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.
