Ghanbari Behzad
Department of Engineering Science, Kermanshah University of Technology, Kermanshah, Iran.
Department of Mathematics, Faculty of Engineering and Natural Sciences, Bahçeşehir University, 34349 Istanbul, Turkey.
Adv Differ Equ. 2020;2020(1):585. doi: 10.1186/s13662-020-03040-x. Epub 2020 Oct 19.
Humans are always exposed to the threat of infectious diseases. It has been proven that there is a direct link between the strength or weakness of the immune system and the spread of infectious diseases such as tuberculosis, hepatitis, AIDS, and Covid-19 as soon as the immune system has no the power to fight infections and infectious diseases. Moreover, it has been proven that mathematical modeling is a great tool to accurately describe complex biological phenomena. In the recent literature, we can easily find that these effective tools provide important contributions to our understanding and analysis of such problems such as tumor growth. This is indeed one of the main reasons for the need to study computational models of how the immune system interacts with other factors involved. To this end, in this paper, we present some new approximate solutions to a computational formulation that models the interaction between tumor growth and the immune system with several fractional and fractal operators. The operators used in this model are the Liouville-Caputo, Caputo-Fabrizio, and Atangana-Baleanu-Caputo in both fractional and fractal-fractional senses. The existence and uniqueness of the solution in each of these cases is also verified. To complete our analysis, we include numerous numerical simulations to show the behavior of tumors. These diagrams help us explain mathematical results and better describe related biological concepts. In many cases the approximate results obtained have a chaotic structure, which justifies the complexity of unpredictable and uncontrollable behavior of cancerous tumors. As a result, the newly implemented operators certainly open new research windows in further computational models arising in the modeling of different diseases. It is confirmed that similar problems in the field can be also be modeled by the approaches employed in this paper.
人类始终面临传染病的威胁。事实证明，一旦免疫系统失去对抗感染和传染病的能力，免疫系统的强弱与结核病、肝炎、艾滋病和新冠病毒等传染病的传播之间存在直接联系。此外，事实证明数学建模是准确描述复杂生物现象的有力工具。在最近的文献中，我们很容易发现这些有效工具为我们理解和分析肿瘤生长等问题做出了重要贡献。这确实是需要研究免疫系统如何与其他相关因素相互作用的计算模型的主要原因之一。为此，在本文中，我们针对一个计算公式提出了一些新的近似解，该公式用几个分数阶和分形算子对肿瘤生长与免疫系统之间的相互作用进行建模。该模型中使用的算子在分数阶和分形 - 分数阶意义上分别是刘维尔 - 卡普托、卡普托 - 法布里齐奥和阿坦加纳 - 巴莱努 - 卡普托算子。还验证了每种情况下解的存在性和唯一性。为了完善我们的分析，我们纳入了大量数值模拟来展示肿瘤的行为。这些图表有助于我们解释数学结果并更好地描述相关生物学概念。在许多情况下，所获得的近似结果具有混沌结构，这证明了癌性肿瘤不可预测和不可控行为的复杂性。因此，新应用的算子无疑为不同疾病建模中出现的进一步计算模型打开了新的研究窗口。可以确认，该领域的类似问题也可以用本文所采用的方法进行建模。
