具有时滞的年龄结构肿瘤免疫模型的稳定性和Hopf分岔分析

Zhong Luo, Zijian Liu, Yuanshun Tan
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引用次数: 0

摘要

本文提出并分析了一种具有时间延迟的年龄结构肿瘤免疫模型。我们把免疫细胞分为两种。一种是生长与肿瘤无关的,另一种是生长依赖于肿瘤模拟的。对于这些细胞,我们考虑了它们的生理年龄。在肿瘤模拟依赖性免疫细胞中引入成熟时间延迟[公式:见文本],以限制参与免疫反应的细胞生长到最小生理年龄。建立了无肿瘤平衡状态的存在性和稳定性阈值[公式:见文]。如果[公式:见文],无肿瘤平衡状态是局部和全局渐近稳定的。而当[公式:见文]时,当[公式:见文]时,肿瘤平衡状态局部渐近稳定,当[公式:见文]通过阈值[公式:见文]时,肿瘤平衡状态发生Hopf分岔。这可以部分解释某些肿瘤的周期性复发。最后通过数值模拟对理论结果进行了验证。
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Stability and Hopf Bifurcation Analysis for an Age-Structured Tumor Immune Model with Time Delay
In this paper, we propose and analyze an age-structured tumor immune model with time delay. We divide immune cells into two kinds. One is those whose growth is independent of tumor and the other is those whose growth depends on the simulation of the tumor. For these cells, their physiological ages are considered. A mature time delay [Formula: see text] is introduced to the tumor-simulation-dependent immune cells to restrict those cells who participate in the immune response to grow to a minimum physiological age. The existence and stability threshold [Formula: see text] is established for the tumor-free equilibrium state. If [Formula: see text], the tumor-free equilibrium state is both locally and globally asymptotically stable. Whereas, when [Formula: see text], the tumor equilibrium state is locally asymptotically stable if [Formula: see text] and a Hopf bifurcation occurs when [Formula: see text] passes through the threshold [Formula: see text]. This may partly explain the periodic recurrence of some tumors. Finally, theoretical results are verified by some numerical simulations.
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