Xi'an Key Laboratory of Human-Machine Integration and Control Technology for Intelligent Rehabilitation, Xijing University, No. 1, Xijing Road, Xi'an, 710123, Shaanxi, China.
Department of Mathematics, Wilfrid Laurier University, 75 University Avenue West, Waterloo, ON, N2L 3C5, Canada.
J Math Biol. 2023 Apr 24;86(5):80. doi: 10.1007/s00285-023-01918-4.
In this paper, we first formulate a system of ODEs-PDE to model diseases with latency-age and differential infectivity. Then, based on the ways how latent individuals leave the latent stage, one ODE and two DDE models are derived. We only focus on the global stability of the models. All the models have some similarities in the existence of equilibria. Each model has a threshold dynamics for global stability, which is completely characterized by the basic reproduction number. The approach is the Lyapunov direct method. We propose an idea on constructing Lyapunov functionals for the two DDE and the original ODEs-PDE models. During verifying the negative (semi-)definiteness of derivatives of the Lyapunov functionals along solutions, a novel positive definite function and a new inequality are used. The idea here is also helpful in applying the Lyapunov direct method to prove the global stability of some epidemic models with age structure or delays.
在本文中，我们首先建立了一个具有潜伏期和差异传染性的 ODE-PDE 系统来对疾病进行建模。然后，基于潜伏个体离开潜伏阶段的方式，推导出一个 ODE 和两个 DDE 模型。我们仅关注模型的全局稳定性。所有模型在平衡点的存在上都有一些相似之处。每个模型都有一个全局稳定性的阈值动力学，完全由基本再生数来刻画。方法是 Lyapunov 直接法。我们提出了一种为两个 DDE 和原始 ODE-PDE 模型构建 Lyapunov 泛函的思路。在验证 Lyapunov 泛函沿解的导数的负（半）定性质时，使用了一个新的正定函数和一个新的不等式。这个思路在应用 Lyapunov 直接法来证明具有年龄结构或时滞的一些传染病模型的全局稳定性时也很有帮助。
