Afzal Zeeshan, Alshehri Mansoor
Department of Mathematics, Lahore Garrison University, Lahore Campus, Lahore, Pakistan.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia.
Sci Rep. 2025 Jul 2;15(1):22992. doi: 10.1038/s41598-025-07646-7.
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.
本研究开发并分析了一个分数阶易感-感染-康复（SIR）流行病学模型，以研究注意力缺陷多动障碍（ADHD）在人群中的传播动态和控制。该模型通过Caputo分数阶导数纳入记忆效应，捕捉ADHD进展的长期依赖性特征。针对分数阶参数α的不同值，使用拉普拉斯留数幂级数（LRPS）和四阶龙格-库塔（RK4）方法进行了数值模拟。结果表明，α值越高，疾病传播和恢复越快，而较低的值对应于疾病状态之间更长的转变。无病平衡和地方病平衡的稳定性分析证实，基本再生数[公式：见原文]决定了ADHD的持续存在或根除，[公式：见原文]表明持续流行。敏感性分析突出了遗传易感性、治疗效果和干预时机对疾病结果的影响。比较误差分析表明，对于分数阶系统，RK4在精度上优于LRPS。该研究还整合了最优控制理论，引入了代表预防和治疗努力的时间依赖控制函数。模拟结果表明，优化干预措施可显著降低ADHD患病率，同时将相关成本降至最低。这些发现强调了早期诊断、有效治疗和持续公共卫生策略的重要性。未来的扩展可能会纳入随机效应、年龄结构人群和自适应控制机制，以提高预测准确性和政策规划。
