A multi-scale model coupling within-host infection and between-host transmission with immune delay, infection age, multiple transmission routes and general incidence is developed based on the complexity of environmentally-driven infectious disease transmission. The model is composed of ordinary differential equations (ODEs), delay differential equations (DDEs), and a partial differential equation (PDE). Firstly, the dynamics of the within-host model are analyzed, including the existence and stability of infection-free equilibrium, immunity-inactivated equilibrium, immunity-activated infection equilibrium and Hopf bifurcation. Further, in the context of the coupled between-host model that disregards immune responses, the basic reproduction number R̃0h, the existence and stability of equilibria, the existence of backward bifurcation and the uniform persistence are obtained. And then, focusing on the within-host model with stable immunity-activated infection equilibrium, the results are achieved regarding the basic reproduction number R0h, the existence and stability of equilibria for the coupled between-host model. In addition, when stable periodic solutions exist for the within-host model, the existence and stability of the disease-free and positive periodic solutions for the coupled between-host model are determined by numerical simulations. Effective disease control is achieved when two crucial factors are met: a robust adaptive immune response in the host, coupled with an optimally shortened latency period for generating immune components following antigen exposure. Finally, numerical simulations are employed to substantiate these primary findings, illustrate the practical application of our model and propose control strategies for mitigating disease transmission.

中文翻译：

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具有感染年龄和一般发病率的延迟多尺度环境病传播模型的复杂动力学


基于环境驱动的传染病传播的复杂性，开发了一种多尺度模型，将宿主感染和宿主间传播与免疫延迟、感染年龄、多种传播途径和一般发病率耦合。该模型由常微分方程 （ODE）、延迟微分方程 （DDE） 和偏微分方程 （PDE） 组成。首先，分析了宿主内模型的动力学，包括无感染平衡、免疫失活平衡、免疫激活感染平衡和 Hopf 分叉的存在和稳定性;此外，在忽略免疫反应的宿主间耦合模型的背景下，获得了基本繁殖数 R̃0h、平衡的存在和稳定性、向后分叉的存在和均匀的持久性。然后，以具有稳定免疫激活感染平衡的宿主内模型为重点，获得了基本繁殖数 R0h、宿主间耦合模型平衡的存在性和稳定性的结果。此外，当宿主内模型存在稳定的周期解时，通过数值模拟确定宿主间耦合模型的无病周期解和正周期解的存在性和稳定性。当满足两个关键因素时，即可实现有效的疾病控制：宿主中强大的适应性免疫反应，以及抗原暴露后产生免疫成分的最佳缩短潜伏期。最后，采用数值模拟来证实这些主要发现，说明我们模型的实际应用，并提出减轻疾病传播的控制策略。