In Spring 2021, the Mathematics Department will be offering a topics course (MATH 490) on
Introduction to Mathematical Modeling. This describes this course and what students can expect from it.
The course ‘Introduction to Mathematical Modeling’ is about the modeling process, how mathematicians translate phenomena in the world around us into a mathematical description, and how they use that description to more deeply understand the phenomena. This process is central to applied mathematics, and many career paths in the mathematical sciences rely on skill in modeling.
The modeling process can be broken down into the following steps:
- identify relevant assumptions about how a phenomena works,
- translate those assumptions and mechanisms into a mathematical description of the phenomena,
- validate the model output, for example using empirical data about the phenomena, and
- use the results of the validation to improve the model by modifying assumptions.
An often overlooked fifth step in the process is communicating your model to others. We will study this five-step modeling process through practice. Because modeling is highly effective when done in teams (where different individuals have different strengths and skills), we will practice modeling both individually and in teams.
Mathematical models can be categorized into types based on the type of mathematics they use (e.g., continuous, discrete, probabilistic, and hybrid). We will survey all these basic types. The mathematics we will use will include basic probability, single variable calculus, and linear algebra. We will also study models that use systems of ordinary differential equations, partial differential equations, and multivariable calculus; because we will often be working in teams, not everyone needs to be experienced with all mathematical topics.
Depending on the mathematics used, a model can be analyzed analytically (using purely theoretical methods) or numerically (using computerized computational methods). We will survey both approaches. This means that familiarity with computer programming (even if at the complete novice level) will be useful. Favored computing environments will include R, Sage, and python, though instruction in these will not be part of the course. Note that becuase we will often be working in teams, not everyone needs to be experienced with computer programming.
Topics and Projects
Topics that we will explore as we learn about the modeling process will likely come from the following list:
- projectile paths (falling objects)
- growth of a single population
- dynamics of two coupled populations (e.g., predator-prey populations)
- dynamics of a structured population
- the spread of infectious disease
- drug metabolism
- optimization of resources
- optimizing a search
Depending on the interests of those in the class, other topics may be explored.
Learning will be hands-on. While the professor will present some foundational material from time to time, most work in class will be done in groups and focus on specific modeling scenarios. In groups, we will build models in response to those scenarios, and we will share the results with others in the class as we deepend our understanding of modeling. Each scenario will culminate in a group written report.
Students will be expected to do work in group outside class meetings.
Student who take this course will be prepared to compete in the international Mathematical Contest in Modeling which happens early each Spring. (We will not be prepared for this year’s competition.) In this competition, each participating school selects up to two teams of 2-3 students to compete. The teams each select one of three competition problems and spends four days developing a model and writing up a contest solution. Solutions are submitted to the contest, and judges evaluate submissions. Submissions are acknowledged as ‘Successful Participant’, ‘Honorable Mention’, ‘Meritorious’, ‘Finalist’, or ‘Outstanding Winner’. Any of these success levels merit inclusion on a person’s resume!
The catalog prerequisite for this course is ‘upper-division standing’, which means you must have completed 60+ units of college credit to be eligible to register for the course. If you think you are ready the course but don’t meet this prerequisite, make an appointment to talk with Dr. Miller and make your case.
This course replaces Operations Research (MATH 429) in the Spring 2022 course schedule for mathematics. That course had a prerequisite of MATH 352 (Probability and Statistics). This course will not lean heavily on probability or statistics, but the knowledge in this area will be useful.
This course also substitutes for the Advanced Research Investigations requirement for the mathematics major’s emphasis in applied mathematics and the emphasis in statistics.