Spatial Heterogeneity, Nonlocal Interactions and Time Delay in Biological and Epidemiological Spread

本项目研究了空间异质性、非局部相互作用和时间延迟对某些生物和流行病学模型时空动态的影响。研究了一类非局部时滞反应扩散方程行波解的存在性、唯一性和稳定性。所得结果可用于研究生物学、生态学、流行病学、医学等领域的行波解。该项目的第二部分旨在开发一个广义的罗斯-麦克唐纳病媒宿主模型，该模型由具有时空延迟的扩散积分微分方程建模，该方程描述了非局部相互作用和潜伏期。研究模型的时空动力学，包括流行病波的存在。空间异质性和时间延迟对疾病建立和传播的影响也将被考虑。该模型可用于研究移民对当地寄生虫流行和灭绝的影响。延迟Ross-Macdonald疟疾模型的结果有助于理解和控制肯尼亚沿海等地区的疟疾传播。项目第三部分为医院耐药菌感染流行模型研究，重点关注感染年龄、宿主异质性和多重耐药菌株。通过研究这些模型的渐近行为，试图找出导致观察到的感染模式的因素，并为设计有效的控制措施提供理论指导。在现代社会中，全球变化导致了更多的人类活动和环境变化，这反过来又影响了人类健康。例如，全球变暖影响了疟疾的传播。在整个有记载的历史中，在新地区到达、建立和传播的非本地媒介引发了人类疾病的流行，如疟疾、黄热病、斑疹伤寒、鼠疫和西尼罗河病毒。新引入的物种或疾病的空间传播一直是理论家和经验主义者感兴趣的主题。本项目拟对某些媒介宿主疾病如疟疾的空间传播进行研究，并希望该研究有助于了解其他媒介传播疾病如西尼罗病毒和禽流感的建立和传播。在世界各地，由耐甲氧西林金黄色葡萄球菌（MRSA）和耐万古霉素肠球菌（VRE）等抗微生物细菌（ARB）在医院引起的流行病正在增加。大约5-10%的住院患者会发生与住院直接相关的感染。在美国，这些感染每年导致9万多人死亡。据估计，70%的致病菌对一种或多种抗菌素具有耐药性。该项目还研究具有抗生素耐药性的医院流行病，这将有助于提供一些控制措施，在医院内防治这种流行病。

This project is a study of the effect of spatial heterogeneity, nonlocal interactions and time delay on the spatiotemporal dynamics of some biological and epidemiological models. The first part of the project focuses on the existence, uniqueness and stability of traveling wave solutions of reaction-diffusion equations with nonlocal delay. The results can be applied to study traveling wave solutions in various models arising in biology, ecology, epidemiology, medicine, etc. The second part of the project seeks to develop a generalized Ross-Macdonald vector-host disease model, modeled by diffusive integro-differential equations with spatio-temporal delay, which describe nonlocal interactions and latent periods. An investigation of the spatio-temporal dynamics of the model, including the existence of epidemic waves, is proposed. The effects of spatial heterogeneity and time delay on the establishment and transmission of diseases will also be considered. The model can be applied to study how immigration affects local epidemic and extinction of the parasite. The results on the delayed Ross-Macdonald malaria model are useful in understanding and controlling the malaria transmission in some regions such as in costal Kenya. The third part of the project is a study of epidemic models of antibiotic-resistant bacteria infection in hospitals focusing on infection age, host heterogeneity and multi-drug resistant bacterial strains. By studying the asymptotic behavior of these models an attempt is made to identify factors responsible for observed patterns of infection, and to provide theoretical guidance for designing efficient control measures.In modern society, global change has resulted in more human activities and environmental changes, which in turn have affected human health. For example, global warming has affected the transmission of malaria. Throughout recorded history, non-indigenous vectors that arrive, establish, and spread in new areas have fomented epidemics of human diseases such as malaria, yellow fever, typhus, plague, and the West Nile virus. The spatial spread of newly introduced species or diseases is a subject of continuing interest to both theoreticians and empiricists. This project proposes to study the spatial transmission of certain vector-host diseases such as malaria and hopes that the study can help understand the establishment and spread of other vector-borne diseases such as West Nile virus and bird flu. Epidemics caused by antimicrobial-resistant bacteria (ARB), such as methicillin-resistant Staphylococcus aureus (MRSA) and vancomycin resistant enterococci (VRE), in hospitals are increasing throughout the world. Approximately 5-10% of patients admitted to a hospital will develop an infection directly related to their hospitalization. These infections contribute to over 90,000 deaths per year in the US. It is estimated that 70% of the causative pathogens are resistant to one or more antimicrobials. The project also studies nosocomial epidemics with antibiotic resistance which will help to provide some control measures to combat such epidemics inside hospitals.