Solution of Time Fractional SIQR Epidemic System and Research with Respect to the Fractional Order

This article deals with the global existence and uniqueness of solutions to a fractional-order SIQR epidemic model, alongside its intricate chaotic and complex dynamics as functions of the fractional order. The well-posedness of the model solutions, including global existence, uniqueness, and positivity, is established by constructing appropriate Lyapunov functions. The local and global stability analyses are conducted for both the disease-free and endemic equilibria of the model. An asymptotic solution of the system in the form of series is derived by the Laplace–Adomian decomposition method (L–ADM), and its convergence is rigorously proved. Subsequently, numerical analysis determines and interprets the optimal truncation order of this asymptotic solution. Numerical simulations are performed based on the asymptotic solution, and the dynamics and chaos of the dynamic system with respect to the fractional order are analyzed and illustrated in terms of the maximum Lyapunov exponent and structural complexity. Finally, a local sensitivity analysis is conducted for each state variable with respect to the model parameters.