Stability and Hopf-type bifurcation of a fractional-order delayed SIRS model with behavioral responses driven by positive and negative media information

We established a five-dimensional SIRS epidemic model coupling information–disease dynamics, featuring behavioral responses driven by competitive interactions between positive and negative media information, where the behavioral responses were concretely characterized by an arctangent vaccination willingness function and a fractional linear incidence rate, together with fractional-order memory effects and a behavioral response delay. After proving the well-posedness of the model (existence, uniqueness, nonnegativity, and boundedness of solutions), defining the basic reproduction number $\mathcal{R}_0$, and establishing the existence and uniqueness of the two equilibria, we investigated the local asymptotic stability of the disease-free equilibrium and the endemic equilibrium, as well as the Hopf-type bifurcation of the endemic equilibrium. Specifically, for the disease-free equilibrium, determinant factorization reduced the 5D problem, proving that if the delay-free asymptotic stability condition holds, the presence of the behavioral response delay ($\tau>0$) does not destabilize the system and no Hopf-type bifurcation occurs. For the endemic equilibrium, using the Sylvester resultant elimination method, a twentieth-degree algebraic criterion provided delay-independent stability conditions and delay-induced Hopf-type bifurcation criteria, yielding an explicit expression for the critical delay. Finally, numerical simulations verified the local asymptotic stability of the disease-free equilibrium, the local asymptotic stability of the endemic equilibrium, and the Hopf-type bifurcation, confirming the theoretical analysis results.