The mathematical modeling of infectious diseases is essential for understanding transmission mechanisms and the long-term persistence of pathogens. This paper investigates the qualitative dynamics of an SIRS epidemic model, particularly relevant for diseases where immunity is temporary. The study provides a rigorous mathematical framework to describe how the interplay between infection rates and the loss of immunity affects the system's behavior, moving from local stability to a global perspective.

The study employs the qualitative theory of planar differential equations to explore the system’s state space. A central component is the application of the Poincaré compactification technique, which extends the polynomial vector field from the finite plane to the Poincaré disc. This enables the analysis of the system’s behavior at infinity. We characterize all singular points by analyzing Jacobian matrices for finite equilibria and utilizing specialized transformations for infinite singularities at the disc's boundary.

The analysis provides a comprehensive characterization of the system’s equilibria, including disease-free and endemic states, and singularities at the horizon of the Poincaré disc. A significant finding is the identification of a Hopf bifurcation occurring at critical biological thresholds. This bifurcation is responsible for the emergence of limit cycles. Global phase portraits are constructed to visualize trajectories and detail the topological structure of the flow under various parameter regimes.

The investigation reveals that the SIRS model possesses complex dynamical features. The occurrence of limit cycles indicates that the disease can exhibit periodic oscillations, with significant implications for predicting recurring outbreaks. By integrating global analysis at infinity, this research provides a complete topological description of the model, offering valuable insights into nonlinear disease dynamics and public health strategies.