Stability and Transient Dynamics of a Distributed-Order Fractional SEIRS Epidemic Model with Temporary Immunity

This paper investigates a distributed-order fractional SEIRS epidemic model with temporary immunity, vaccination, disease-induced mortality, and density-dependent natural mortality. The distributed-order formulation provides a flexible framework for incorporating heterogeneous memory effects into epidemic dynamics. The basic reproduction number is derived, the disease-free and endemic equilibria are characterized, and their local stability properties are investigated through the distributed-order characteristic equation. In addition, a sufficient global disease extinction criterion is established using a distributed-order Lyapunov argument, yielding global asymptotic stability of the disease-free equilibrium under an explicit transmission bound. Numerical spectral analysis shows that no oscillatory stability transition occurs within the considered parameter ranges and that the endemic equilibrium remains asymptotically stable within these ranges. Numerical simulations examine the effects of the memory interval and memory density shape on epidemic transients. Broadening the memory interval toward lower fractional orders slows convergence and increases transient persistence. For a fixed memory interval, densities concentrated near lower fractional orders produce the slowest relaxation, whereas those concentrated near higher orders, closer to the classical first-order limit, yield the fastest decay. These results show that distributed-order memory primarily regulates transient epidemic dynamics while preserving endemic equilibrium stability within the investigated parameter regimes, providing new insight into heterogeneous memory effects in fractional epidemic models.