Comprehensive Dynamic Analysis of a Time-Delayed SIR Epidemiological Model

In this paper, the Hopf bifurcation of the time-delay epidemic model is deeply studied, and the stability and bifurcation conditions are analyzed by combining theory and numerical simulation, so as to provide a scientific basis for the prevention and control strategy. Firstly, the epidemic dynamics model with time delay is constructed. Then by applying stability theory and bifurcation theory, we have conducted a detailed analysis of the local stability of the model equilibrium point and identified the delay threshold τ0 that causes Hopf bifurcation to occur. Using mathematical theory and numerical calculation, the time-delay dynamic figure of the system is drawn, which provides theoretical support for the prevention and control of infectious diseases. The analysis of complex mathematical phenomena has deepened our understanding of infectious disease prevention, enriched the delay model theory, and helped formulate scientific prevention and control strategies.