The diversity of transmission modes and the heterogeneity of populations in the transmission of infectious diseases are issues that have to be faced in the current disease protection. In this paper, an infectious disease model incorporating age structure and horizontal and environmental spread, along with asymptomatic infection, is proposed to describe diversification of disease transmission routes and population heterogeneity. The expression of the basic reproduction number is derived by a linear approximation method, and it is concluded that if , the disease‐free steady state is globally asymptotically stable; if , the model has a unique endemic steady state which is locally asymptotically stable under certain conditions. Further, the uniform persistence of disease is proved using the persistence theory of the infinite‐dimensional dynamical system when . In additional, the issue of optimal control problem according to this model is investigated, and the representation of optimal control with respect to state variables and adjoint variables is obtained which also imply the existence and uniqueness of optimal control. Finally, numerical simulations are carried out to interpret the theoretical results and to discuss the impact of age and control measures in disease transmission.

中文翻译：

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无症状感染、多种传播途径的年龄结构流行病模型的动力学分析与最优控制

传染病传播方式的多样性和人群的异质性是当前疾病防护不得不面对的问题。本文提出了一种结合年龄结构、水平和环境传播以及无症状感染的传染病模型来描述疾病传播途径的多样化和人群异质性。采用线性逼近法推导出基本再生数的表达式，得出若 ，则无病稳态全局渐近稳定；如果 ，模型具有独特的地方性稳态，在某些条件下局部渐近稳定。此外，当 时，使用无限维动力系统的持续性理论证明了疾病的一致持续性。此外，根据该模型研究了最优控制问题，得到了最优控制关于状态变量和伴随变量的表示，这也暗示了最优控制的存在唯一性。最后，进行数值模拟来解释理论结果并讨论年龄和控制措施对疾病传播的影响。