Stability and bifurcation analysis of a delayed information-induced SIR model with generalized incidence function and Holling Type II treatment

The motive of this study is to investigate nonlinear dynamical properties of an SIR delay model that incorporates a novel and generalized incidence rate, Holling Type II treatment to reflect limited medical resources, and the effects of delayed information-induced behavioral changes. A dual-term generalized incidence rate function is introduced to measure the influence of both psychological factors and individual behavioral responses arising from awareness, economic consequences, and environmental factors. The local and global stability properties are investigated to determine the rich behavior of multiple equilibrium points. The conditions for the occurrence of codimension one bifurcations, such as Transcritical, Saddle–node and Hopf bifurcation, are derived under no delay effect in information’s growth. Further, the direction and stability of oscillations via Hopf bifurcation further lead to the codimension-two Generalized Hopf (Bautin’s) bifurcation, while normal form theory determines the occurrence of Cusp and Bogdanov–Takens bifurcations. Due to limitations in medical resources and a nonlinear incidence rate, the model admits multiple equilibria with successive stability switches, giving rise to endemic bubbles with different frequencies that closely mimic the case of the recent COVID-19 outbreak. For the delay model, stability results of endemic equilibria and conditions for the emergence of Hopf bifurcation, along with their direction and stability, are established. Due to the delayed information’s effect, multiple Hopf bifurcations cause repeated stability and instability switches (stable to unstable and vice versa) of endemic equilibria, leading to alternating stable and unstable states and waves. Extensive numerical simulations suggest that delays in information dissemination can significantly alter equilibrium stability and generate complex disease dynamics. Thus, our findings demonstrate the importance of generalized incidence, treatment and timely information dissemination in designing effective strategies to reduce disease spread.