Dynamical Systems in Host-Parasite and Structured Population Models

新出现和再出现的疾病引起了人们对传染病和宿主-寄生虫系统及其数学建模的新兴趣。不仅人类及其食物来源（家畜和农用植物），而且自然动植物种群也受到影响（两栖动物数量减少），这一事实引起了人们对寄生虫在生态系统中重要而迷人的作用的关注。由于它们的快速更替，寄生种群是研究进化原理的理想对象；反过来，了解这些原则对有效控制传染病也很重要。数学模型及其分析对于更深入的理解和作为设计控制和管理策略的理论实验室是非常需要的。虽然寄生物与宿主相互作用的健康方面是一个直接的关切，但它们与人口结构相互作用对生物多样性的影响可能同样重要。真菌和病毒病原体在两栖动物数量下降甚至消失中的作用只是寄生虫在物种多样性中可能发挥负面作用的一个例子，另一个例子是寄生虫介导的竞争排斥。下列数学建模和模型分析项目是该项目的重点：(i)水平和垂直传播疾病模型的稳定性和周期性振荡；阶段结构寄主种群中的寄生虫；（iii）猎物-捕食者-寄生虫系统的介导共存和灭绝；（iv）宿主内朊病毒增殖的大小结构模型。研究寄主-寄生虫的数学模型将有助于理解寄生虫和种群结构在维持生物多样性中的作用以及它们之间的相互作用。它将增强我们对可能导致物种减少和消失的众多因素之间相互作用的洞察力。这些模型将有助于设计环境管理策略，促进或至少与物种保护相兼容。它们可以作为研究生和本科生教学的教育工具，并与科学界、公共卫生官员和受过教育的公众进行有效沟通。

Emerging and re-emerging diseases have led to a renewed interest in infectious diseases and host-parasite systems, and in their mathematical modeling. The fact that not only humans and their food sources (domestic animals and agronomic plants), but also natural animal and plant populations are afflicted (amphibian decline), has directed attention to the important and fascinating role of parasites in ecosystems. With their rapid turn-over, parasitic populations are ideal objects to study the principles of evolution; in turn, it is important to understand these principles to control infectious diseases effectively. Mathematical models and their analysis are very much needed for a deeper understanding and as a theoretical laboratory to devise control and management strategies.While the health aspects of parasite-host interactions are of immediate concern, their impact on biological diversity, in interaction with population structures, may be as important. The role of fungal and viral pathogens in the decline or even disappearance of amphibians is only one example of concern for the negative role parasites can play in species diversity, another would be parasite-mediated competitive exclusion. The following projects in mathematical modeling and model analysis are the focus of the project: (i) Stability and periodic oscillations in disease models with horizontal and vertical transmission; (ii) Parasites in stage-structured host populations; (iii) Mediated coexistence and extinction in prey-predator-parasite systems; (iv) Size-structured models for in-host prion proliferation.Studying these mathematical host-parasite models will greatly contribute to the understanding of the role of parasites and population structure as well as their interaction in the maintenance of biological diversity. It will enhance our insight in the interplay of the possible multitude of factors which cause species decline and disappearance. These models will help to design environmental management strategies that foster or at least are compatible with species preservation. They can be used as educational tools in graduate and undergraduate instruction and for effective communication with the scientific community, public health officers, and an educated public audience.