Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA.
Math Biosci. 2012 Jan;235(1):98-109. doi: 10.1016/j.mbs.2011.11.002. Epub 2011 Nov 13.
Mathematical models have made considerable contributions to our understanding of HIV dynamics. Introducing time delays to HIV models usually brings challenges to both mathematical analysis of the models and comparison of model predictions with patient data. In this paper, we incorporate two delays, one the time needed for infected cells to produce virions after viral entry and the other the time needed for the adaptive immune response to emerge to control viral replication, into an HIV-1 model. We begin model analysis with proving the positivity and boundedness of the solutions, local stability of the infection-free and infected steady states, and uniform persistence of the system. By developing a few Lyapunov functionals, we obtain conditions ensuring global stability of the steady states. We also fit the model including two delays to viral load data from 10 patients during primary HIV-1 infection and estimate parameter values. Although the delay model provides better fits to patient data (achieving a smaller error between data and modeling prediction) than the one without delays, we could not determine which one is better from the statistical standpoint. This highlights the need of more data sets for model verification and selection when we incorporate time delays into mathematical models to study virus dynamics.
数学模型在帮助我们理解 HIV 动力学方面做出了重要贡献。在 HIV 模型中引入时滞通常会给模型的数学分析以及将模型预测与患者数据进行比较带来挑战。在本文中，我们将 HIV-1 模型中的两个时滞（一个是病毒进入后感染细胞产生病毒所需的时间，另一个是适应性免疫反应出现以控制病毒复制所需的时间）引入到模型中。我们通过证明解的正定性和有界性、无感染和感染平衡点的局部稳定性以及系统的一致持久性来开始模型分析。通过开发几个李雅普诺夫泛函，我们得到了确保平衡点全局稳定性的条件。我们还将包括两个时滞的模型拟合到 10 名原发性 HIV-1 感染患者的病毒载量数据中，并估计参数值。尽管延迟模型比没有延迟的模型更能拟合患者数据（在数据和建模预测之间的误差更小），但我们无法从统计学角度确定哪个更好。这突出表明，当我们将时滞引入到研究病毒动力学的数学模型中时，需要更多的数据集来验证和选择模型。
