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Abstract：
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.
摘要：
人类总是面临传染病的威胁。事实证明,免疫系统的强弱与结核病等传染病的传播有着直接的联系,肝炎，艾滋病,而新冠肺炎一旦免疫系统没有能力对抗感染和传染病。此外，事实证明，数学建模是准确描述复杂生物现象的重要工具。在最近的文学中，我们很容易发现，这些有效的工具为我们理解和分析肿瘤生长等问题提供了重要贡献。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，我们提出了一些新的近似解的计算公式，模型之间的相互作用肿瘤生长和免疫系统与几个分数和分形算子。该模型中使用的运算符是Liouville-Caputo，卡普托-法布里齐奥,和Atangana-Baleanu-Caputo在分数和分形分数意义上。还验证了每种情况下解的存在性和唯一性。为了完成我们的分析，我们包括许多数值模拟来显示肿瘤的行为。这些图表帮助我们解释数学结果，更好地描述相关的生物学概念。在许多情况下，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无法控制的行为的复杂性。因此,新实施的运营商肯定会在不同疾病建模中出现的进一步计算模型中打开新的研究窗口。可以肯定的是，该领域的类似问题也可以通过本文采用的方法进行建模。
