具有潜伏期传染与年龄结构的SEI模型动力学分析

Dynamic Analysis of an Age-StructuredSEI Model with Latent Infectivity

摘要: 本文研究了一类具有年龄结构的SEI 模型。该模型强调潜伏者人群对易感者的感染机制，并刻画了疾病在潜伏期的转化过程。在给定初值与边界条件下，讨论了模型解的非负性和最终有界性，并确立了决定疾病最终是否爆发的关键阈值:基本再生数R₀。通过Lyapunov直接法，建立了无病平衡点P⁰以及地方病平衡点P^*的全局稳定性判据。结果表明:当R₀ < 1时，无病平衡点P⁰全局渐近稳定，疾病最终将会消亡;当R₀ > 1时，系统存在唯一地方病平衡点P^*且其全局渐近稳定，疾病将长期存在并形成地方病。

Abstract: This paper investigates an SEI epidemic model with age structure, in which the infection mechanism from exposed individuals to susceptible individuals is emphasized, and the transition process during the latent period is characterized. Under given initial and boundary conditions, the nonnegativity and ultimate boundedness of solutions are established. Moreover, the basic reproduction number R₀, which determines whether an outbreak will occur, is derived. By employing the Lyapunov direct method, global stability criteria for the disease-free equilibrium P⁰ and the endemic equilibrium P^* are obtained. The results show that when R₀ < 1, the disease-free equilibrium P⁰ is globally asymptotically stable, implying that the disease will eventually die out; when R₀ > 1, there exists a unique endemic equilibrium P^* which is globally asymptotically stable, indicating that the disease will persist and evolve into an endemic state.