Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (6): 1950-1968.

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

Dandan Sun1,*(),Yingke Li1(),Zhidong Teng2,Tailei Zhang3   

  1. 1 School of Mathematics and Physics, Xinjiang Agriculture University, Urumqi 830052
    2 School of Mathematics and System Science, Xinjiang University, Urumqi 830046
    3 School of Sciences, Changan University, Xi an 710064
  • Received:2020-07-03 Online:2021-12-26 Published:2021-12-02
  • Contact: Dandan Sun;
  • Supported by:
    the China Postdoctoral Science Foundation(2020M683714XB);the NSF of Xinjiang(2021D01A98);the Natural Science Basic Research Plan in Shaanxi Province(2021JM-445)


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 well-possdness of solutions of the model is obtained, including non-negativity, boundedness, asymptotic smoothness, etc. Then the equilibria and the basic reproduction number ${{\cal R}}_{0}$ of the model is derived, and it is proved that the epidemic is uniformly persistent when ${{\cal R}}_{0}>1$. 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 ${{\cal R}}_{0}<1$; if ${{\cal R}}_{0}>1$, 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.

Key words: Age-structured measles model, Well-posedness of solutions, Basic reproduction number, Uniform persistence, Stability of equilibrium

CLC Number: 

  • O175.1