How R Naught (R₀) Shapes an Epidemic

Background Information for Teachers

R₀ (pronounced R naught) is the basic reproduction number of an infectious disease. It quantifies how many people, on average, an infected person can infect. For example, an R₀ of 3 means that an infected person will, on average, infect three more people. R₀ is the measure of how spreadable the disease is and can be used to model and understand the speed and scale of an outbreak.

Three factors contribute to the R₀ for an infectious disease:

1. The infectious period of the disease. This is the average length of time, usually measured in days, during which an infected individual is able to pass the infection to another person.
2. The mode of transmission of the disease. Diseases can be passed from one person to another in many ways including through direct contact with bodily fluids like blood, droplet spray from coughing or sneezing, or airborne diseases whose small particles stay suspended in the air for minutes or hours, even after the infected person has left the area.
3. The contact rate between individuals. This is the most variable factor and depends on the population. In densely populated cities, individuals tend to have contact (interactions) with more people per day than in sparsely populated, rural areas.

Since some of the factors, like contact rate, are variable, R₀ is often given as a range. When a new contagious disease appears in a population, like COVID-19, scientists look at the data to determine the R₀ range. Knowing the R₀ helps policy makers, public health officials, and scientists determine if an epidemic is coming and what that epidemic might look like.

A disease with an R₀ below 1 will not expand into an epidemic. Instead, it dies out because, on average, not all infected individuals pass along the disease. In contrast, if R₀ is above 1, an epidemic is likely. The larger the R₀, the faster the pace of the epidemic and the larger the scope. Compared to epidemics from diseases with low R₀'s, epidemics from diseases with high basic reproduction numbers tend to be shorter in duration, but have higher numbers of infected individuals per day. Overall, the total number of individuals infected over the course of the epidemic increases as R₀ increases.

Students will learn all of this by using SimPandemic to model what a COVID-19 epidemic looks like with different R₀ values. They will observe that the infection curve appears sooner and is steeper and narrower with a high R₀ .

You will see that students obtain slightly different results, even when they have the same inputs for SimPandemic. Many events in the real world and in a simulation of the real world are based on chance. When you become infected in the real world, you often don't know when or where the infection occurred. Perhaps someone sneezed when you were randomly walking by them. A simulation cannot predict that you were going to get infected (except in special cases), but it can predict fairly well that someone would get infected. In SimPandemic there are many events where the simulation literally rolls virtual dice to determine when an infected individual will transmit the disease to another (all within the bounds specified by the input parameters). To better understand the assumptions and parameters involved, read the SimPandemic FAQ.

Additional Background Links

• Understanding Predictions: What is R-Naught?, Harvard Global Health Institute
• R₀: How scientists quantify the intensity of an outbreak like coronavirus and predict the pandemic's spread, The Conversation

Prep Work (15 minutes)

1. Familiarize yourself with the SimPandemic online software. Read through the SimPandemic FAQ and try out the SimPandemic Notebook associated with this lesson plan.

Engage (15 minutes)

Remote learning adaptation: This lesson plan can be conducted remotely. Students can work independently on the Explore section of the lesson plan using the Student Worksheet as a guide. The Engage and Reflect sections can either be dropped entirely, done in writing remotely, or be conducted over a video chat.

1. Start by describing the following scenario to your students and asking them for a scientific explanation. Record their answers to revisit at the end of the lesson.

   Imagine this scenario: Doctors are tracking COVID-19 infections in two communities, H and L, that both started out with similar numbers of infected patients. Neither community is taking any preventative measures like wearing masks, social distancing, or quarantining. So, it isn't surprising that the rate of infection is high in community H. However, despite the same lack of preventative measures, the rate of infection in community L is much lower. What is your scientific explanation for this phenomena?

   Students may be puzzled. Encourage them to give their best guess, regardless of whether they are right or wrong. Record all answers to revisit at the end of the lesson.

2. Tell students that they'll be learning more about the basic reproduction number, also called R naught and written as "R₀", and how it can shape the timing and intensity of an epidemic. At the end of the lesson, they will have a chance to use their new knowledge.
3. See what students already know about R₀.

   Raise your hand if you have heard of R₀ before. What do you think R₀ is?

   Students may or may not know what R₀ is. Listen to students' answers, then move right into the Explore section where they can see the definition in the SimPandemic Notebook for this lesson.

Explore (40 minutes)

1. Distribute either the Student Worksheet (PDF) or the Student Worksheet (Quiz assignable in any LMS) to each student. Break the class into pairs if they will be collaborating on the worksheet.
2. Set up Part 1 of the lesson by telling students they will be using an online tool called SimPandemic to learn about R₀ and how it shapes epidemics. Navigate to the SimPandemic Notebook that accompanies this lesson plan and review the information in the first section with students. In particular:
   1. Explain to students that this is what a model of the COVID-19 epidemic looks like if no interventions (no masks, no social distancing, no shutdowns, etc.) are used.
   2. Go over the For Simulated Populations of 100,000 outcomes table and how to read the accompanying graph. Make sure every student understands the type of data being presented before continuing.
   3. Emphasize that the outcomes are listed per 100,000 individuals to enable direct comparisons between simulations of different scenarios.
   4. Point out the FAQ button. If students get stuck or have questions about the program, the SimPandemic FAQ is likely to have answers.
   5. Explain that if the simulation was run twenty times, each time the results would be slightly different. This is because the simulation mimics real life in that it looks at the chance of getting infected and the chance of having symptoms or dying. In SimPandemic there are many events where the code underlying the simulation literally rolls virtual dice to determine when an infected simulant will transmit the disease to another person (all within the bounds specified by the input parameters). Just like rolling dice in a board game, the results can vary within a predictable statistical range.
3. Direct students to work independently, or in pairs, to read through the information in sections 1 and 2 of the SimPandemic Notebook and answer the questions in Part 1 of the Student Worksheet.
4. Set up Part 2 of the lesson by having a student read out loud the first paragraphs of section 3 of the SimPandemic Notebook. Poll your students about their hypotheses.

   What is your hypothesis? Will the epidemic progress differently if we run simulations with different R₀ values? How do you think it will change? How do you think it will stay the same?

5. Tell students that they will have the opportunity to test their predictions by exploring and running their own simulations. They will use the Sandbox in SimPandemic Notebook section 3 to alter the R₀ of COVID-19, then run the simulation several times at each R₀ value and compare the results. Before letting students explore, consider going over with them:
   1. How to make changes to the settings.
   2. How to save their work in SimPandemic.
   3. Remind them that if they have questions they can always consult the FAQ.

Reflect (10 minutes)

1. Remind students of the scenario you laid out for them about communities H and L, which had similar starting levels of infection and lack of preventative measures, but different rates of infection as the epidemic progressed (see the Engage section for details). Read back the explanations they originally came up with. Ask students what they now think the scientific explanation is.

   What is your scientific explanation for why community H's rate of infections is significantly higher than community L's?

   Based on their new knowledge, students should be able to conclude that community H has a higher R₀ for COVID-19 than community L. They should be able to articulate that a higher R₀ means more interactions between residents in community H. When prompted, they should be able to list a number of reasons why a community might have higher levels of interaction, such as living in larger family units or multigenerational homes, higher population density, more-frequent or more-attended large social gatherings, etc.

Assess

Students' worksheets can be used to assess their learning over the course of the lesson. Alternatively, you can have students write out their scientific explanations for the scenario reviewed in the Reflect section.

Make Career Connections

Discussing or reading about these careers can help students make important connections between the in-class lesson and STEM job opportunities in the real world.

Epidemiologist

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Career Profile

Epidemiologists work to study, understand, and contain outbreaks. They often work in public health roles to help devise and implement procedures, like vaccination campaigns, that prevent outbreaks. When an outbreak occurs, they are on the front lines, dealing with the outbreak and communicating life-saving information to the public, as well as calculating important information like the herd immunity threshold for a disease. Read more

Physician

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Career Profile

Physicians are often the first to recognize an uptick in local cases of a disease, or the emergence of a new disease. They play a big role in sounding the alarm at the beginning of an outbreak, in addition to treating infected patients. Read more

Lesson Plan Variations

• This lesson may be combined with the Preventing Outbreaks with Herd Immunity lesson plan to have students evaluate the mathematical relationship between R₀ and the herd immunity threshold for a disease.
• Several other SimPandemic lesson plans are available.