Course Description
Mathematical models are an essential tool in epidemiology. Models are used in a number of ways, including: to predict the course of an epidemic; to estimate key epidemiological parameters; to determine an optimal vaccination strategy; or to gain a general, qualitative understanding of the salient features of the dynamics of a disease. This course will focus on differential equations, a class of mathematical models used across the sciences to model quantities that change continuously in time. The course will begin with a short introduction to differential equations. We will then step through the full “pipeline” of mathematical modeling using epidemiology as a case study. We will begin with a first-principles derivation of several models and will consider how the natural history of an infectious disease influences modeling choices. We will then see how to solve and analyze differential equations computationally. We will then cover how to calibrate a model to epidemiological data and estimate key model parameters in a principled way. Finally, we will see how to create model projections for different intervention scenarios and will explore the strengths and limitations of a mathematical model in relation to given research and policy questions. Through this process, students will gain skills and experiences that will apply to mathematical modeling in a range of settings across the natural and social sciences.
In addition to learning the mathematics of differential equations applied to epidemiology, a central goal of this course is to gain skills necessary for research in the mathematical, natural, and social sciences. This includes conceptualizing and framing a research question, formulating a model to explore that question, engaging with peer-reviewed research publications, and giving a research presentation and preparing a short technical report.
Evaluation will be based on class participation, problem sets and programming exercises, and a term-long project culminating in a research presentation and short technical report.
Additional Course Info
- Meets the following requirements: QR, ES
- Prerequisites: Calculus II or the equivalent or permission of instructor. Prior experience with either R or Python.
- Level: Intermediate
- Course limit: 16
- Lab fee: No lab fee
Learning Objectives
This course is designed as a community learning journey. Together, we will:
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Gain a basic understanding of ordinary differential equations (ODEs), including systems of coupled ODEs. Learn
what ODEs are, what it means to find a solution to an ODE, and different ways of visualizing solutions, including phase space methods. Learn and implement (by hand and in R) Euler’s method, a numerical approach to solving ODEs. Learn to use ODE integrators that are part of standard packages in R.
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Gain an understanding of the different ways mathematical and computational models are used in the sciences. Gain experience deriving a model from first principles, using models to explore and test ideas, and developing your own models.
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Gain skills in identifying key characteristics of a disease including who, what, when, where, and how the disease is spread. Develop skills in translating and incorporating these characteristics into epidemiological models.
It is also our hope that we will:
- Create a shared environment where we can investigate mathematical and computational ideas creatively, using a growth mindset which values revision and experimentation. Demonstrate community leadership skills as a collaborator who shares strengths, helps each other learn, and contributes to a broader shared understanding. (Participate in teamwork in respectful ways that allow people to relax and explore ideas.)
Course Materials:
Books and Resources:
Course Outline
This course has several distinct components, each of which may have a somewhat different feel.
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Introduction to Differential Equations: We’ll talk about what differential equations are and what it means to solve a differential equation. We’ll learn about some techniques for determining the qualitative behavior of solutions to a differential equation. We’ll then learn Euler’s method, a simple but powerful numerical technique for solving differential equations. This part of the course will feel sorta like a math class. There will be a few problem sets and you’ll be doing some math.
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R Basics: In parallel with item 1, we’ll cover some of the basics of the R environment for statistical computing and visualization. You’ll learn how to work with data files and plot data. We’ll then see how to use R to plot numerical solutions differential equations.
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Basic Epidemiological Models: We then turn our attention to SIR models, a simple compartmental model commonly used in epidemiology (and elsewhere). We will build up this model carefully, starting with first principles and discussing the assumptions embedded in the model. This part of the course will seem less like a math class, since we’ll be relying almost exclusively on computational solutions to models. We’ll also consider a number of variants of the basic SIR model.
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More Epidemiological Modeling Topics: Our attention will turn more toward modeling. We’ll read and think about the different sorts of ways that models are used in epidemiology and in the sciences more generally. We’ll also learn about how to estimate model parameters using data. AS time permits, we will cover additional topics of interest to students.
Technology:
- Bring a laptop to every class, we will be programming most days.
- If you are in need of a term long loaner laptop, please contact the IT department at helpdesk@coa.edu. Mention that you are taking a computer science class, and pick up the laptop in A&S right by the whale skull.
Course components
Weekly structure
The class meets on Tuesday, Wednesday and Friday from 14:35-16:00.
In addition to the 4.5 hours of scheduled class time every week, we expect that between readings, going over notes, and doing assignments you will spend at least an additional 10.5 hours a week on this course, for a total of at least 150 hours over the term devoted to this class.
Getting Help
The question forum on google classroom will be used for online Q&A and collaborative discussion. You can post questions and answers can be posted by the instructor, TA, and/or other students. Please check google classroom to see whether a question has already been asked/answered before starting a new post. However, instead of sharing code on google classroom, describe the issue or post the error message - if we need to troubleshoot outside google classroom, I will let you know!
We will make an effort to check the question forum daily during the week to answer any questions. We will check the board on Monday to answer any questions posted over the weekend.
Online resources (e.g., StackOverflow) can be a good source for understanding and inspiration. These should not be an immediate go-to without your having struggled with the problem first. You should also take any such online posting with a grain of salt, as they can sometimes be misleading or in a different context than your own or, in some cases, simply wrong. Moreover, take special note of the “Sharing and Reusing Code” section below.
Help sessions
The Teaching Assistant and us will have a handful of help sessions every week. You are warmly invited and encouraged to attend these sessions. Help sessions are relaxed, informal, and hopefully fun. Things that happen at help sessions:
- The TAs and/or we are around to offer help on the assignments.
- Some students do most of the assignment while at a help session. They work through problems alone or with others, and find it comforting to know that help is at hand if needed.
- Others do the problems at home and come to the help session with specific questions.
- Help sessions are also a chance to ask general questions about the course.
- Help sessions are a great way to meet other students in the class.
Everyone is welcome at help sessions! Attending these sessions help students do well in class and get as much out of it as possible.
Grades:
A growing body of research indicates that traditional approaches to grading fail to produce the sorts of meaningful learning desired by both teachers and students. Such approaches often reinforce inequitable power dynamics between teachers and students, promote faulty reward systems that disincentive creativity and risk-taking, and devalue important aspects of learning (including revision and feedback). Given this context, instead of a traditional approach to grading in which you do work that is evaluated singularly by me, this course assumes that you opt to take ownership and responsibility over your performance and engagement with the class. To make this happen, this course uses a “contract grading” scheme, which gives you a voice in the grading process, provides you with the agency to specify your intended course performance, and also share in the responsibility for evaluating whether or not you fulfilled your intended obligations. Please see the contract grading document (on Google Classroom) for a more-fleshed-out explanation of this approach and how it will operate in the course.
We will meet with each of you individually to set goals at the start of the course.
The work in this course will be comprised of the following components and their weights:
- Homework and Labs: 40%
- Final Project: 40%
- Class Participation and Attendance: 10%
- Reflections: 10%
Policies
Collaboration policy
Unless indicated otherwise on a specific assignment, your HW and lab assignments may, and should, be discussed with others in this course. Each assignment is subject to the “empty-hands policy:”
- You may freely discuss ideas with other students, but each student must leave the discussion without any written or otherwise recorded material.
Any manifestation of copying another student’s work for your own (whether digital, hand-written, oral, etc.) is not permitted. This includes, for example, looking at or screen-capturing another student’s implementation and then writing “your own” version of that implementation.
Sharing / reusing code
I am well aware that a huge volume of code is available on the web to solve any number of problems. You can, and should, consult online resources (e.g., StackOverflow) when you get stuck. Recommendations there are often useful for helping you to get unstuck. However, you are not permitted to copy-and-paste any solutions directly from online resources. Rather, use them to guide your own solution and implementation. Caveat: answers on StackOverflow are sometimes wrong and/or can be misleading.
- Any online source referenced in your solution must be documented in the comment block at the top of your submission. (We all refer to help online — do not consider doing so a badge of dishonor, but do give credit to the source of help.)
- If you have collaborated with any other students (subject to the guidelines above), please also cite those students in your comment block at the top of your submission.
Academic Integrity (excerpt from Course Catalog)
By enrolling in an academic institution, a student is subscribing to common standards of academic honesty. Any cheating, plagiarism, falsifying or fabricating of data is a breach of such standards. A student must make it their responsibility to not use words or works of others without proper acknowledgement. Plagiarism is unacceptable and evidence of such activity is reported to the provost or their designee. Two violations of academic integrity are grounds for dismissal from the college. Students would request in-class discussions of such questions when complex issues of ethical scholarship arise.
Most often, violations of academic integrity are due to perceived timeline pressures and a corresponding feeling of desperation. Please understand that, if you find yourself in such a situation, I want you to come talk to me, rather than making a rash (and poor) decision that can have negative consequences. We can discuss appropriate accommodations and alternative timelines for an assignment.
Universal Learning and Learning in Community
Many of us learn in different ways. For example, you may process information by speaking and listening, so while lectures are quite helpful for you, some of the written material may be difficult to absorb. You might have difficulty following lectures, but are able to quickly assimilate written information. You may need to fidget to focus in class. You might take notes best when you can draw a concept. For some of you, speaking in class can be a stressful or daunting experience. For some of you, certain topics or themes might be so traumatic as to be disruptive to learning. The principle of Universal Design for Learning calls for our classrooms, our virtual spaces, our practices and our interactions to be designed to include as many different modes of learning as possible, and is a principle I take seriously in this class.
It is also my goal to create an inclusive classroom, which depends on community building, and which requires everyone to come to class with mutual respect, civility, and a willingness to listen to and observe others. As such the syllabus serves as a contract of some expectations between all members of the class, including myself.
If you anticipate or experience any barriers to learning in this course, please reach out to me and your student support advisor. If you have a disability, or think you may have a disability, COA’s Disability Support Services located within the Office of Student Life in Deering Commons to develop a plan for your academic accommodations. You can find out more information in the course catalog under Accommodating students with disabilities. If you have already been approved for accommodations through the Disability Support Services please let me know! We can meet 1-1 to explore concerns and potential options.
Late work, extensions, and special circumstances
All work is due on the stated due date. Due dates are there to help guide your pace through the course and they also allow me to return feedback to you in a timely manner. However, sometimes life gets in the way and you might not be able to turn in your work on time.
If you intend to submit work late for an assignment or project, you must notify me before the original deadline and as soon as the completed work is submitted. This allows me to return feedback to you and let’s me know when to check your work. Lab work cannot be submitted late.
Learning during a pandemic
We want to make sure that you learn everything you were hoping to learn from this class. If this requires flexibility, please don’t hesitate to ask.
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You never owe us personal information about your health (mental or physical) but you’re always welcome to talk to us. If we can’t help, we likely know someone who can.
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We want you to learn lots of things from this class, but we primarily want you to stay healthy, balanced, and grounded.
