In this paper, we study ultimate dynamics and derive tumor eradication conditions for the angiogenic switch model developed by Viger et al. This model describes the behavior and interactions between host ([Formula: see text]); effector ([Formula: see text]); tumor ([Formula: see text]); endothelial ([Formula: see text]) cell populations. Our approach is based on using the localization method of compact invariant sets and the LaSalle theorem. The ultimate upper bound for each cell population and ultimate lower bound for the effector cell population are found. These bounds describe a location of all bounded dynamics. We construct the domain bounded in [Formula: see text]- and [Formula: see text]-variables which contains the attracting set of the system. Further, we derive conditions imposed on the model parameters for the location of omega-limit sets in the plane [Formula: see text] (the case of a localized tumor). Next, we present conditions imposed on the model and treatment parameters for the location of omega-limit sets in the plane [Formula: see text] (the case of global tumor eradication). Various types of dynamics including the chaotic attractor and convergence dynamics are described. Numerical simulation illustrating tumor eradication theorems is fulfilled as well.

中文翻译：

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血管生成开关和免疫治疗的癌症模型：终极动力学分析中的肿瘤根除

在本文中，我们研究了 Viger 等人开发的血管生成开关模型的最终动力学并推导出肿瘤根除条件。该模型描述了宿主之间的行为和交互（[公式：见正文]）；效应器（[公式：见正文]）；肿瘤（[公式：见正文]）；内皮（[公式：见正文]）细胞群。我们的方法基于使用紧不变量集的定位方法和拉萨尔定理。找到每个细胞群的最终上限和效应细胞群的最终下限。这些边界描述了所有有界动态的位置。我们构造了以[公式：见文本]-和[公式：见文本]-变量为界的域，其中包含系统的吸引集。进一步，我们推导出模型参数上施加的条件，用于平面中欧米茄极限集的位置[公式：见文本]（局部肿瘤的情况）。接下来，我们提出了施加在模型和治疗参数上的条件，用于平面中欧米茄极限集的位置[公式：见正文]（全局肿瘤根除的情况）。描述了各种类型的动力学，包括混沌吸引子和收敛动力学。还实现了说明肿瘤根除定理的数值模拟。描述了各种类型的动力学，包括混沌吸引子和收敛动力学。还实现了说明肿瘤根除定理的数值模拟。描述了各种类型的动力学，包括混沌吸引子和收敛动力学。还实现了说明肿瘤根除定理的数值模拟。