Effects of heterogeneity on population and epidemic dynamics

异质性是许多生物过程中的一个重要因素。它可以是空间的和/或时间的。空间异质性是指每个物种在一个区域内不同密度的不均匀分布。空间异质性主要有两种：斑块结构和空间扩散。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，在疾病的传播中，由于社会的演变，人类的行为和相互之间的接触具有明显的异质性。建模异构性无疑会产生一些新的动力系统，这是非常具有挑战性的。 这项研究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/病毒传播过程的潜在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组、时滞微分方程组、反应扩散方程以及常微分方程组和一阶偏微分方程组来描述。此外，大多数模型是高维或无限维动力系统，因此通常很难研究。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。预期结果包括系统的适定性、稳态的存在性和稳定性、渐近传播速度、行波、后向分叉、Hopf分叉等。 在这些结果的基础上，我们将进一步分析异质性对这类模型动力学的影响。这项研究不仅将促进应用动力学理论的进步，而且将为决策者在疾病控制和资源管理方面制定适当和有效的战略提供指导。此外，它还将加强HQP的培训，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.