Mathematics: Mathematical Biology&Ecol

Mathematical modeling in the life sciences. Topics include population modeling, dynamics of infectious diseases, reaction kinetics, wave phenomena in biology, and biological pattern formation. Prerequisites: Graduate student or Instructor permission; knowledge of linear algebra and differential equations required.

Prereqs: MATH 2522 or MATH 2544; MATH 3230 or MATH 3201; or Instructor permission; Open to Degree and PACE students; Colocated with BME 5990 B; Total combined enrollment: 35

Math 5788A/BME 5990B: Mathematical Biology & Ecology Course Outline and Learning Objectives This course will be an introduction to the interplay of mathematics with several disciplines, namely biology, ecology, pharmacology, and some physiology, and it will be taught within the context of continuous and mean-field models for complex systems. The applications we shall discuss will range from population biology to chemical kinetics and reaction-diffusion systems. No previous knowledge of these areas will be assumed. The biological background to each problem will be described in sufficient detail to construct and analyze models. The lectures will show how models are built up and will provide the mathematical tools indispensable for studying their dynamics. With each topic discussed the scenario will consist of (i) a description of the biological problem; (ii) development of the mathematical model and an assessment of its realism; (iii) mathematical analysis of the model and clues to numerical computations; (iv) biological interpretation of the results from a modeling viewpoint; (v) Seminal papers and Project work. This course is aimed to be accessible to both senior undergraduates and graduate students who have taken introductory courses in linear algebra, ordinary and partial differential equations, and some programming skills looking to broaden their knowledge in biological and ecological application areas. The course extends the range of usage of mathematical models in biology, ecology and physiology. At the end of the course, students should be able to  formulate and solve mathematical models in biology, ecology and biochemistry.  use techniques from ordinary and partial differential equations to describe spread of diseases and understand biological pattern formation mechanisms.  explain how these techniques are applied in biology, ecology, biochemistry, and other scientific studies. The main emphasis of this course will be introductory lectures, problem solving, analysis of seminal papers, and computer (dry lab) assignments. It will focus on techniques of mathematical modeling in biology and ecology in the context of mean-field type modeling of complex systems (not agent-based modeling of complex systems). More specifically, a selection of modeling problems from the following areas will be studied: I. Continuous and discrete models for single and interacting populations II. Reaction kinetic modeling; Pharmacokinetic-Pharmacodynamic (PKPD) modeling III. Diffusion in biology; Biological waves IV. Dynamics of infectious diseases; Waves of disease spread and control V. Seminal Papers & Project Work We shall cover book chapters from Murray and allude to other topics, the detailed topics of which will include: Population Modeling: Continuous and discrete models for single species and interacting populations. Nondimensionalization; Analysis of an insect outbreak model. Cobwebbing, stability, periodic solutions, bifurcations, and chaos. Phase plane methods for two coupled ODEs and an introduction to the use of nullclines. Applications to symbiotic systems, competing systems (illustrating competitive exclusion), and predator prey systems (illustrating Hopf bifurcation). Delay differential equations: Introduction to DDE models in ecology and physiology. Derivation of critical delay for stability in a single DDE. Construction of periodic solutions for piecewise constant negative feedback. Regulation of hematopoiesis. Reaction Kinetics: Basic enzyme-substrate reaction modeling. Transient time estimates and Nondimensionalization. Michaelis-Menten quasi-steady state analysis (illustrating singular perturbation techniques.) Applications to suicide substrates, cooperative phenomena, and autocatalysis; Analysis of pharmacokinetic and pharmacodynamic (PKPD) models (illustrating fractional differential equation modeling in pharmacology.) Epidemiology: Simple models of infectious disease. Threshold density for an epidemic. Conditions for a disease to be endemic. Assess the relationship between disease persistence and the basic reproduction rate of a disease. Age-dependent epidemic models and threshold analysis. Diffusion in Biology: Introduction to reaction-diffusion models. Chemotaxis. Wave fronts for scalar equation with cubic kinetics; existence of unique wave speed, and derivation of wave form. Biological pattern formation models. Turing instability and bifurcations. Mechanochemical models for biological pattern formation. Learning Outcomes: Students completing this course will be equipped with the skills necessary to enter the fast-growing field of mathematical biology or pursue graduate or advanced work in biological and other interdisciplinary fields in which modeling skills are expected. Indeed, they will: I. understand a range of biological and ecological questions for which mathematical approaches have been utilized. II. translate biological problems into mathematical models using appropriate mathematical and computational tools, such as Mathematica and Matlab. III. validate mathematical models using appropriate biological experimentation and sensitivity analysis. IV. practice communication skills in interdisciplinary studies. Assessment: Assignments will be given during the course. It will be combined with two midterm examinations, oral presentations, and final project work. Textbooks: Mathematical Biology, Volumes I (3rd Edition): Murray JD. Mathematical Models in Biology: Edelstein-Keshet, L