MATHEMATICAL BIOLOGY

Department of Applied Mathematics

Overview

Mathematical biology is a large and well-established branch of applied mathematics. The size of the field reflects both the importance of the biological and biomedical sciences and an appreciation for the mathematical subtleties and challenges that arise in modelling complex biological systems. Our interest, as a group, lies in understanding the spatial and temporal patterns that arise in dynamic biological systems and in understanding how these patterns affect biological function. Our mathematical activities range from nonlinear and chaotic dynamics, to reaction-diffusion equations, to optimization. We employ a variety of tools and models to study problems that arise in biomechanics, cell biology, development, ecology, epidemiology, neuroscience, and resource management. We maintain collaborations with a large number and variety of biologists and with biological and biomedical departments both here and elsewhere.

Courses

We teach a number of courses in mathematical biology including:

Undergraduate/Graduate:

• Amath 422/522: Computational Modeling of Biological Systems
• Amath 423/523: Mathematical Analysis in Biology and Medicine

Graduate:

• Amath 504: Mathematical Epidemiology
• Amath 531: Mathematical Theory of Cellular Dynamics
• Amath 532: Mathematics of Genome Analysis and Molecular Modeling
• Amath 533: Neural Control of Movement
• Amath 534: Dynamics of Neurons and Networks
• Amath 535: Mathematical Ecology
• Amath 536: Spatial Models in Ecology and Epidemiology

AMATH 422/522

Computational Modeling of Biological Systems

Fundamental models that arise in biology and their analysis through modern scientific computing. Discrete and continuous-time dynamics, in deterministic and stochastic settings, with applications from molecular biology to neuroscience to population dynamics. Statistical analysis of experimental data. MATLAB or R programming taught from scratch.

Prerequisite: working knowledge of calculus.

AMATH 423/523

Mathematical Analysis in Biology and Medicine

This course focuses on developing and analyzing mechanistic, dynamic models of biological systems and processes, to better understand their behavior and function. Applications are drawn from many branches of biology and medicine. Students will gain experience in applying differential equations, difference equations, and dynamical systems theory to biological problems.

Prerequisite: Background equivalent to AMATH 351, AMATH 422, or Math 307

AMATH 504

Mathematical Epidemiology

Focuses on the construction and analysis of mathematical models for infectious disease transmission and control. Emphasizes evaluation and comparison of vaccination programs. Applications are presented for a variety of diseases such as measles, rubella, smallpox, rabies, etc.

Prerequisite: AMATH 351 or equivalent.

AMATH 531

Mathematical Theory of Cellular Dynamics

Biological cells are biochemical systems that obey the laws of physics. This course develops a coherent mathematical theory for processes inside living cells. It focuses on analyzing dynamics leading to functions of cellular components (gene regulation, signaling biochemistry, metabolic networks, cytoskeletal biomechanics, epigenetic inheritance) using deterministic and stochastic models.

Prerequisites: 402 and 403 and working knowledge of probability

AMATH 532

Mathematics of Genome Analysis and Molecular Modeling

Genome analysis, i.e., bioinformatics, and molecular modeling in terms of molecular dynamics (MD) and Brownian dynamics are now fast growing areas of applied mathematics in molecular biology. This course introduces the fundamentals of these approaches in terms of discrete probability, classical mechanics, theory of diffusion, and Monte Carlo simulations.

Prerequisites: 506 or 572, or with permission from the instructor

AMATH 533

Neural Control of Movement

This class provides a comprehensive view of how the brain controls movement. It brings together elements of biomechanics and muscle physiology, neuroanatomy and neurophysiology of the motor system, sensorimotor psychophysics and kinesiology, and movement disorders. Empirical data are interpreted in the context of control-theoretic models whenever possible.

No prerequisites

AMATH 534

Dynamics of Neurons and Networks

Mathematical analysis and computational modeling on three interconnected scales \(em neurons, networks, and populations \(em including (1) oscillations and synchrony, (2) role of network structure and symmetry, (3) statistical mechanics tools for large-scale models, (4) bifurcation and reduction methods for biophysical models. Emphasizes links between system dynamics and signal processing.

Prerequisite: CSE/NBB 528, or knowledge of differential equations and probability at similar level.

AMATH 535

Mathematical Ecology

This course considers models, methods, and issues in population ecology. Topics include the effects of density dependence, delays, demographic stochasticity, and age structure on population growth; population interactions (predation, competition, and mutualism); and applications of optimal control theory to the management of renewable resources.

Prerequisite: AMATH 402 or AMATH 423 or permission of the instructor

AMATH 536

Spatial Models in Ecology and Epidemiology

This course considers models for the growth and dispersal of biological populations. Topics include population persistence, climate-induced range shifts, and rates of spread of invading organisms. We will consider reaction-diffusion equations, integrodifference equations, branching random walks, and other relevant classes of models.

Prerequisite: AMATH 403 or permission of the instructor

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Schedule

Schedule of Mathematical Biology Courses:

Academic Year:

• 2010-2011
• 2011-2012

2010-2011

Fall	Winter	Spring
	Amath 422/522	Amath 423/523
Amath 531	Amath 533	Amath 535

2011-2012

Fall	Winter	Spring
	Amath 422/522	Amath 423/523
Amath 532	Amath 534	Amath 536

Journal Club

The Mathematical Biology Journal Club (MBJC) consists a group of students and faculty members in mathematics, biological and physical sciences, medicine, as well as engineering who are interested in the interface of biology and medical science and mathematics. MBJC meets once a week in an informal atmosphere and encourages participants to explore and develop various topics of common interest.

Faculty

Andrea Barreiro

• Andrea Barreiro's Web Page

Mark Kot

• Mark Kot's Web Page

Hong Qian

• Hong Qian's Web Page

Eric Shea-Brown

• Eric Shea-Brown's Web Page

Emanuel Todorov

• Emanuel Todorov's Web Page

Adjunct Faculty