Stability analysis and Hopf bifurcation in a time-delayed fractional epidemic model

• This paper proposes a fractional-order SIR epidemic model with time delay and saturated incidence rate, and analyzes the stability of its equilibrium points. • Hopf bifurcation analysis is conducted separately with time delay τ and fractional order α as bifurcation parameters. • The critical bifurcation delay τ 0 * decreases as the fractional order α increases, where larger α values reduce the stability region and slow down system convergence. • Analysis shows that the critical bifurcation order α * decreases monotonically with increasing time delay τ . This paper investigates a time-delayed fractional SIR epidemic model incorporating a saturated incidence rate. First, the existence of endemic equilibria is investigated under different conditions, and the basic reproduction number is calculated utilizing the next-generation matrix method. Then, the global and local asymptotic stability of equilibrium points are analyzed. For the time delay τ = 0 , global asymptotic stability conditions are obtained by constructing a Lyapunov function. Finally, through Hopf bifurcation analysis, the influence of the fractional order on the system’s dynamic behavior is discussed. This study provides epidemiological insights that inform practical measures for managing infectious disease outbreaks: (i) During the initial epidemic stage, it is crucial to minimize the response delay to prevent exceeding the critical bifurcation threshold; (ii) The stability of epidemic containment systems can be significantly enhanced by adjusting the fractional order appropriately.