Modeling wildlife disease spread with the SIR model

Overview

In this lab, you'll explore how infectious diseases spread through animal populations. We begin with a hands-on simulation of a small outbreak to help you visualize disease transmission. Then, you’ll investigate a real-world case study: the spread of canine distemper virus (CDV) in wild fox populations. You’ll use a professional disease simulator based on the SIR model to make and test predictions about how population density and vaccination affect disease spread.

Background Reading

Scitable: The origins of viruses

Objectives

Students should be able to

Describe how models may use simple assumptions to simulate population dynamics
Discuss the basic pieces of a transmission model
Use simulations to explore how various scenarios impact virus spread

Introduction

Although we often don’t think about it, humans are a part of the natural world. One unfortunate example of this is the use of models to predict disease dynamics. While epidemiologists and public health experts use specific models to predict spread and impacts of diseases on human communities, these models share a common conceptual background with species interactions models and are very similar to models used to predict disease dynamics in other organisms.

Regardless of the species (or even the interaction), modeling interactions is useful for a number of reasons. Modeling species interactions can help predict the growth of various populations. In the case of diseases, models can predict spread under various scenarios and allow potential impacts of different mitigation strategies to be compared. While models are never perfect, they also force scientists to clearly state their assumptions so they can be tested, identify data and knowledge gaps, and may help in identifying unforeseen interactions among various factors.

Models can be developed in several ways. Some are mechanistic, meaning they develop basic interactions among various groups or agents and then parameterize (give a value to) those relationships using available data. For example, today will be using a derivative of a SIR model, which is a transmission model for disease. These models consider disease dynamics by dividing the population into susceptible (S, those that can catch the disease), infected (I, those that have and can spread the disease), and recovered (R, those that are immune to the disease since they have survived an infection). These groups correspond to a basic understanding of disease transmission and immune systems. When a new disease enters a population (e.g., via transmission from another group or another species (zoonotic diseases for humans)), the infection will spread as infected individuals come into contact with susceptible individuals. Models must parameterize how likely the disease is to “jump” from an infected to a susceptible individual. Infected individuals may suffer mortality (another value to parameterize), but those that survive transition to the recovered group are no longer susceptible. This is because immune systems often work by having general (e.g. inflammation) and specific (i.e. disease-specific) responses. Disease-specific responses develop after an organism is exposed to a disease agent and their body learns to identify and respond to it specifically, often limiting the impact of future exposure.

Notice our basic model makes several assumptions. We have no variation within our 3 groups (all infected individuals are the same), so we can’t consider how disease transmission or impacts differ by age, gender, or other factors (like jobs, which may impact probability of exposure!). We also assume all recovered individuals are immune to subsequent infections, when in fact infection may only lower your chances of subsequent infections and these effects may degrade over time. On a practical note, in the model we can also tell the difference among these groups; that may not be true for a disease where some infected individuals are asymptomatic or for a disease that is contagious before hosts show symptoms.

While these are major assumptions, simple models like these are tractable and often produce useful results. This type of model may be deterministic (based solely on equations,so you get the same result every time) or stochastic (randomness in the model makes each “run” a little different). An example of a stochastic implementation is shown in this simulation from the Washington Post. It models disease (in this case, coronavirus) spread by showing agents (dots, or organisms) bouncing around an enclosed area (a community) and transitioning from healthy to infected to recover.

Since the movement of the dots may be random, every simulation run will be slightly different. However, the average, or typical, outcome will be the same.

This model leads to several useful results (and explanations of impacts). First, we could (but we won’t ) set up and solve a system of equations related to this model (that just put what we noted before in math terms) and find that disease agents must balance the ability to spread with mortality of their host. This is due to the fact that the infected population will only grow if the density of susceptible hosts exceeds a critical, threshold density. In the example above, if there are too few susceptible individuals, the infected individuals will never encounter them! This explains how vaccines work and why infections tend to decrease over time. Over time exposure effectively moves individuals from the susceptible to recovered group, lowering the density of susceptible individuals and thus chances for the disease to spread. Quarantines work in similar ways - by isolating infected individuals, they are less likely to infect others, and the number of infected individuals should decrease.

Quarantines bring up several interesting points that you can explore in the Post's simulation. If they are extremely effective (e.g., contact tracing works) then few people get the disease. However, this also means most of the community remains susceptible, which may be an issue if the disease later enters the community again (i.e., from a traveler!). You can also observe the effects of social distancing (when dots are less likely to interact with others).

Other models ignore the mechanisms behind the disease and instead focus on using equations to fit the observed data. These curve-fitting models extrapolate current trends to the future using a variety of techniques and equations. Models can also be a hybrid of these approaches.

Today we will use another model to consider coronavirus spread. Viruses are infectious agents that, unlike bacteria or other microbes, only replicate in the cells of other living organisms. For this reason they are sometimes argued to not really even be alive, and are sometimes described as replicators. Viruses contain genetic material contained in a protein capsid; some may also have a lipid shell. They infect living cells by injecting their genetic material and then using the cell's own machinery to produce new copies. In this way they are obligate intracellular parasites.

The model we will be using today is a derivative of the SIR model. It differs by adding “exposed” individuals. These individuals have been exposed to an infected individual but may not have enough of the disease agent to actually show symptoms or be contagious(high load). They are effectively transitioning among the susceptible and infected states. This added states slow the growth of the infected population but also may lead to some mitigation strategies (like quarantines) being harder to implement.

PLEASE NOTE THIS IS A LAB DEMONSTRATION TO AID YOU IN UNDERSTANDING DISEASE DYNAMICS. MANY MORE SOPHISTICATED MODELS AND UPDATED DATA ARE EMPLOYED TO SET REGULATIONS AND RECOMMENDATIONS! THE MODEL ALSO ONLY CONSIDERS VIRUS SPREAD AND NOT ASSOCIATED HEALTH, ENVIRONMENTAL, OR ECONOMIC IMPACTS.

Disease dynamics (student)

Methods

Model Exploration

First let's get to know our model. Click on the Background Information to learn more about COVID-19. Read through each statement and click on the ? to the right of each statement for more information.

Are you able to briefly summarize the background information?

Now click on Model Overview. This guides you through the boundaries of the simulation and thus includes some important assumptions. Read through each model assumption and click on the ? to the right of each statement for more information.

Can you briefly summarize the assumptions in the Model Overview section?

Next click on Disease Assumptions. Click on Rest to Default Assumptions. Notice that this tab includes parameter values.Look at each tab (Severity and Infectivity, Behavior, Duration, Mortality). Be sure to click the ? next to each assumption/parameter.

Can you briefly summarize the assumptions in the Disease Assumptions section?

Look at the Infection Curve graph.

What is on the x-axis?
What is on the y-axis?
What do the colors represent?

Look at the Population Status graph.

What is on the x-axis?
What is on the y-axis?
Why are there so many different numbers on the y-axis?
What is the maximum number of presumed new infections (estimate)?

Look at the Cumulative Measures graph.

What is on the x-axis?
What is on the y-axis?
Why are there so many different numbers on the y-axis? How many total people were tested by day 292?
Can you explain the relationship among these three graphs? Hint: It's the same data but graphed in different ways - how?

Next click on Policies and Results. This is where you can modify responses to the virus. This includes

Quarantine start date
- Our model runs for a year, and the virus enters on day one (in one carrier). This toggle allows you to institute a global (for everyone) quarantine. Note when you slide the bar it shows the start date in days since beginning of simulation.
Quarantine duration
- This is how the quarantine lasts. Note when you slide the bar it shows the duration in days.
Quarantine effectiveness
- This ranges from 0 (nothing changes in behavior) to 1 (a perfect quarantine where no contact occurs)
Contact tracing effectiveness
- If you implement a contact tracing program, this notes how well the test does at identifying positive (exposed or infected) individuals.
Symptomatic test rate
- This is the number of tests per day each sick person receives. If this number is less than 1 it implies it takes multiple days to get a test.
Daily testing capacity
- How many tests can be run each day. Note this determines if test rates can be met.

The model has many more variables you could change, like mortality rate (Disease assumptions > Mortality), how long each stage of infection lasts (Disease assumptions > Duration), and how people who are infected change their behavior (Disease Assumptions > Behavior tab). Some of these interact while others don’t. For example, the global quarantine is different from how individuals respond to being sick. We’ll next explore some example scenarios. To compare them, we should leave the other assumptions the same among all simulations.

Example Scenarios

Click on The COVID-19 Simulator title at the top of the simulator and select the Do Nothing scenario.

Go to Policies and Results. Describe why the parameter values listed represent ‘Do Nothing’.
Describe what happens to the number of Sick individuals over time with the default parameters. When is the peak number of infected individuals? What is that peak?

Click on The COVID-19 Simulator title at the top of the page and select the Effective Quarantine scenario.

Go to Policies and Results. Describe why the parameter values listed represent ‘Effective Quarantine’.
At the maximum, are there more people too ill to work or who need hospitalization? How do you know?

Click on The COVID-19 Simulator title at the top of the page and select the Additional Testing scenario.

Go to Policies and Results. Describe why the parameter values listed represent ‘Additional Testing’.
How many people were sick when the maximum number of people were sick? (You can and should interact with the graph. Be careful, the answer is not 90.8K.)

Click on The COVID-19 Simulator title at the top of the page and select the Containment scenario.

Go to Policies and Results. Describe why the parameter values listed represent ‘Containment’.
Explain why the Infection Curve looks the way that it does. Do you think this graph is an effective way to display the data? Why or why not.

Compare the outcomes from each four strategies. Support your conclusions below with graphs (that have clear captions and are referenced in your text).

How do the Infection Curves vary for each of the four strategies? Consider Hospitalized, Sick, Healthy, Dead, and Recovered.
How do the Population Status graphs vary for each of the four strategies? Specifically consider Too Ill to Work, and Needing Hospitalization.
How do the number of Cumulative Deaths, Hospital Bed Days, and Cumulative Testing vary for each of the four strategies?
What are the overall important takeaways from comparing the four strategies?

Strategy Design

Design a strategy to reduce or eliminate the virus and its impacts.

Describe your strategy.
Describe how you implemented your strategy (what parameter values did you use).
How did your strategy compare to the four strategies you explored above? You should include screenshots of graphs that support your conclusions.
How realistic (given real world constraints) is your strategy compared to the four strategies you explored above?

Review Questions or Points to ponder (optional)

How did your strategy compare to the four strategies you explored above? You should include screenshots of graphs that support your conclusions.
How realistic (given real world constraints) is your strategy compared to the four strategies you explored above?
What are the tradeoffs among the strategies you explored and developed? What other factors would you like to see included in comparing strategies?

References

Adapted from

Hews, S. (2020). COVID-19 Simulator (ISEE) Module. Interdisciplinary Mathematics Education, QUBES Educational Resources. doi:10.25334/GA88-NM81

Google Sites

Report abuse