Dynamical Systems in Host-Parasite and Structured Population Models

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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.