传染病的爆发对人类社会和经济造成很大的影响。现有的大部分数学方法和工具仅适用于简单的传染病模型，很多关键因素如时滞、空间异质、周期和多传播途径被忽略或简化，所得的理论结果与现实应用可能产生偏差。为此，本项目将利用时滞微分方程、时滞偏微分方程和时滞周期系统来建立更符合实际应用的传染病动力学模型，研究传染病在空间上的传播机制、在时间上的演变规律以及防控策略的优化，揭示时滞、多传播途径、控制函数等关键因子对传染病传播速度、爆发强度和多爆发高峰的影响，以减小基本再生数和空间传播速度为目标确定用药的最佳组合，通过构造新的Lyapunov泛函和上下解、发明新的谱方法和数值计算方法等来证明平衡态的全局稳定性和行波解的存在性、计算疾病的基本再生数、求解最优的疾病防控措施。本项目将拓展基于时滞和多传播途径的传染病模型动力学的基本理论，丰富传染病模型的研究方法，为传染病的防控和治疗提出有效合理的方案。

The outbreaks of infectious diseases have made serious impacts on human society and economics. Most of the theoretical tools and mathematical methods are only applicable to simplified infectious disease models, where many key factors such as time delay, spatial heterogeneity, period and multiple transmission mechanisms are ignored or simplified. The corresponding theoretical results may diverge from real applications. In this project, we will use delay differential equations, delay partial differential equations, and delay periodic systems to construct dynamical models for infectious disease models which are more relevant with real applications, investigate the spatial propagation, time evolution and optimal control of infectious diseases, reveal the impacts of critical factors such as delay, multiple transmission mechanisms and control functions on the propagation speed, outbreak strength and multiple peaks, find the optimal drug therapy which minimizes basic reproduction number and spatial propagation speed, construct new Lyapunov functional and upper and lower solutions, develop new spectral method and numerical algorithm, prove global stability of steady states and existence of traveling wave solutions, calculate basic reproduction number of the disease, and find the optimal control strategy. The research outcomes will help to extend the existing theory in dynamics of infectious disease models with multiple transmission mechanisms and time delays, enrich the analytic methods of infectious disease models, and design effective and realistic control strategies of infectious diseases.