A tumor is most dangerous disease of medical science which is a mass or lumps of tissue that’s formed by an accumulation of abnormal cells. A famous fractional tumor-immune model is interpreting the dynamics of tumor and effector cells. In this work, we provide a comparative and chaotic study of tumor and effector cells through fractional tumor-immune dynamical model. A new arbitrary operator based on the Mittag-Leffler law is assumed for this study. Again, we examine the interactions among distinct tumor cell inhabitants and immune structure through a model of real world problem of medical science. We First investigate the dynamical effect of the activation of the effector immune and tumor cells by using Adams-Bashforth-Moulton and Toufik-Atangana methods. Furthermore, this paper analyses the existence and uniqueness of given tumor-immune model of arbitrary order. Further, we have examined the dynamical behaviors of the fractional tumor-immunne model and obtained results are compared with exiting results by other methods. Numerical simulations are executed by Adams-Bashforth-Moulton and Toufik-Atangana methods using popular Atangana-Baleanu fractional derivative. Our obtained results will be useful for biologists to the treatment of cancer disease.

肿瘤是医学上最危险的疾病，是由异常细胞堆积形成的组织块或团块。著名的肿瘤免疫分数模型正在解释肿瘤和效应细胞的动力学。在这项工作中，我们通过肿瘤免疫分数动态模型对肿瘤和效应细胞进行了比较和混沌的研究。本研究假设基于Mittag-Leffler律的新的任意算子。再次，我们通过医学现实世界问题模型研究了不同肿瘤细胞居民与免疫结构之间的相互作用。我们首先研究使用Adams-Bashforth-Moulton和Toufik-Atangana方法激活效应免疫细胞和肿瘤细胞的动力学效应。此外，本文分析了给定任意顺序的肿瘤免疫模型的存在性和唯一性。此外，我们检查了部分肿瘤免疫模型的动力学行为，并通过其他方法将获得的结果与现有结果进行了比较。使用流行的Atangana-Baleanu分数导数通过Adams-Bashforth-Moulton和Toufik-Atangana方法执行数值模拟。我们获得的结果将对生物学家治疗癌症疾病有用。