MATH 315: Mathematical Modeling

Course Description

Fall Term 2024

Course Title: Mathematical Modeling

Course Description: An introduction into the process of developing and interpreting mathematical models within the framework of numerous applications. We will utilize discrete, continuous, and probabilistic approaches to explore applications in a wide range of fields with an emphasis on mathematical models of proven usefulness in problems arising in the  life and social sciences.  We will explore specific models in  population dynamics, epidemiology,  ecology, political science, ecology, sociology, anthropology, psychology, and economics. We will use MATLAB to implement and analyze several of these models. If time permits, we will examine simulation models using NETLOGO.

Mathematical Modeling is a core course in the Applied Mathematics track of the Mathematics major; it is a prerequisite for the Senior Seminar MATH 715 to be offered in Spring 2026.

Prerequisites: MATH 200 (Linear Algebra) and MATH  226 (Differential  Equations) or by instructor's approval.  

CW  Option: Students interested in using the  course to satisfy the college's writing requirement should sign up for the CW  section (MATH  315B).

Instructor: Michael Olinick, 202 Warner, Phone: 443-5559. Home telephone: 388-4290; email: molinick@middlebury.edu. Usual Office Hours: Monday and Wednesday: 9:30 –11 AM and 12:10 - 1 PM; Thursday: 10 AM – Noon, and Friday: 9:30 – 11 AM. I am happy to make an appointment to see you at other mutually convenient times..

Meeting Times: MWF 11:15 AM – 12:05 PM, Warner 11.

Textbooks: The basic text will be my book Mathematical Modeling in the Social and Life Sciences (Wiley, 2014). There will be a copy of this book, as well as the Brannan and Boyce Differential Equations text, on reserve in the Davis Family Library. In addition, there will be many additional notes distributed as well as possibly some readings on reserve in the library.

Additional Course Materials: See the course webpage at http://f24.middlebury.edu/math0315a and/or the course folder on the file server Classes/Fall2024/math0315a.

Requirements: There will be one evening mid-term examination and a final examination in addition to required daily homework assignments and an independent project. Ideally, the project would be the creation, analysis and testing of a mathematical model of a real world problem of interest to the student, but it might consist of more extended reading and problem solving or a critical review of some of the literature in mathematical model building.

The various components of the course work and their approximate weight in determining a final grade will be these:

A. Homework Assignments/ Lab Reports [25% of course grade]

B. Mid-term Examination (Monday Evening, October 21) [25%]

C. Final Examination [30%] (9 AM – Noon, Wednesday, December 11)

D. Independent Project [20%]. This project is due no later than Friday, December 6; this is a very firm deadline. More information about this project will be distributed next week.

Comments: Topics that I would like to cover include arms race models, population dynamics, mathematical ecology, epidemic modeling, cultural stability, criminal justice systems, residential segregation, opera and the Bible. Mathematical techniques that will be introduced will include differential equations, autonomous systems, Markov processes, game theory and computer simulation.

Extensive use will be made of MATLAB. We will also introduce NetLogo, the premiere software tool for agent based modeling.

Accommodations: Students who have Letters of Accommodation in this class are encouraged to contact me as early in the semester as possible to ensure that such accommodations are implemented in a timely fashion. For those without Letters of Accommodation, assistance is available to eligible students through Student Accessibility Services. Please contact Jodi Litchfield or Courtney Cioffredi, the ADA Coordinators, for more information: Courtney Cioffredi can be reached at ccioffredi@middlebury.edu or 802-443-2169 and Jodi Litchfield can be reached at litchfie@middlebury.edu or 802-443-5936. All discussions will remain confidential.

People who are offended by strong language should skip this note. I am going to use the strongest language I know, apart from poetry, which is mathematics. In fact, its terseness, immense analogical power and frequent difficulty make mathematics the poetry of the sciences. If you haven’t read some mathematics or some poetry lately you’re not having as much fun in life as you could be.

-Joel Cohen, How Many People Can The Earth Support

MATH 315: Fall, 2024

Further Course Information

Tentative Course Outline (Time may not permit covering all topics; some substitutions may occur to reflect student/faculty interests)

I. Introduction: What is a Mathematical Model?

A. The Analytical Approach

B. The Simulation Approach

II. Deterministic Models

A. Richardson's Arms Race Model

B. Population Dynamics

1. Single Species

2. Interacting Species: Predation and Competition

C. Deterministic Epidemic Models

III. Probabilistic Models

A. Markov Chains

B. Hoffman's Models of Cultural Stability

C. Blumstein-Larson Models of Recidivism in Criminal Justice System

D. Stochastic Epidemic Models

IV. Models of Decision Making Under Uncertainty or Conflict

A. Models of Economic and Social Justice

B. Game Theory Models of Old Testament Stories

C. Prisoner’s Dilemma and Tosca

D. Evolutionary Game Theory

VI. Computer Simulation Models

A. A Hospital Planning Model

B. Agent-Based Modeling and Residential Segregation

VII. A Deeper Dive into Differential Equations

A. Existence – Uniqueness Theorems

B. Poincare – Bendixson Theorem