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We place here and in the Supporting Documents all the materials in support of the SIMIODE Remote Teaching Module.

**Introduction to Modeling**

*This Remote Teaching Module introduces modeling with first order differential equations and motivates students to fully engage in the solution and analytical techniques for such equations. Students model data generated via group simulation over Zoom, analyze that data, and present their results. This project is based upon the SIMIODE modeling scenario*:

R.C. Harwood (2016), “**1-037-S-CommonColdSpread**”

*Through this project, students investigate the impact of heightened hygiene and decreased interactions on the spread of an infectious disease. While the simulation focuses on the common cold, the analysis and conclusions can be extended to serious infectious diseases like COVID-19.*

This module contains:

1) A brief Teaching Guide (copied below) with an overview of the content and activity, including any necessary prerequisite material.

2) Videos and materials to assign to students.

3) Video and materials for the instructor to guide implementation of the module in an online format. All sources files for adapting the materials are available in the Supporting Documents.

*Each video is available for instant access and versatile playback on SIMIODE's YouTube Channel as well as downloadable files in the Supporting Docs.*

4) An Assessment Guide with learning goals, guiding questions, and an analytic version of the project rubric which is easily implemented in a Learning Management System.

Additionally, we will be hosting live Q&A sessions in the future and posting FAQ guides.

*For the Q&A session we will use ZOOM meeting software without recording, but we will post FAQ information based on Chat and meeting conversation. We will discuss the lesson experience and issues about teaching, interactions, and online experience.*

**Teaching Guide for SIMIODE Remote Teaching Module:**

**Introduction to Modeling**

Prepared by Corban Harwood, George Fox University, Newberg OR USA

rharwood@georgefox.edu

**Overview of Content**

*This Remote Teaching Module introduces modeling with first order differential equations and motivates students to fully engage in the solution and analytical techniques for such equations. Students model data generated via group simulation over Zoom, analyze that data, and present their results. This project is based upon a SIMIODE modeling scenario (***www.simiode.or****g***).*

**Teaching Materials: **These files and their sources versions are available in the Supporting Documents for customization, editing, and use by individual faculty

**RMT_IntroModeling_TeachingGuide.mp4**:*Video walk through of this guide, a discussion of the module’s purpose, and an overview of the available resources.*- Simulating the Spread of the Common Cold:
**1-037-T-CommonColdSpread**-*the original Teacher Version of the referenced SIMIODE modeling scenario which includes discussions of solution strategies and pedagogical issues.* - RMT_IntroModeling_SimulationGuide.pdf:
*Modified instructions for online simulation with this SIMIODE modeling scenario.* - RMT_IntroModeling_slides.pdf:
*Presentation slides for this module’s video series.* - RMT_IntroModeling_slides.tex:
*LaTeX Beamer structure which permits production of PowerPoint-like presentation using the mathematical typesetting capabilities of LaTeX.* - RMT_IntroModeling_Spreadsheets.xlsx:
*Excel spreadsheet with “Project Template” tab for creating class shared spreadsheet (e.g. Google Sheets) and several other tabs that demonstrate a complete analysis for teacher reference.* - RMT_IntroModeling_AssessmentGuide.pdf:
*Guide for assessing student learning with learning goals, guiding questions, example exam questions, and analytic project rubric.*

**Videos and Materials to Assign to Students **

- Simulating the Spread of the Common Cold:
**1-037-S-CommonColdSpread**–*the original Student Version of the referenced SIMIODE modeling scenario.* - RMT_IntroModeling_SimulationGuide.pdf:
*Modified instructions for online simulation* **RMT_IntroModeling_Part1.mp4**:*Introduction and project preview***RMT_IntroModeling_Part2.mp4**:*Interpretations of differential equations***RMT_IntroModeling_Part3.mp4**:*Characterizations of differential equations***RMT_IntroModeling_Part4.mp4**:*Visualizations of differential equations***RMT_IntroModeling_Part5.mp4**:*Evaluation of numerical solution and model fit to data***RMT_IntroModeling_Part6.mp4**:*Project Review and Supplemental material*

**Prerequisite Material**

No differential equation knowledge is needed before this project begins, but the separable method is needed in completing it and a familiarity with spreadsheet software is helpful. The following free software is very useful for plotting slope fields:

- Java App dfield.jar (
**https://www.cs.unm.edu/~joel/dfield/**), requiring updated Java Runtime Environment (**http://www.java.com/en/download/help/download_options.xml**)

We suggest the following schedule to weave this module throughout a 9 day chapter (50-60 minute days) on first order diff. eq. to motivative chapter topics and provide a strong narrative.

**Schedule **(See **SIMIODE YouTube Channel** for the Remote Teaching Modules playlist)

Day 1: *Topic*: Separable method, introduce module project

*Follow up*: watch RMT_IntroModeling_Part1 (project preview) and

RMT_IntroModeling_Part2 (interpretations)

Day 2: *Topic: *Run simulation during class time or separately

*Follow up: *watch RMT_IntroModeling_Part3 (characterizations)

Day 3: *Topic: *Integrating factor method (and Bernoulli substitution if time)

*Follow up:* watch RMT_IntroModeling_Part4 (visualizations)

Day 4: *Topic: *Slope field and phase line

* Follow up: *watch RMT_IntroModeling_Part5 (evaluations)

Day 5: *Topic: *Fitting model parameters: discussion of project and other model equations

* Follow up: *watch RMT_IntroModeling_Part6 (project review)

Day 6: *Topic:* Existence & uniqueness theorem investigation or activity

e.g. B. Winkel (2015), “**1-015-T-Torricelli**”

*An activity fitting a model with uniqueness issues to data collected from a video.*

*Follow up:* Recordings of project presentation are due.

Day 7:* Topic:* Numerical methods derivation/programming activity

e.g. R.C. Harwood (2019), “**1-005-Text-T-NavigatingNumericalMethods**” .

*A role-playing activity involving a rescue at sea where students develop numerical
methods through analyzing slope fields and guided derivations. *

Day 8:

*Topic:*Review day: Interpret, Characterize, Visualize, and Evaluate

*Follow up:*What are differential equations and why do we care?

Day 9:

*Topic:*Exam over first order differential equations

__One-Day Abridged Version__

This project can be abridged to fit one 50-minute class period by doing the following. For this version, I suggest that students be familiar with the separable method beforehand (solving or assigning the solution of potential model equations beforehand) and have access to Zoom and shared Project Template in GoogleSheets.

- Assign 1-2 simulations per pair, recording data directly into the class spreadsheet.
- Instructor shares spreadsheet via Zoom and leads class in a discussion to select the model to agree with the shape of the derivative data and estimate the parameters by the shape characteristics, slope trend line.
- Apply local optimization on a key parameter to decrease RMSE against the solution of the differential equation. Point out how the two curves converge.
- Assign discussion questions for homework.

**Nature of the Project**

In the project for this module, students model data that is generated via group simulation over Zoom, analyze that data, and present their results. This project is a good discovery technique to introduce the first few weeks of differential equations. The online Simulation motivates the need for the separable method and develops visual analysis for dealing with slope fields and phase lines. Looking at the graph of the infected populations across all simulations, students can estimate qualitative features of the governing equation such as the existence (and stability) of equilibrium points and the constancy of slopes at fixed population values to determine if the equation is autonomous. Completion of this project is a great jump-off point for numerical methods and more advanced methods and models.

Before the Online Simulation, it is helpful to have a class discussion about what the simulation represents (the two-week spread of a cold throughout a residence hall) and what students think should happen. The focus of this simulation is on the common cold, a recurring, but mild, disease. The analysis and conclusions, however, can be extended to more serious infectious diseases such as COVID-19. A shareable spreadsheet (e.g. Google Sheets) is a helpful way for students to collaborate on their own schedule and present together in class. To reduce grading, I suggest a single sheet shared with the whole class with independent tabs for each group. Having access to each other's work is a risk, but this is an opportunity for students to practice respectful sharing of resources. A wiki can also be used to consolidate group work for class presentations. A private wiki, accessed through a university course management system like Moodle (**www.moodle.com**) or separately like PBWorks (**www.pbworks.com**), is a web environment, similar to Wikipedia, where students can create and link together multiple web pages in a more controlled environment. Then students can present directly from their browser tab.

The Analysis portion begins with a graphical fitting of the group's model and then a hands-on optimization of the main parameter(s) using either a local or global optimization method. It may be helpful to demonstrate examples of writing a model in a form that spreadsheets can do least squares regression on:

linear Y=mX+b, and

polynomial Y=a_n X^n+...+a_1 X+a_0.

See the example in the student instructions along with the following.

Model x'=a(t-b)(t-c)

can be rewritten in the polynomial form

x'=aX^2-a(b+c)X+abc

with X=t and each parameter can be estimated through the regression constants a=a_2, -a(b+c)=a_1, abc=a_0.

Note, this kind of regression optimizes the parameter(s) for the slope function instead of the solution of the differential equation, so there will be room for further parameter optimization of the RMSE and R^2 fit of the model to the solution.

The Local Optimization described in the modeling scenario is an adaptation of the bisection method to finding a minimum (instead of a root). At every step, students compare the root mean square error (RMSE) on the left and right and subdivide their search interval sequentially to better estimate the parameter value which minimizes the RMSE. Global Optimization is a basic global search method which evaluates the RMSE at a regularly-spaced set of values for a given parameter. This set is then centered at the lowest RMSE value and compressed for better precision.

After the presentations, I suggest a class discussion of the various models derived and the range of values used for parameter(s) to help wrap up the project. One of the main things I bring up is the difference between our simple simulation for the spread of the common cold and a more realistic SIR model for the spread of the disease. Note that the logistic model (model four below) can be written as I'=aIS since S+I=N. This model assumes no recovered population (R=0), which can be justified as the recovery time of the common cold is longer than the time it takes for the infection to spread throughout this high-density residence hall. For a week-long cold and a simulation which ends in about seven rounds, this seems realistic. It is helpful to point out that a true SIR model would be a system of differential equations instead of just one and such systems are analyzed later on in the course.

I tend to guide groups through the final selection of their model using their data comparison graph by pointing out patterns in the data and shapes of their models. A less open-ended approach would be to offer them the following potential models (based upon initial ideas from students). Each of these slope functions can be argued to share a shape with the infected derivative from the change in infected data, but the first two do not match the value dependence and cannot be extended in time for predictive purposes. Model four is the traditional logistic growth model usually attributed to capacity-limited population growth. The models are all separable, the first three are also linear, and they are ordered with increasing difficulty of their solution (solution to the fourth model requires partial fraction decomposition, while the fifth requires a cosecant integral).

x’=a sin(bt)

x’=a(t-b)(t-c)

x’=a(N-x)

x’=ax(N-x)

x’=a sin(bx)

**Related Activities in Infectious Disease Modeling** **for Follow-Up:**

**6-001-T-Epidemic**– Analysis of data from an epidemic in an English Boarding School**6-007-T-FunctionsAndDerivativesInSIRModels**– Graphical analysis of relationships between function and derivative in the SIR model system**6-011-T-HumansVsZombies**– SIR modeling of a Zombie outbreak that mirrors the epidemiology of infectious diseases.**6-016-T-PandemicModeling**- Comparative modeling of pandemics: Ebola, COVID-19, and many more.

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