Fractional-order analysis and optimal control of the NERA model: stability, sensitivity, and numerical validation

In this study, the four-dimensional non-linear dynamical behavior of the NERA model is modeled using fractional calculus. The existence of the solution is examined through the integration of the Caputo operator. The existence, uniqueness, and the conditions ensuring the uniqueness of the solution of the NERA model are investigated. The fixed points are determined, and the stability of the proposed model is demonstrated. The fractional-order Lyapunov stability of the system is analyzed. The numerical scheme of the Adams-Bashforth-Moulton method is also implemented with the Caputo operator. Time series and phase portraits are employed to analyze the behavior of the fractional-order system under different derivative orders and parameter variations. In addition to these aspects, the basic reproduction number $$(R_{0})$$ is derived to determine the threshold conditions for disease persistence and eradication. A detailed sensitivity analysis is performed to identify the key parameters that most significantly influence the system’s dynamics. To further explore effective disease control strategies, an optimal control framework is formulated and implemented by incorporating multiple control variables. Alongside the ABM method, the fractional Runge-Kutta fourth-order (RK4) method is employed to validate the numerical findings and enhance the accuracy of the simulations. The interplay of analytical results with numerical simulations is emphasized, highlighting how fractional-order dynamics capture more realistic behaviors compared to classical integer-order models.