Global Dynamics and Complete Phase Portrait Classification of a Nonlinear SIRS Model with Saturated Incidence
TL;DR
The analysis proves the absence of Hopf bifurcations, thereby excluding the existence of limit cycles and sustained oscillations, and provides a rigorous global description of the system and contributes to the qualitative theory of nonlinear epidemic models.
Abstract
This paper investigates the global dynamics of a nonlinear Susceptible–Infective–Recovered–Susceptible (SIRS) epidemic model with a nonmonotonic saturated incidence rate. By reducing the model to a planar cubic polynomial differential system, we perform a complete qualitative analysis in the biologically relevant positive quadrant. By using Poincaré compactification and blow-up techniques, all finite and infinite equilibrium points are characterized, and their local and global behaviors are determined. A complete classification of the topologically distinct global phase portraits is established. The analysis proves the absence of Hopf bifurcations, thereby excluding the existence of limit cycles and sustained oscillations. From an epidemiological perspective, the results reveal how behavioral responses and psychological inhibition effects shape the long-term dynamics, leading to either disease extinction or persistence depending on parameter regimes. These findings provide a rigorous global description of the system and contribute to the qualitative theory of nonlinear epidemic models.