Bifurcation analysis and chaotic dynamics in an SIR model with nonlinear incidence and constrained healthcare capacity

Abstract

This study develops and analyses an $SIR$ epidemic model incorporating a non-monotone incidence rate that captures the combined effects of stringent governmental interventions, evolving social behavior, and public response. A distinctive feature of this work lies in examining how intra-specific competition among susceptible individuals for limited resources shapes the course of disease dynamics. Both autonomous and non-autonomous frameworks are examined, with special emphasis on the periodic variation in hospital bed availability—a critical yet often overlooked driver of epidemic dynamics. Analytical results establish conditions for the existence and stability of trivial, infection-free and endemic equilibria in relation to the basic reproduction number. Local stability shifts induced by transcritical, Hopf, and saddle–node bifurcations are explored, while two-parameter bifurcation analysis reveals regions of coexisting equilibria. Advanced bifurcation structures, including Bogdanov–Takens, generalized Hopf, and cusp points, are characterized, exposing transitions to complex and chaotic dynamics of the system. Sensitivity analysis identifies key parameters influencing outbreak intensity and continued persistence of the disease. Numerical simulations illustrate theoretical findings, indicating how healthcare capacity constraints, especially fluctuating hospital beds, profoundly shape epidemic trajectories. These results highlight the intricate interplay between medical resources, nonlinear transmission effects, and public health measures, offering strategic insights for designing adaptive, resource-aware intervention policies capable of mitigating epidemic impact under varying socio-behavioral and infrastructural conditions.