基于免疫状态依赖的免疫流行病学模型全局动力学分析
Global Dynamics of anImmuno-Epidemiological Modelwith Immune-State Dependence
DOI: 10.12677/PM.2025.1512299, PDF,   
作者: 邓幽兰：中国地质大学（武汉）数学与物理学院, 湖北 武汉
关键词: 免疫流行病模型；Lyapunov泛函；全局稳定性；Immuno-Epidemiological Model； Lyapunov Functional； Global Stability
摘要: 本文研究了一类状态依赖的的免疫流行病学模型，其中感染个体的分布由其体内免疫状态所刻 画，并允许染病者体内的免疫状态在感染过程中连续变化，其增长率由连续函数描述。 通过对系 统平衡点附近的线性系统进行稳定性分析，并通过构建合适的Lyapunov泛函，我们证明了无病 平衡点和地方病平衡点的全局动力学性质，平衡点的稳定性态仅依赖于基本再生数。 结果表明， Lyapunov 直接法是分析具有免疫或年龄结构的流行病模型全局动力学的有效工具。 文中进一步 给出了数值模拟结果以验证主要理论结果。
Abstract: In this paper, we investigate a class of state-dependent immune-epidemiological mod- els, in which the distribution of infected individuals is characterized by their immune status. The model allows the immune status of infected individuals to vary dynami- cally throughout the infection period, with the growth rate described by a continuous function. By conducting local stability analysis of the linearized system near equilibria and constructing appropriate Lyapunov functionals, we establish the global stability of both the disease-free equilibrium and the endemic equilibrium, which depends solely on the basic reproduction number. Our results demonstrate that the Lyapunov direct method is a powerful and effective tool for analyzing the global dynamical behavior of epidemic models incorporating immune or age-structured components. Furthermore, numerical simulations are presented to validate the theoretical findings.
文章引用：邓幽兰. 基于免疫状态依赖的免疫流行病学模型全局动力学分析[J]. 理论数学, 2025, 15(12): 104-118. https://doi.org/10.12677/PM.2025.1512299

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