Abdo Mohammed S, Alghamdi Najla, Alzumi Hadeel Z, Shammakh Wafa
Department of Mathematics, Hodeidah University, Al-Hudaydah, 3114, Yemen.
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia.
Comput Biol Med. 2025 Sep;195:110514. doi: 10.1016/j.compbiomed.2025.110514. Epub 2025 Jun 21.
This paper proposes a fractional-order model using the Atangana-Baleanu-Caputo derivative to study the co-dynamics of tuberculosis and diabetes mellitus among susceptible (S), TB-infected (I), DM-infected (D), and co-existence (C) populations. The model's well-posedness is established via the Banach fixed-point theorem, ensuring the uniqueness and positivity of solutions. Basic reproduction numbers (R,R,R) are derived, with values exceeding unity indicating the instability of the disease-free equilibrium and progression toward endemicity. Sensitivity analysis highlights key parameters (β,β,δ,δ,δ) affecting co-existence dynamics. Numerical simulation is conducted over T=365 days (1 year) with a unit step h=1 day, using the Adams-Bashforth method to reveal that lower fractional orders α∈(0,0.8] slow disease decay. The model is validated against real data over 90 days at α=0.5 using logistic growth for C(t). Results underscore the effectiveness of fractional calculus in modeling chronic co-existence and guiding control strategies.
本文提出了一个使用阿坦加纳-巴莱努-卡普托导数的分数阶模型，以研究易感人群（S）、结核病感染人群（I）、糖尿病感染人群（D）和共存人群（C）中结核病和糖尿病的共同动态。通过巴拿赫不动点定理建立了模型的适定性，确保了解的唯一性和正性。推导了基本再生数（R、R、R），其值超过1表明无病平衡点不稳定并向地方病发展。敏感性分析突出了影响共存动态的关键参数（β、β、δ、δ、δ）。使用亚当斯-巴什福思方法在T = 365天（1年）内进行数值模拟，步长h = 1天，结果表明较低的分数阶α∈(0, 0.8]会减缓疾病衰退。在α = 0.5时，使用C(t)的逻辑增长对90天的真实数据进行了模型验证。结果强调了分数阶微积分在模拟慢性共存和指导控制策略方面的有效性。
