A Novel Framework for Proving Global Stability of Age-Structured Models: Application to a Heroin Epidemic Model

Jianquan Li ^[1] ; Junyuan Yang ^[2] ; Nini Xue ^[1] ; Yao Chen ^[1]
1. [1] Xijing University
  
  Xijing University
  
  China
2. [2] Shanxi University
  
  Shanxi University
  
  China
Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 5, 2025
Idioma: inglés
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Resumen
- The global stability of age-structured dynamical models holds significantly practical implications. At present, the Lyapunov direct method is one of the most effective techniques for proving such global properties. In this article, we introduce a method for establishing the global stability of the positive steady state of an age-structured model. This method involves constructing a suitable Lyapunov functional and demonstrating the negative (semi-)definiteness of its derivative along the solutions of the model. The appropriate Lyapunov functional is systematically derived based on the structure of the model and its corresponding boundary conditions. Meanwhile, the negative (semi-)definiteness of the derivative is proven by expressing it in precise form and applying a series of inequalities. The application of this method to an age-structured heroin epidemic model demonstrates its efficacy and generalizability. This approach significantly simplifies the analysis of the global stability of the positive steady state of age-structured models with integral boundary conditions.