具有年龄结构的麻疹传染病模型的稳定性分析

Analysis of the Stability for Measles Epidemic Model with Age-Structured

该文研究一类具有年龄结构的SVEIR麻疹模型.首先将模型化为所谓的Volterra型积分方程,得到了模型解的适定性,包括非负性、有界性、渐近光滑性等.其次得到了模型的平衡点和基本再生数R_(0),并证明了当基本再生数R_(0)>1时疾病的一致持续性.进一步通过分析特征方程和构造适当的Lyapunov函数得到了:若R_(0)<1,则模型仅存在全局渐近稳定的无病平衡点;若R_(0)>1,则无病平衡点不稳定,地方病平衡点存在且全局渐近稳定.这些理论结果应用在关于全国麻疹传染病数据的趋势分析方面.

In this paper,a kind of SVEIR measles epidemic model with age structure is established.Firstly,the model is transformed into Volterra integral equation and the wellpossdness of solutions of the model is obtained,including non-negativity,boundedness,asymptotic smoothness,etc.Then the equilibria and the basic reproduction number R_(0)of the model is derived,and it is proved that the epidemic is uniformly persistent when R_(0)>1.Further by analyzing the characteristic equations and selecting suitable Lyapunov functions,we get the model only has the disease-free equilibrium that is globally asymptotically stable if R_(0)<1;if R_(0)>1,the disease-free equilibrium is unstable,the endemic disease equilibrium exist and is globally asymptotically stable.These main theoretical results are applied in the analysis of the trend in data on measles infectious diseases across the country.