Effects of heterogeneity on population and epidemic dynamics

异源性是许多生物过程中的重要因素。它可以是空间的和/或时间的。空间异质性是指一个区域内每种植物的不同浓度的不均匀分布。空间异质性包括斑块结构和空间扩散两种类型。例如，时间异质性反映了流行病和病毒动力学中易感性和传染性的差异。此外，在疾病传播中，由于社会的演变，人与人之间的行为和接触具有明显的异质性。 对异质性进行建模无疑会产生一些新的动力系统，这是非常具有挑战性的。 本研究计划的目的是了解异质性对一些人口和流行病模型的动力学的影响。 随着对种群进化和疾病/病毒传播过程的机制有了更好的理解，我们将修改现有的模型，并提出新的模型来纳入异质性。这些模型可以用常微分方程、延迟微分方程、反应扩散方程以及常微分方程与一阶偏微分方程耦合的形式来描述。此外，大多数模型是高维或无穷维动力系统，因此通常是相当困难的研究。我们将利用现有的动力系统理论，并开发新的技术来解决这些问题。期望的结果包括系统的适定性，稳定状态的存在性和稳定性，渐近传播速度，行波，向后分支，Hopf分支，等等。 基于这些结果，我们将进一步分析异质性对这类模型动力学的影响。本研究不仅将促进应用动力学理论的发展，而且将为政策制定者制定适当和有效的疾病控制和资源管理策略提供指导。此外，它将加强对HQP的培训，并促进我们的合作。

Heterogeneity is an important factor in many biological processes. It can be spatial and/or temporal. Spatial heterogeneity refers to the uneven distribution of various concentrations of each species within an area. There are two kinds of spatial heterogeneity: patch structure and spatial diffusion. Temporal heterogeneity, for example, reflects the differentiation in the susceptibility and infectivity in epidemic and viral dynamics. Moreover, in the spread of diseases, behaviors of the human being and contacts among each other have obvious heterogeneity due to the evolution of society. Modeling heterogeneity will undoubtedly yield some new dynamical systems, which are very challenging to deal with. The purpose of this research program is to understand the effects of heterogeneity on the dynamics of some population and epidemic models. With better understanding of the mechanisms underlying processes on population evolution and disease/virus spread, we shall modify existing models and propose new models to incorporate heterogeneity. The models may be described by ordinary differential equations, delay differential equations, reaction-diffusion equations, and ordinary differential equations coupled with first-order partial differential equations. In addition, most of the models are high dimensional or infinitely dimensional dynamical systems and hence usually are quite difficult to study. We shall use existing theory of dynamical systems and develop new techniques to tackle these problems. Expected results include the well-posedness of the systems, existence and stability of steady states, asymptotical spread speeds, traveling waves, backward bifurcation, Hopf bifurcation, among others. Based on these results, we will further analyze the effects of heterogeneity on the dynamics of such models. This study will not only lead to advancements in the theory of applied dynamics but also provide guidelines for policy makers to draw appropriate and effective strategies in disease control and resource management. Moreover, it will enhance the training of HQP and further our collaborations.