Faculty of Arts and Sciences, Department of Mathematics, Near East University, Northern Cyprus, Turkey; Department of Computer Science and Mathematics, Lebanese American University, 1107-2020, Beirut, Lebanon.
Mathematics Research Center, Near East University, Nicosia, North Cyprus, 99138, Turkey.
Comput Biol Med. 2024 Aug;178:108756. doi: 10.1016/j.compbiomed.2024.108756. Epub 2024 Jun 19.
Tuberculosis, a global health concern, was anticipated to grow to 10.6 million new cases by 2021, with an increase in multidrug-resistant tuberculosis. Despite 1.6 million deaths in 2021, present treatments save millions of lives, and tuberculosis may overtake COVID-19 as the greatest cause of mortality. This study provides a six-compartmental deterministic model that employs a fractal-fractional operator with a power law kernel to investigate the impact of vaccination on tuberculosis dynamics in a population.
Some important characteristics, such as vaccination and infection rate, are considered. We first show that the suggested model has positive bounded solutions and a positive invariant area. We evaluate the equation for the most important threshold parameter, the basic reproduction number, and investigate the model's equilibria. We perform sensitivity analysis to determine the elements that influence tuberculosis dynamics. Fixed-point concepts show the presence and uniqueness of a solution to the suggested model. We use the two-step Newton polynomial technique to investigate the effect of the fractional operator on the generalized form of the power law kernel.
The stability analysis of the fractal-fractional model has been confirmed for both Ulam-Hyers and generalized Ulam-Hyers types. Numerical simulations show the effects of different fractional order values on tuberculosis infection dynamics in society. According to numerical simulations, limiting contact with infected patients and enhancing vaccine efficacy can help reduce the tuberculosis burden. The fractal-fractional operator produces better results than the ordinary integer order in the sense of memory effect at diffract fractal and fractional order values.
According to our findings, fractional modeling offers important insights into the dynamic behavior of tuberculosis disease, facilitating a more thorough comprehension of their epidemiology and possible means of control.
结核病是一个全球性的健康问题，预计到 2021 年将新增 1060 万例病例，其中耐多药结核病的病例将会增加。尽管 2021 年有 160 万人死亡，但目前的治疗方法挽救了数百万人的生命，结核病可能会超过 COVID-19 成为最大的死亡原因。本研究提供了一个六 compartmental 确定性模型，该模型采用具有幂律核的分形分数算子来研究疫苗接种对人群中结核病动态的影响。
考虑了一些重要的特征，如疫苗接种和感染率。我们首先证明了所提出的模型具有正有界解和正不变区域。我们评估了最重要的阈值参数，基本再生数的方程，并研究了模型的平衡点。我们进行了敏感性分析，以确定影响结核病动态的因素。不动点概念表明了所提出模型的解的存在性和唯一性。我们使用两步牛顿多项式技术来研究分数算子对幂律核广义形式的影响。
已经证实了分形分数模型在 Ulam-Hyers 和广义 Ulam-Hyers 类型下的稳定性分析。数值模拟显示了不同分数阶值对社会中结核病感染动态的影响。根据数值模拟，限制与感染患者的接触并提高疫苗的效力可以帮助减轻结核病的负担。在分数阶值的分形和分数阶意义上，分形分数算子比普通整数阶产生更好的记忆效果。
根据我们的发现，分数模型为结核病疾病的动态行为提供了重要的见解，有助于更深入地理解其流行病学和可能的控制手段。
