Part 1 R-Lab DiffEq1 DiffEq2 DiffEq3 DiffEq4

Introduction to SIR Modeling

Exploring Epidemic Models with Visualizations

Part 1A: Introduction

On December 31, 2019, health authorities worldwide received a sobering wake-up call: a mysterious virus had emerged in Wuhan, China, and was beginning to spread. Within weeks, this unidentified threat crossed borders, reaching countries across the globe. Case numbers surged dramatically, overwhelming hospitals, filling intensive care units, and stretching healthcare professionals to their breaking point as they fought to contain the escalating crisis.

When a new virus appears—whether it's a familiar nuisance like the common cold or a worldwide emergency like a pandemic—scientists rely on a critical tool: epidemic modeling. This approach provides answers to pressing questions:

How quickly will this virus spread?
How many people will be affected?
What can we do to slow or stop the outbreak?

Part 1B: What Influences the Spread of a Viral Disease?

Our goal is to explore how the Susceptible-Infectious-Recovered (SIR) model can address the above questions. It is a compartment-based model, meaning each person in the population is considered to be within one of the following three groups:

Susceptible (S): People who haven’t been infected yet.
Infected (I): People currently infected and able to pass the disease on.
Recovered/Removed (R): People who have recovered from the disease or have died and been removed from the population. Either way, they are no longer infected or susceptible to infection.

While simplified, the SIR model serves as the foundation for more complex models used by government organizations to monitor and analyze the progression of a pandemic. It helps predict the trajectory of an epidemic, providing valuable insights that inform decisions to protect public health.

Four key parameters influence how many people move from one of the SIR compartments to another.

Population (N):The total number of people in the population.
(i.e. S + I + R = N)
Initial Infected (I₀):The number of people infected with the disease on the day that the model "starts".
Transmission Rate (β):A measure of how quickly the disease will spread. This is the expected number of susceptible people that each infected person will infect per day.
Recovery Rate (γ):This is the expected proportion of people who are currently infected who will move to the recovered/removed compartment per day. 1/γ is the expected amount of time a person is sick.

In reality, the transmission and recovery rates are unknown, especially at the beginning of an epidemic. Thus, researchers must estimate each of the values. To start exploring this process, we can use the visualization app below to see how these parameters can influence how quickly the virus spreads and the number of people infected.

Part 1C: Visualizing the SIR Model

The simulation below illustrates how SIR models are used to track the spread of disease. At any given time, each person (depicted as a colored dot) is placed into one of the SIR compartments. In this simple model, we start with a population of 100 people. This could represent the number of people on a cruise ship or a small island. Two infected people interact with the 98 susceptible people on the island, and the number of new infections rises quickly.

Click the "Play" button to see how the number of infected people quickly rises and then decreases as more people move to the recovered compartment.

Your goal is to use the model simulation tool to answer some exploratory questions about how an epidemic may spread in practice. Try experimenting with different parameter values in the left-hand panel.

Part 1E: Discussion Questions

The SIR model is shown below.

As discussed in the video above, this model can also be written as a system of equations. These equations were used in creating the SIR Model App shown above. It can be very helpful to create models that allow us to estimate the effects of a virus on the population. However, researchers can not know the actual transmission rate or recovery rates at the beginning of an epidemic.

\( S_{t+1} = S_t - \beta S_t \frac{I_t}{N} \)

\( R_{t+1} = R_t + \gamma I_t \)

\( I_{t+1} = I_t + \beta S_t \frac{I_t}{N} - \gamma I_t \)

In addition to the equations, the SIR model we have been exploring has several assumptions that help us keep the mathematics simpler:

The transmission rate is constant, we assume that everyone in the population is intermixing and everyone has an equal chance of contracting the disease.
The population is constant, no people are entering or leaving the population during the period we are studying.
Everyone who has the disease is sick for the same amount of time.
The virus stays constant, it doesn't mutate in any way.
People can only go from susceptible to infected and from infected to removed/recovered. In other words, once a person is in the removed compartment, they cannot infect others and they cannot contract the disease again.

More complex compartment models are created to address each of these issues. However, it is essential to realize that all models are based on assumptions.

The first assumption may not be very realistic for the COVID-19 pandemic. For example, the transmission rate when people were wearing masks is likely different from that for those who were not wearing masks. People who practiced social distancing likely had a lower transmission rate than others in the population. The answers to questions such as these can have massive implications on the impact of an epidemic. Choose another assumption listed above and discuss why it may not be realistic when modeling the COVID-19 pandemic.
During the COVID-19 pandemic, there was a lot of discussion in the national news media about "flattening the curve." What do you think was meant by this phrase in the context of the SIR model? What "curve" do you think people were referring to? Do you think this was a good description of the problem for a person with no technical understanding of the SIR model? Why or why not?
A key measure used in epidemic modeling is
\( R_0 = \frac\beta\gamma\),
the Reproducibility Number. This is a unitless approximation of the total number of people each infected person will infect. Try various transmission rates and recovery rates in the app above and explain how a higher or lower Ro value will impact an epidemic.
What other questions do you have about understanding what is happening with this modeling process and the tools we are using to explore the model?

Part 1E: Discussion Questions

Part 1F: Can you determine the transmission and recovery rates?

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