Dynamics analysis and optimal control of a fractional-order SEAIQRM model with media effects

As a crucial research tool, epidemic models have played an important role in predicting disease progression. In this study, we delineated the full dynamics of epidemics by extending the susceptible–exposed–infectious–recovered (SEIR) model to include media effects ( $M$), unascertained cases ( $A$), and case isolation in a hospital ( $Q$), generating a model that we called SEAIQRM. Dual time delays, aware susceptibles, and fractional order were also incorporated, synergistically enhancing the model's accuracy in depicting the real dynamic process of disease transmission. We calculated a specific expression for the basic reproduction number $R_0$, proved the existence and uniqueness of the model solution, and analyzed the local and global stability of two types of equilibrium points. To investigate the optimal control of the model, the sensitivity indices of the parameters in $R_0$ were computed, and vaccination rate and isolation rate were selected as control variables, exerting the strongest effects on $R_0$. Finally, the effectiveness of the model in illustrating and controlling the spread of infectious diseases is verified through numerical simulation. For a fractional order $\alpha$ in the interval [0.7, 0.9], the peak sizes of the asymptomatic ( $A$) and symptomatic ( $I$) compartments decreased significantly relative to the $\alpha$ = 1 benchmark, corresponding to reductions of 32%–37% and 28%–33%, respectively. Implementing an optimal control strategy with vaccination ( $u$ = 0.4) and quarantine ( $q$ = 0.5) minimized implementation costs while achieving the most effective reduction in disease spread. The model can provide information regarding intervention timing in a setting with similar parameters. However, its use in the real-world requires calibration and validation.