Grant: Discovery Grants Program - Individual

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.