# Materials for Teaching the SIR and SEIR Epidemic Models

## Overview

This web page contains materials created by faculty of the University of Nebraska-Lincoln Department of Mathematics to teach basic fundamentals of mathematical epidemiology.

The original motivation for these materials was not the emergence of COVID-19, but rather the need for education about the value of vaccination. Epidemic modeling cannot address the unsupported claims of harm caused by vaccines, but it can very clearly show the tremendous benefit that vaccination has in combating disease. This message will be critically important once a vaccine is available for COVID-19. In a poll taken in May 2020, only half of Americans said they would get a COVID-19 vaccine, while 20% said they would not. This is not surprising, given the support the anti-vaccination movement has at the highest levels of the United States government, but it is, of course, alarming. With somewhere between 50% and 80% of a community receiving a COVID-19 vaccine, it is quite possible that there will be insufficient herd immunity to protect those people who cannot take a vaccine, leaving the public health threat of COVID-19 to continue indefinitely.

Given our motivation, the material we present has two primary objectives:

• Help students understand the patterns of epidemic disease outbreaks and the challenges they present;
• Help students explore the effect of public health policies, such as isolation of symptomatic patients and prior vaccination, on the course of the outbreak.

We approach these objectives through materials for study of the standard SIR and SEIR disease models. The SIR model is ideal for general education in epidemiology because it has only the most essential features, but it is not suited to modeling COVID-19. The SEIR model is the logical starting point for any serious COVID-19 model, although it lacks some very important features present in COVID-19. See COVID-19 educational module for material designed specifically for COVID-19.

The spreadsheet-based versions do not require any background knowledge other than basic algebra and spreadsheet skills. The program-based versions use the differential equation forms of the models, but do not require any specific background in differential equations or computer programming.

## Materials for Computational Modeling

The materials presented here were created by Glenn Ledder as tools for students to explore the predictions made by the standard SIR and SEIR epidemic models. There are 4 modules:

1. S1 SIR is a spreadsheet-based module that uses the SIR epidemic model.
2. S2 SEIR is a spreadsheet-based module that uses the SEIR epidemic model.
3. P1 SIR is a program-based module that uses the SIR epidemic model.
4. P2 SEIR is a program-based module that uses the SEIR epidemic model.

The Excel versions use workbooks that can be created by students from template workbooks following step-by-step instructions. (Instructors can do some or all of the setup for their students.)

The Matlab/R versions use suites consisting of one function program and three driver programs. These are included in the distribution, along with directions for how to use them and modify them for specific experiments.

All versions contain 5 primary components, along with brief instructor notes:

1. A brief Powerpoint introduction;
2. A Student Notes document that outlines the model and some details about the module;
3. An Excel workbook template or a suite of Matlab/R programs;
4. An Instructions document that explains how to setup and use the workbook or how to use and modify the programs;
5. An Experiments document that describes several experiments and contains a large number of questions. Many of these explore the impact of public health policies.

The completed Excel workbooks and answer documents for all modules are available to instructors from gledder@unl.edu by request.

All versions were posted on November 18, 2020.

#### Feedback

Feedback is welcome on these educational modules, as well as contributions of additional experiments. Send comments to gledder@unl.edu.

## Materials for a Simulation Activity

created by Glenn Ledder and Michelle Homp

### Description

A “speed dating” protocol is used to randomly pair students and a die rolled to determine if transmission occurs. This strategy implements a version of the SIR model that couples a low contact rate with a high transmission probability. While this is the opposite of reality (with high contact rates and low transmission probability), the resulting simulation generates data that roughly matches what would be obtained from the standard SIR model. With no incubation period, a one-day asymptomatic infectious stage, and a one-day symptomatic infectious stage, the epidemic runs its course in about 10 to 14 days. The activity runs quickly and can be repeated to illustrate how random variation can affect results. Directions are given for both online and in-person classes.

The standard version of the activity is designed so that it can easily be adapted to incorporate isolation of symptomatic infectives or prior vaccination of susceptibles.

This activity was originally created for a middle school summer program and then adapted for Math 203, which is a general education mathematics course with minimal prerequisites. All materials are accessible to students with no background other than basic algebra. This activity would also be suitable for high school mathematics or biology courses.