Dynamics of Delayed Systems with Applications to Ecoepidemiology

生态流行病学是研究疾病传播的相互作用的物种。理论研究和实地观察已经确定了各种传染病在大多数生态系统人口中的流行。这就需要研究流行病参数在生态领域的影响。另一方面，延迟在许多过程中非常常见，例如人口动态学和流行病学。不幸的是，对时滞生态传染病模型的研究并不多。这激发了拟议的研究。 在本项目中，将提出一些新的时滞生态传染病模型。以已有的和新建立的时滞生态传染病模型为原型，我们将研究一些抽象时滞方程的局部和全局动力学。在这些方程中，延迟可以是离散的、分布的或状态相关的。我们关心的平衡点的稳定性，霍普夫分支，向后分支，传播速度和行波。将采用线性化、李雅普诺夫泛函方法、规范形理论、单调算子理论等技术。这将对时滞微分方程理论的发展作出重大贡献。此外，理论结果将应用于具体问题。我们希望所获得的结果将有助于指导公共卫生决策和控制传染病。

Ecoepidemiology is the study of interacting species in which diseases spread. Theoretical research and field observations have established the prevalence of various infectious diseases amongst the majority of ecosystem populations. This has necessitated the study of the impact of epidemiological parameters in the ecological domain. On the other hand, delay is very common in many processes such as population dynamics and epidemiology. Unfortunately, not much has been done for delayed ecoepidemic models. This motivates the proposed research. In this project, some new delayed ecoepidemic models will be proposed. With existing and these newly built delayed ecoepidemic models as prototypes, we will study the local and global dynamics of some abstract delayed equations. In these equations, delay can be discrete, distributed, or state-dependent. We concern with the stability of equilibria, Hopf bifurcation, backward bifurcation, spreading speed, and travelling waves. Techniques like linearization, Lyapunov functional method, normal form theory, monotone operator theory will be employed. This will contribute greatly to the theory of delayed differential equations. Moreover, the theoretic results will be applied to concrete problems. We hope that the obtained results will be useful to guide public health policy decisions and to control infectious diseases.