Department of Mathematics, University of Management and Technology, Lahore, Pakistan.
Department of Mathematics, Faculty of Sciences, University of Central Punjab, Lahore, 54500, Pakistan.
Sci Rep. 2024 Oct 12;14(1):23871. doi: 10.1038/s41598-024-74767-w.
The diabetes mellitus model (DMM) is explored in this study. Many health issues are caused by this disease. For this reason, the integer order DMM is converted into the time delayed fractional order model by fitting the fractional order Caputo differential operator and delay factor in the model. It is proved that the generalized model has the advantage of a unique solution for every time t. Moreover, every solution of the system is positive and bounded. Two equilibrium states of the fractional model are worked out i.e. disease free equilibrium state and the endemic equilibrium state. The risk factor indicator, R is computed for the system. The stability analysis is carried out for the underlying system at both the equilibrium states. The key role of R is investigated for the disease dynamics and stability of the system. The hybridized finite difference numerical method is formulated for obtaining the numerical solutions of the delayed fractional DMM. The physical features of the numerical method are examined. Simulated graphs are presented to assess the biological behavior of the numerical method. Lastly, the outcomes of the study are furnished in the conclusion section.
本研究探讨了糖尿病模型（DMM）。这种疾病会导致许多健康问题。因此，通过在模型中拟合分数阶 Caputo 微分算子和延迟因子，将整数阶 DMM 转换为时滞分数阶模型。证明了广义模型对于每个时间 t 都具有唯一解的优势。此外，系统的每个解都是正的且有界的。计算了分数模型的两个平衡点，即无病平衡点和地方病平衡点。计算了系统的风险因素指标 R。对基础系统在两个平衡点进行了稳定性分析。研究了 R 在疾病动力学和系统稳定性中的关键作用。提出了用于获得时滞分数 DMM 数值解的混合有限差分数值方法。检查了数值方法的物理特性。呈现了模拟图以评估数值方法的生物学行为。最后，在结论部分给出了研究结果。
