Bifurcation analysis and phase-locked invariant circles in a discrete SIR system

This paper investigates the nonlinear dynamics of a discrete-time SIR epidemic model. We establish the existence of transcritical, flip, and Neimark-Sacker bifurcations, along with Arnold tongues and 1:2 strong resonance, and derive practical and actionable strategies for effective epidemic control. Numerical simulations confirm the theoretical findings and illustrate period-doubling cascades, invariant circles, and homoclinic structures. Moreover, when parameters lie within the Arnold tongue regions bounded by weak phase-locking, the system admits an invariant circle composed of one stable periodic orbit and one unstable periodic orbit. Based on theoretical analysis, we finally obtain several infectious disease control strategies.