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Abstract

Understanding the evolution of infectious diseases within a dynamic population is essential for formulating effective public health strategies. This work introduces an enhanced SEIRS epidemic model that incorporates demographic variation and a single-dose vaccination strategy, supporting that immunity can be reduced over time. The model reflects diseases in which individuals may return to the susceptible state after recovery. Extending prior reduced three-dimensional models, this study develops a complete four-dimensional SEIRS framework, thereby increasing the model’s applicability to real-world scenarios. The inclusion of the full system presents greater analytical challenges. To overcome this, we extend the second additive compound matrix and establish the global stability of equilibrium points using appropriate Lyapunov functions. The threshold transmission parameter is derived analytically, and the steady states are examined through both local and global analyses. By linking the reduced and full systems, the model offers deeper insights into the long-term behavior of disease dynamics. Numerical simulations implemented in MATLAB (R2024a) validate the analytical results. A detailed sensitivity analysis identifies the parameters that most substantially affect disease spread. The results of this study emphasize key factors in the modeling of epidemics and offer practical guidance to improve vaccination strategies and health policy planning.

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