具有潜伏期和分布时滞的传染病模型及动力学分析

Dynamic Analysis on Latent Period and Distributed Delay Epidemiological Model

摘要: 本文研究了一类潜伏期具有感染性的疾病传播模型, 其中引入分布时滞刻画了潜伏期患者转症为感 染者的概率. 通过分析相应的特征方程, 讨论了无病平衡点和地方病平衡点的局部稳定性. 再通过 构造合适的 Lyapunov 泛函, 应用 LaSalle 不变性原理, 得到当疾病传播阈值即基本再生数小于 1 时, 无病平衡点是全局渐近稳定的, 疾病将会消失; 当基本再生数大于 1 时, 地方病平衡点是全 局渐近稳定的, 疾病将持续存在. 最后用数值模拟验证了理论结果。

Abstract: In this paper, a model of disease transmission with infectious incubation period is studied, in which a distributed time delay is introduced to characterize the probability of patients in latency becoming infected. By analyzing the corresponding character- istic equations, the local stability of disease-free equilibrium and endemic equilibrium is discussed. Then, by constructing an appropriate Lyapunov functional and applying LaSalle invariance principle, we obtained that the disease-free equilibrium point is globally asymptotically stable and the disease will disappear when the disease trans- mission threshold, namely the basic regeneration number, is less than 1. When the basic reproduction number is greater than 1, the endemic equilibrium point is globally asymptotically stable and the disease will persist. Finally, the theoretical results are verified by numerical simulation.