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<p>This study investigates an epidemic framework governed by differential equations featuring delays of generalized piecewise constant structure (DEGPCD). The primary objective is to construct an invariant region and to demonstrate the existence and uniqueness of solutions by employing integral equation techniques under appropriately defined conditions. Additionally, an auxiliary lemma is derived, establishing a precise connection between the functional values of the state variable within the delay argument and the temporal domain. To investigate the model’s dynamic behavior, the Lyapunov–Razumikhin framework is utilized, tailored to accommodate the structural features introduced by the DEGPCD. The stability of the infection-free state is rigorously examined, while the positive steady state is transformed into an equivalent zero equilibrium to facilitate the analysis. Sufficient criteria are then formulated to guarantee uniform asymptotic stability for both equilibria, offering theoretical insights that enhance the applicability of DEGPCD-based epidemic modeling.</p>

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Existence, Uniqueness, and Stability Analysis Results for an Epidemic Model with Piecewise Constant Delay of Generalized Type

  • 期刊论文

  • 发表日期：

    2026年3月23日

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  • Kuo-Shou Chiu

摘要

This study investigates an epidemic framework governed by differential equations featuring delays of generalized piecewise constant structure (DEGPCD). The primary objective is to construct an invariant region and to demonstrate the existence and uniqueness of solutions by employing integral equation techniques under appropriately defined conditions. Additionally, an auxiliary lemma is derived, establishing a precise connection between the functional values of the state variable within the delay argument and the temporal domain. To investigate the model’s dynamic behavior, the Lyapunov–Razumikhin framework is utilized, tailored to accommodate the structural features introduced by the DEGPCD. The stability of the infection-free state is rigorously examined, while the positive steady state is transformed into an equivalent zero equilibrium to facilitate the analysis. Sufficient criteria are then formulated to guarantee uniform asymptotic stability for both equilibria, offering theoretical insights that enhance the applicability of DEGPCD-based epidemic modeling.