2007 Arizona Summer Program on Mathematical Modeling

The 2007 Arizona Summer Program has been an exciting 4-week research experience for undergraduates interested in mathematical modeling. Participants worked in teams on modeling projects inspired by articles recently published in the research literature. Each team was supervised by a graduate or faculty mentor. The program also included

• Lectures on LaTeX, Beamer, MATLAB, HTML, and PowerPoint;
• A workshop on how to write a resume and prepare for graduate school;
• Seminars by researchers using modeling in their work;
• Visits to the Applied Mathematics Laboratory;
• Visits to local scientific institutions, such as the Biosphere, the UA Mirror Laboratory, and Kitt Peak Observatory;
• Social actitivities and field trips to the Arizona Sonora Desert Museum and Kartchner Caverns.

Proposed Modeling Projects

All of the proposed projects were based on recent research articles, listed below. These papers formed the starting point of the research activities. Each group then decided in which direction(s) to take its project. Please follow the links below to explore the main ideas behind each project. Web pages prepared by the participants during the Summer Program are available from the group reports page.

West-Nile virus epidemics

The first two papers describe models of infection between birds and mosquitoes. The third article discusses experimental results showing that healthy mosquitoes may become infected by co-feeding with infected mosquitoes. The last link points to a review of mathematical models of infectious diseases.

• An epidemiological model for West Nile virus: invaton analysis and control applications by M.J. Wonham, T. de-Camino-Beck, M.A. Lewis


• Modeling the dynamics of West Nile virus, by G. Cruz-Pacheco, L. Esteva, J.A. Montano-Hirose, and C. Vargas


• Nonviremic transmission of West Nile virus, by S. Higgs, B.S. Schneider, D.L. Vanlandingham, K.A. Klingler, and E.A. Gould


• The mathematics of infectious diseases by H.W. Hethcote

Collective behaviors

The first two links point to a general discussion of schooling behaviors. The next four papers discuss models for collective behaviors, and how they are affected by the presence of noise. The last link describes Boids, which are simulated creatures that can collectively reproduce flocking behaviors.

• Complexity, Pattern, and Evolutionary Trade-Off in Animal Aggregation by J.K. Parrish & L. Edelstein-Keshet


• Self-Organized Fish Schools: An Examination of Emergent Properties by J.K. Parrish, S.V. Viscido, and D. Grünbaum


• Mutual interactions, potentials, and individual distance in a social aggregation by A. Mogilner, L. Edelstein-Keshet, L. Bent, and A. Spiros


• Effective leadership and decision-making in animal groups on the move by I.D. Couzin, J. Krause, N.R. Franks and S.A. Levin


• Self-Propelled Particles with Soft-Core Interactions: Patterns, Stability, and Collapse by M.R.D'Orsogna, Y.L. Chuang, A.L. Bertozzi, and L. S. Chayes


• Phase Transitions in Systems of Self-Propelled Agents and Related Network Models by M. Aldana, V. Dossetti, C. Huepe, V.M. Kenkre, and H. Larralde


• C. Reynolds Boids page.

Gene genealogies and DNA sequence evolution

The first link is to an online book that provides the mathematical background and a description of the coalescent model. The second link is to a review article that describes the use of the coalescent in the analysis of DNA. The third link points to a review article that describes how DNA is used to reconstruct human origins. The bibliography at the end of this article provides a starting point for more focused project ideas.

• Probabilistic backgound, described in a book by John Wakeley (Harvard University), to be published in September 2007 (the first 3 chapters are freely available online).


• Genealogical trees, coalescent theory, and the analysis of genetic polymorphism by N.A. Rosenberg and M. Nordborg


• Reconstructing human origins in the genomic era by D. Garrigan and M.F. Hammer

Formation and dynamics of sand dunes

The first article discusses a discrete model for the time evolution of the height of the sand bed in aeolian ripples. The second paper presents a mesoscopic model of sand dune formation. The next two articles are concerned with a continuous model for the formation of barchan dunes. The last link points to a paper describing longitudinal dunes on Mars.

• Computer simulation of aeolian sand ripples and dunes by T.-D. Miao, Q.-S. Mu and S.-Z. Wu


• Eolian dunes: Computer simulations and attractor interpretation, by B. T. Werner.


• Solitary behavior of sand dunes by V. Schwämmle and H.J. Hermann



• The morphology of dunes by H.J. Herrmann, G. Sauermann, and V. Schwämmle


• The Sand Seas of Titan: Cassini RADAR Observations of Longitudinal Dunes by R.D. Lorenz et al.