带有饱和常数和布朗运动的HIV随机分析

Stochastic Analysis of an HIV Model with Saturation Incidence and Brownian Motion

摘要: 本文在经典HIV感染动力学模型基础上，提出了一种同时考虑外部随机扰动与饱和常数的新型随机微分方程模型。在模型构建中，将随机噪声引入靶细胞感染速率，以刻画外界不确定因素对感染过程的干扰；同时引入饱和常数a描述高靶细胞密度下的感染饱和效应，从而提升模型的生物学合理性。针对所建立的随机模型，本文首先利用停时技巧、Lyapunov函数及Itô公式，严格证明了全局正解的存在唯一性，并分析了解的正性与有界性。进一步，通过对数Lyapunov函数结合停时方法，推导了感染细胞与病毒的均值持久性条件，以及系统发生随机灭绝的充分条件。最后，结合数值模拟验证理论分析，并针对噪声强度开展参数敏感性研究，划分了系统的持久区与灭绝区。结果表明，适度的随机扰动可显著影响感染细胞稳态水平，且饱和感染机制能够有效避免非生物学意义的无限增长。本文为HIV感染的随机动力学分析提供了理论依据与数值参考。

Abstract: Based on the classical HIV infection dynamical model, this paper proposes a new stochastic differential equation model considering both external stochastic perturbations and saturated infection rate. In the modeling process, random noise is introduced into the infection rate of target cells to characterize the disturbance of external uncertain factors. Meanwhile, the saturation constant a is introduced to describe the infection saturation effect at high target cell density, which improves the biological rationality of the model. For the established stochastic model, we first rigorously prove the existence and uniqueness of the global positive solution by using the stopping time technique, Lyapunov functions and Itô’s formula, and analyze the positivity and boundedness of the solution. Furthermore, using logarithmic Lyapunov functions combined with the stopping time method, we derive the conditions for the mean persistence of infected cells and viruses, as well as the sufficient conditions for stochastic extinction of the system. Finally, numerical simulations are conducted to verify the theoretical results, and parametric sensitivity analysis is performed on the noise intensity, which divides the persistence region and extinction region of the system. The results show that moderate random perturbations can significantly affect the steady-state level of infected cells, and the saturated infection mechanism can effectively avoid non-biologically meaningful unbounded growth. This work provides a theoretical basis and numerical reference for the stochastic dynamic analysis of HIV infection.