Fractional-order SIR model for ADHD as a neurobiological and genetic disorder.

This study develops and analyzes a fractional-order Susceptible-Infected-Recovered (SIR) epidemiological model to investigate the transmission dynamics and control of Attention Deficit Hyperactivity Disorder (ADHD) within a population. The model incorporates memory effects via the Caputo fractional derivative, capturing long-term dependencies characteristic of ADHD progression. Numerical simulations are carried out using the Laplace Residue Power Series (LRPS) and Runge-Kutta 4th Order (RK4) methods for different values of the fractional-order parameter α. Results reveal that higher values of α lead to faster disease spread and recovery, while lower values correspond to more prolonged transitions between disease states. Stability analysis of disease-free and endemic equilibria confirms that the basic reproduction number [Formula: see text] governs the persistence or eradication of ADHD, with [Formula: see text] indicating sustained prevalence. Sensitivity analysis highlights the influence of genetic susceptibility, treatment efficacy, and intervention timing on disease outcomes. A comparative error analysis shows that RK4 outperforms LRPS in accuracy for fractional-order systems. The study also integrates optimal control theory, introducing time-dependent control functions representing prevention and treatment efforts. Simulation results demonstrate that optimized interventions significantly reduce ADHD prevalence while minimizing associated costs. These findings emphasize the importance of early diagnosis, effective treatment, and sustained public health strategies. Future extensions may incorporate stochastic effects, age-structured populations, and adaptive control mechanisms to enhance predictive accuracy and policy planning.