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Description
This study develops and analyzes an SIRS epidemic model with convex incidence and saturated treatment under both autonomous and nonautonomous frameworks. For the autonomous system, we characterize the disease-free and endemic equilibria and perform a detailed bifurcation analysis, revealing backward and saddle-node bifurcations, as well as Hopf bifurcations that generate endemic bubbles. Furthermore, the bifurcation structure uncovers a codimension-two double-zero bifurcation arising from the interaction between saddle-node and Hopf bifurcations. The nonautonomous extension incorporates seasonal variations in transmission
and recovery rates, capturing realistic periodic forcing observed in infectious diseases such as influenza. Using epidemiological data from the Democratic Republic of the Congo, we identify December as the peak influenza season. Analytical results establish conditions for the existence and global stability of a positive periodic solution, while numerical simulations demonstrate that seasonality can induce complex dynamics, including multiperiodic and chaotic oscillations. Low seasonal intensity sustains disease coexistence, whereas strong seasonal forcing may lead to population extinction. The emergence of quasiperiodic (torus) and chaotic (strange) attractors highlights how seasonal forcing can transform regular epidemic cycles into irregular outbreaks, providing new insights into the role of seasonality in infectious disease dynamics and control.
