Multicycle dynamics and high-codimension bifurcations in SIRS epidemic models with cubic psychological saturated incidence

TL;DR

This paper explores complex bifurcation phenomena in an SIRS epidemic model with cubic psychological saturation, revealing multiple limit cycles and intricate dynamics influenced by infection rates and psychological effects.

Contribution

It rigorously demonstrates the existence of codimension-three Bogdanov-Takens bifurcations and multiple limit cycles using singularity theory, advancing understanding of epidemic model complexity.

Findings

Existence of three limit cycles in certain parameter regimes

Identification of codimension-three Bogdanov-Takens bifurcations

Demonstration of complex dynamics influenced by psychological effects

Abstract

This study investigates bifurcation dynamics in an SIRS epidemic model with cubic saturated incidence, extending the quadratic saturation framework established by Lu, Huang, Ruan, and Yu (Journal of Differential Equations, 267, 2019). We rigorously prove the existence of codimension-three Bogdanov-Takens bifurcations and degenerate Hopf bifurcations, demonstrating the coexistence of three limit cycles within a single epidemiological model, a phenomenon that is rarely documented and exhibits significant dynamical complexity. Our analysis reveals that both the infection rate $\kappa$ (through specific inequality conditions) and psychological effect thresholds critically govern disease dynamics: from complete eradication to various persistence patterns, including multiple periodic oscillations and coexistent steady states. By innovatively applying singularity theory, we characterize the topology of the bifurcation set through the local unfolding of singularities and the identification of nondegenerate singularities for fronts. Numerical simulations verify the emergence of three limit cycles in monotonic parameter regimes and two limit cycles in nonmonotonic regimes. This work advances existing bifurcation research by incorporating higher-order interactions and comprehensive singularity analysis, thereby providing a mathematical foundation for decoding complex transmission mechanisms critical to the design of public health strategies.