In this paper, mathematical modeling of the dynamics of Human T-cell lymphotropic virus type I (HTLV-1) infection and the development of adult T-cell leukemia (ATL) cells is investigated by a time scale approach. The proposed models, constructed by nonlinear systems of first-order difference equations and -difference equations, characterize the relationship among uninfected, latently infected, actively infected CD4 cells, and ATL cells, where the growth of leukemia cells is described by discrete logistic curves. The stability results are established based on basic reproduction number . When , infected T-cells always die out and there exist two disease-free equilibria depending on the proliferation rate and the death rate of leukemia cells. When , HTLV-1 infection becomes chronic and spreads, and there exists a unique endemic equilibrium point. The stability results of disease-free and endemic equilibrium points are obtained when and , respectively. Furthermore, the sensitivity analysis discovers the key parameters of the models related to . Estimated parameters are applied based on the experimental observation. The numerical analysis also shows the equilibrium level of ATL cell proliferation is higher when the HTLV-I infection of T-cells is chronic than when it is acute. Moreover, our mathematical modeling by a time scale approach yields a new parameter to an HTLV-1 infection model which determines data frequency.

中文翻译：

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分析与 HTLV-1 感染相关的 ATL 发育发病机制的时间尺度方法


在本文中，通过时间尺度方法研究了人类 T 细胞嗜淋巴细胞病毒 I 型 (HTLV-1) 感染和成人 T 细胞白血病 (ATL) 细胞发育动力学的数学模型。所提出的模型由一阶差分方程和-差分方程的非线性系统构建，表征了未感染、潜伏感染、主动感染的 CD4 细胞和 ATL 细胞之间的关系，其中白血病细胞的生长由离散 Logistic 曲线描述。稳定性结果是根据基本再生数建立的。当 时，受感染的 T 细胞总是死亡，并且存在两个无病平衡，具体取决于白血病细胞的增殖率和死亡率。当 时，HTLV-1感染转为慢性并扩散，存在一个独特的流行平衡点。无病平衡点和地方病平衡点的稳定性结果分别在 和 时获得。此外，敏感性分析发现了与 相关的模型的关键参数。根据实验观察应用估计参数。数值分析还表明，当T细胞的HTLV-I感染为慢性时，ATL细胞增殖的平衡水平高于急性感染时的水平。此外，我们通过时间尺度方法进行的数学建模为 HTLV-1 感染模型产生了一个新参数，该参数决定了数据频率。