A fractal-fractional mathematical model for COVID-19 and tuberculosis using Atangana–Baleanu derivative

This study aims to develop a compartmental epidemiological model for the co-infection of COVID-19 and tuberculosis, incorporating a Holling type II treatment rate for individuals with tuberculosis, COVID-19, and dual infections while considering incomplete treatment in some TB cases. The model analysis examines the sub-models for COVID-19, TB, and the combined co-infection model. Using the fixed-point method, the research investigates the existence and uniqueness of solutions for the proposed model. It also explores a stability analysis to evaluate Ulam-Hyer’s reliability. Furthermore, it discusses and validates Lagrange’s interpolation polynomial through a specific case study to numerically compare different fractal and fractional orders.

本研究旨在构建针对新冠病毒（COVID-19）与结核病（Tuberculosis，TB）合并感染的分室流行病学模型，纳入针对结核病患者、新冠感染者及双重感染人群的霍林II型（Holling type II）治疗率参数，同时考虑部分结核病病例存在治疗不彻底的情况。本研究对该模型的三类子模型——新冠感染子模型、结核病子模型及合并感染综合模型——开展分析。研究采用不动点法（fixed-point method），探究所提出模型的解的存在性与唯一性。同时开展稳定性分析，以评估乌拉姆-海尔斯（Ulam-Hyer）稳定性的可靠性。此外，本研究通过特定案例研究讨论并验证拉格朗日（Lagrange）插值多项式，以对不同分形阶数与分数阶数开展数值对比分析。