IMM Final Project

Meeting due: Schedule a 10-20 minute meeting with me or Hassan by Friday, December 1, 2023

Project due: Sunday, December 17, 2023, midnight

Assignment: Investigate a model for either a problem from the list provided at the end of this document, or one of your choosing, with approval from me or Dave. Like the mini-project, it should:

1. be centered on a question (e.g. “How many filter devices are needed to reduce the threat of covid spread to some specified low probability” for the mini-project), and

2. rely on one or two scientific papers as its basis (e.g. Bazant & Bush 2021 for the mini-project).

The form of the project will be a short technical paper that will describe the problem and background, provide sources to data used, rationalize your modeling choices, set up the equations, explain the parameters in the model, present the analysis and solutions of the model, and address the central question asked. If the model is numerical, your code should be included.

The report should include an abstract, introduction, model description, results, conclusions, references. There is no exact page requirement for the project, but shorter reports are encouraged! Please consider using LaTeX to typeset the report (if you’re not familiar with it, you could use this assignment as an opportunity to learn it), but that’s not required. Regardless of what software you use, the final product must be a PDF.

References should be provided. Please refer to references in your text as, for example, Smith & Jones (1902), and use these in-text citations to indicate which facts or formulas come from which source. Include in your references section the authors, year published, title, journal, volume and page numbers (or just the authors, year, and DOI).

The chosen research problem should be at a level of mathematical sophistication consistent with the level of the course.

You are strongly encouraged to work together on a topic with a partner, but each partner should submit an independent report, perhaps investigating different subtopics or parameter regimes. Please convince us that equal independent effort was given by each partner.

 A list of project ideas is included below. If none are sufficiently interesting to you, you may propose your own. Both the Strogatz and Tung texts are full of references to models. You could also look at the papers posted in Resources/PapersEtc/ (but be aware that this set of papers is a bit random, and not all topics are equally accessible). Whatever subject you’re interested in, you must meet with me or Dave to discuss your project plan and get approval.

I am interested in seeing evidence of learning and engagement, using tools we’ve covered in class to investigate something of interest to you, not in ranking you or setting up arbitrary or hidden criteria.

Some important points:

● Focus on the question(s) you're asking, and shape the model and report to address that.

● Don't plagiarize anything. Text from other papers should be placed in quotes, with a

citation following the quote.

● Iterate between writing and computing/analyzing. Writing helps shape your thoughts and

hones your question, and analyzing tells you want you can get from the data and/or

model, and that may lead you to revise the question.

● The final product should be a clean PDF report that includes a title, abstract,

introduction, analysis, discussion, and a reference list with complete details (Authors, year, journal; volume & pages or doi). At least two references should be peer-reviewed papers (unless I've told you otherwise).

● Plots that are part of the story should be included in the report and referenced in the text. You can download images you create in a Python notebook, then insert them in your document. Label your axes and explain clearly what's plotted in captions.

● If you've used code for your project, include it as a second file, either zipped, or as a PDF, or a .ipynb or .py file. Clean your code of irrelevant parts before submitting it.

● Each partner should turn in a separate report. The reports may have a great deal of overlap, but make sure it's clear that each partner has done something individually. Not trying to be nit-picky and there are many ways to make this clear.

● I will be so grateful if you demonstrate any interest and enthusiasm for your project, and exceptionally happy if you can make the report an interesting read! :D

Grading: [25] setting of idea: question asked, clarity/completeness of introduction, relevance/realism of the problem, sources and citations

[25] setup of problem: correctness, appropriateness of tools, setup, methods [25] analysis of problem: code and/or analysis, plots/visualizations, etc..

[25] interpretation of results: answering question, supported by analysis

Project ideas [citations below can be found in Resources/PapersEtc, and each is just a suggestion]

 ● Investigate intervention strategies (lockdown, quarantining infecteds, etc) to stop the spread of a disease (e.g. Covid) in a confined community (Murray 2000 ch 10, Hethcote 2000, Newman 2002). See project guidelines from Spring 2021.

● Use an energy balance model for radiative forcing on Earth (Budyko 1969, Held & Suarez 1974) to investigate the use of surface reflectors to offset warming from anthropogenic carbon (meer.org, and COP27 talk)

● A question regarding interacting oscillators (Kuramoto model, e.g. Strogatz 2000), e.g. as a model for our circadian rhythm (e.g. Lu et al 2018) or spatially distributed systems like fireflies (e.g. Wuster et al 2020)

● How to eradicate an invasive species eradication — choose an invasive species, design a model, use data to set parameters, make a recommendation (Alford et al 2019)

● Timescale over which orbits become unpredictable in chaotic 3-body interactions of planets and suns (Wikipedia page and references therein)

● Model morphogenesis of stripes in an animal (Turing 1952)

● Design and either make or numerically model a toy that exhibits chaotic dynamics (e.g.

Moon 2004)

Another resource for ideas is COMAP, for example this problem from the 2015 Mathematical

Contest in Modeling**:

PROBLEM A: Eradicating Ebola

The world medical association has announced that their new medication could stop Ebola and cure patients whose disease is not advanced. Build a realistic, sensible, and useful model that considers not only the spread of the disease, the quantity of the medicine needed, possible feasible delivery systems (sending the medicine to where it is needed), (geographical) locations of delivery, speed of manufacturing of the vaccine or drug, but also any other critical factors your team considers necessary as part of the model to optimize the eradication of Ebola, or at least its current strain. In addition to your modeling approach for the contest, prepare a 1-2 page non-technical letter for the world medical association to use in their announcement.

** Note that these MCM problems are more challenging that I’m generally expecting for the class.

Some past projects:

● Three-body problem (example question: for how long can you predict a trajectory within

a given error?)

● Random walks on mazes, e.g. a regular grid like NYC city blocks

● Glider flight paths

● Prisoner game dilemma

● Epidemic modeling as stochastic process on a network

● Maximum heart rate while running