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Twenty-first century science is driven, in large part, by challenges at interfaces, including those between the environmental and life sciences--public health, ecology, genomics, cell biology, epidemiology, immunology, neurobiology, physiology, evolutionary biology ... and the mathematical sciences. Progress at these interfaces has made it possible to address problems at levels of resolution and at temporal and spatial scales that were not possible just a decade ago. Together with computational advances and hardware improvements, these changes have accelerated scientific change to the point that no university model or structure is flexible enough to accommodate the rapid pace. Yet programs that augment the nation's capacity to train the next generations of scientists, scientific leaders, and teachers are essential. Programs are needed in theoretical, mathematical, and computational biology--programs that attract the best young minds--if we are to keep our nation at the forefront of research and innovation in the biological sciences. One key to meeting these needs is flexibility.

There was a time when the basic scientific needs of mathematical biologists could be met--minimally--by typical graduate school curricula and traditional conferences. But today's active mathematical biologists must participate in--in addition to their traditional conferences--annual meetings of major biological professional societies, specialty conferences, as well as in focus groups (see Next Wave's article by Fred Roberts, and specialty workshops. Furthermore, a critical part of the training of today's interdisciplinary scientists takes place at summer schools, workshops, and more focused conferences. These nontraditional resources must not be overlooked--indeed, they must be expanded--if we are to meet the nation's hunger for interdisciplinary scientists and to enhance the diversity of the scientific workforce.

Flexibility needed

Despite the success of training and mentorship models like University of Tennessee, Knoxville's Mathematical Ecology Program, in an age of interdisciplinary science no single university model can meet national and international scientific and educational needs. Diverse, well supported, and above all flexible training and research programs must be one of our nation's scientific priorities, as flexibility often leads to innovation. One such flexible, innovative program exists at Arizona State University (ASU).

At ASU, work in mathematical, theoretical, and computational biology is carried out in a variety of settings, including the department of mathematics and statistics, my current home. Our interdisciplinary applied mathematics program offers a Master in Natural Science and a Ph.D. in differential equations and mathematical biology.

One key to our program is that it offers the flexibility required to meet the needs of students with a wide range of interests. Students are encouraged to fulfill half of the Ph.D. course requirements by taking approved courses in the biological sciences. Recent graduates of our program include Purdue University associate professor Zilan Feng, University of Louisville associate professor Bingtuan Li, and University of Nebraska, Lincoln, assistant professor Irakli Loladze.

ASU also offers a Professional Science Master's degree within its Computational Biosciences degree program. With its emphasis on quantitative approaches to biological problems, this program aims to produce students capable of meeting the demands of today's biotechnical and biomedical industries.

Our Ph.D. program in differential equations and mathematical biology offers a wide range of opportunities. The strengths of this program lie in a group of mathematical biologists with overlapping interests and strong research groups in areas that provide excellent training opportunities for our students. These scientists and research groups have complementary expertise in numerical analysis, computational science, dynamical systems, statistics, and stochastic processes. In addition, our School of Life Sciences and Cancer Institute, among other components, provide plenty of opportunities for the training of undergraduates, graduate students, and postdocs interested in theoretical, mathematical, and computational biology. Below, I describe some of the research opportunities that are available at ASU to qualified graduate student and postdoctoral candidates.

Faculty achievements

Recent work by Hal Smith and collaborators involves the construction and analysis of mathematical models of biofilms in fluid environments. Analysis of these models has yielded answers to the question of why it is so difficult to eradicate biofilms using biocides. More recently, their research has turned to modeling the process of gene transfer between bacteria in biofilms. Gene transfer typically occurs when a host bacteria passes a plasmid, a small circular DNA not part of the host genome, to another, not necessarily related bacteria, following conjugation. Smith with graduate student M. Imran and collaborator D. Jones have constructed a simple model of gene transfer in an immersed biofilm. Genetic transfer in biofilms--with its relevance to the transfer of antibiotic resistance--is an example of important phenomena that have received little attention from mathematical modelers.

Horst Thieme studies rapidly reproducing parasitic populations, which, as he notes, "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, he notes, are essential in our efforts to gain a deeper understanding of these phenomena and as a theoretical laboratory to devise control and management strategies. Such models are also needed to elucidate the role of parasites in biocomplexity, as mediators of competition and coexistence. Thieme works at the interface of differential equations, integral equations, and dynamical systems (on the one hand), and ecology, population biology, and epidemiology (on the other).

Yang Kuang's research focuses, in part, on the identifications and characterization of mechanisms of species coexistence or extinction. Individual heterogeneity is critical to the evolutionary process, and yet most population models tend to exclude or limit the level of heterogeneity. Stoichiometry-based population models carefully imbed the natural chemical heterogeneity that is innate to all life forms. Kuang, in collaboration with Jim Elser and others, have used these models to address specific biocomplexity and biodiversity questions. Kuang is also involved in efforts to model various aspects of tumor growth and management as part of ASU's efforts on cancer research. Kuang has also maintained an active mathematics research program on dynamical systems--particularly in the area of functional differential equations.

One of Rosemary Renaut's projects focuses on the development of reliable and sensitive neuroimaging analysis tools for use with positron emission tomography and magnetic resonance imaging. Her work is used in the assessment and detection of functional and anatomical change in the human brain through the course of Alzheimer's disease. The tools developed are being enhanced and extended with the goal of developing a flexible software package that provides an automated approach for neuroimaging studies by Alzheimer's dementia researchers at the Good Samaritan Medical Center in Phoenix.

Steve Baer's research is in computational neuroscience. Current projects include modeling the biophysical mechanisms underlying synaptic plasticity and learning in dendritic trees, the dynamics of neuronal networks in the outer plexiform layer of the retina, and modeling the integration of multiple synaptic inputs in muscle fiber.

My research program lies at the interface of the natural and social sciences, with its emphasis on the role of dynamic social landscapes on pathogens' evolution. In collaboration with many researchers (graduate students, postdocs, and faculty elsewhere), we have examined the role of cross-immunity on the evolution and dynamics of influenza; the impact of behavioral changes, long periods of infectiousness, variable infectivity, co-infections, prostitution, social networks, and vaccine efficacy on HIV dynamics; the role of exogenous re-infection, variable progression rates, vaccination, public transportation, close and casual contacts on tuberculosis dynamics and control; the impact of life-history vector dynamics on dengue epidemics; and on the identification of time-response scales for epidemics of foot and mouth disease.

More recently, I have worked on the role of dispersal and disease as mechanisms that help support and maintain ecological diversity. Most recently, we have started work on problems at the interface of homeland security and disease invasions (natural or deliberate) and on models for the spread of social "diseases" like alcoholism and ecstasy. We have also worked on models for the spread of extreme ideologies and their impact on cultural norms. The work on homeland security is briefly described in my February column, "Beyond Numbers and Proofs." The recently organized ASU School of Life Sciences (SOLS) has as part of its mission the instigation of collaborative research and training in areas that need strong links between diverse groups, including experimental, computational, theoretical, and mathematical biologists. The school brings together a range of scholars from different disciplines ranging from law, philosophy, ecology, and biogeochemistry to biomedicine, bioinformatics, and genomics.

The school offers several opportunities for interdisciplinary research involving the mathematical, statistical, and life sciences--some provided by faculty with joint affiliation with the mathematics department. The work of J. Marty Anderies at the interface of biology, economics, and mathematics is but one example of the type of collaborations that already exist between SOLS and the mathematics department.

Carlos Castillo-Chavez is a Joaquin Bustoz Jr. Professor of Mathematical Biology in the Department of Mathematics and Statistics at Arizona State University. He can be reached at chavez@math.asu.edu.