with convex incidence and saturated treatment under both autonomous and non-autonomous frameworks. For the autonomous system we characterize the disease-free and endemic equilibria and perform bifurcation analysis, showing backward and saddle-node bifurcations, Hopf bifurcations that generate endemic bubbles. Also, bifurcation analyses reveal a codimension-two double-zero bifurcation arising from the intersection of saddle-node and Hopf bifurcations. The non-autonomous extension incorporates seasonal variations in transmission and recovery rates, capturing realistic periodic forcing observed in diseases such as influenza. Using data from the Democratic Republic of the Congo, we confirm December as the peak influenza season. Analytical results establish conditions for the existence and global stability of a positive periodic solution, while numerical simulations show that seasonality induces complex behaviors, including multi-periodic and chaotic oscillations. Low seasonal intensity sustains coexistence, whereas high-intensity forcing leads to population extinction. The emergence of quasi-periodic (torus) and chaotic (strange) attractors demonstrates how seasonal forcing can transform regular epidemic cycles into irregular outbreaks, offering new insights into the role of seasonality in infectious disease dynamics and control.