**Note: Most experts use a value of roughly 2.6 for the basic reproductive number. The correct value is probably about 5.**

This value is computed from three assumptions:

- the average incubation period is about 5 days;
- the average recovery time for patients who do not require hospitalization is about 10 days;
- the average amount of time required for hospitalizations to double in the earliest stage of the epidemic is about 4 days.

## Overview

Are you looking for materials for online teaching in mathematics, biology, or public health? This COVID-19 educational module by Professor Glenn Ledder can fill this niche while helping students learn about what is going to be a defining event in their lives and in human history. It was designed to be accessible to high school science and mathematics students and to college students in courses for non-majors or in a course at the level of calculus and below.

The module embeds a simple model designed specifically for the COVID-19 pandemic into an environment where students can easily design experiments and observe outcomes. The primary files contained in the model are an Excel workbook, a document containing an overview and directions for getting started, and a document that presents a number of pre-designed experiments along with questions that guide students in interpreting the experiment results and judging the accuracy of public statements that have been made about the pandemic.

**Watch the 3-minute overview**.

## The Workbook

The principal component of the workbook is a “sandbox” top sheet that serves as an MUI (model-user-interface). On it, you define an experiment consisting of up to three scenarios, and you get the results in the form of three numerical quantities and two graphs. Ten of the parameters are fixed for the whole experiment, leaving six parameters that can be varied to differentiate the scenarios:

- The contact factor, which measures social distancing. This parameter ranges from a value of 1 for interpersonal contacts at the normal rate and a value of 0 for total isolation.
- The fraction of infected people who are asymptomatic. This is important because transmission from asymptomatics makes COVID-19 harder to stop.
- The fraction of symptomatic people who are isolated without hospitalization, roughly equivalent to the fraction who are tested for the virus. This parameter plays a large role in determining the outcome because isolating people known to have the virus decreases the transmissions from those people.
- The fraction of asymptomatic people who are isolated without hospitalization, which is only meaningful if these cases are found through contact tracing.
- The number of days required for the numbers of infections and hospitalizations to double. This parameter is the best measure of the growth rate of the infection, and it can be used to estimate the basic reproductive number.
- The fraction of the population that is initially immune, which is useful for exploring the effect of a future vaccination or sufficient spread of the virus to allow for some herd immunity.

Three outcomes are reported in the sandbox sheet for each scenario:

- The percentage of the population that remains susceptible at the end of the year-long simulation. This is a broad measure of the risk that still remains at the end of the simulations.
- The percentage of the population that dies from COVID-19. This is a good measure of the human cost of the epidemic.
- The maximum number of people who need to be hospitalized at any one time. This is one possible measure of the stress the epidemic places on the health care system.

There are also two graphs, one for the percentage of susceptibles and one for the number of hospitalizations per million people.

**Watch the 5-minute demonstration video**.

### Module Materials

**Version 1.1.1**, posted on **May 13, 2020**.

The module will be updated as needed, with changes recorded in a version history (see below).

**The answer key is available only to teachers. Send your request to gledder@unl.edu.**

### Pedagogy

This module uses a structure and pedagogical strategy developed by Michael McConnell, founder of Spreadsheet Lab Manual. SLM offers a large number of educational modules in physics, chemistry, mathematics, and biology that were designed for the high school classroom and make excellent tools for online teaching. This module will be incorporated into SLM as Module 403.

## The Model

Simple differential equation models in epidemiology are typically designated by a set of letters indicating the different classes into which the population is divided. The most basic model is SIR, for Susceptible, Infective, and Removed. A slightly better model for many diseases is SEIR, where the extra class, called Exposed, is for people who have contracted the disease but cannot yet transmit it to others. More complicated models can be created by adding more classes or additional features. The COVID-19 model used in the module is of a type we could call "SEAIHRD". Compared to SEIR, the infective class is divided into three subgroups: A for asymptomatic infectives, I for symptomatic infectives, and H for patients requiring hospitalization. These groups differ in their contact rates, the extent to which they can transmit the virus, and the course of their disease. The traditional Removed class is subdivided into a Recovered class and a Deceased class. This does not change the model structure, but it does remind us that the number of people who die is a very important outcome.

#### Feedback

Feedback is welcome on both the model and the educational module that uses it, as well as contributions of additional experiments for the module. Send comments to gledder@unl.edu.

#### Version History

- 1.0, April 8, 2020
- 1.0.1, April 10, 2020: fixed typos and added data tables to the Experiment document.
- 1.0.2, April 29, 2020: fixed error in data for Experiment 6.
- 1.1, May 10, 2020: changed some parameters, added testing of asymptomatics, added a contact tracing experiment.
- 1.1.1, May 13, 2020: corrected a slight error in the workbook and updated the instruction file.

#### Other Materials

- An introduction to compartmental modeling for the budding infectious disease modeler by Julie Blackwood and Lauren Childs is a good place to start reading about disease modeling.
- Materials for Teaching the SIR Epidemic Model contains materials created by mathematics faculty at the University of Nebraska-Lincoln to teach disease modeling to students with no background other than basic algebra.