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

Disease is an important ecological force that shapes population dynamics, species interactions, and conservation outcomes. Just like predators or competition, pathogens can limit population growth and even trigger dramatic declines. Ecologists use models to study these processes because models allow us to test how diseases might spread under different scenarios and evaluate the potential impact of interventions like vaccination. 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.  

The SIR model is one of the most widely used frameworks for studying disease dynamics. It groups individuals into three categories:

• Susceptible (S) – individuals that can be infected

• Infected (I) – individuals that currently have and can spread the disease

• Recovered (R) – individuals that are no longer susceptible, either because they survived infection or were vaccinated

In many disease models, especially the ones used in this lab, the category R is labeled as “Removed” rather than “Recovered.” This is because the model isn’t only tracking individuals who recover with immunity. Instead, “Removed” includes anyone who is no longer able to spread the disease, either because:

• They recovered and gained immunity,

• They were vaccinated and became immune, or

• They died from the disease and are no longer part of the susceptible or infectious pool.

By tracking how individuals move among these categories, scientists can predict how fast a disease spreads, how long an outbreak lasts, and how many individuals remain uninfected. The movement of individuals between categories can be modeled and plotted using the differential equations below:

Where:

• β is the transmission rate

• γ is the recovery rate

Several strategies can reduce disease spread in both human and wildlife populations:

• Quarantines – isolating infected individuals so they cannot transmit the disease further.

• Contact tracing – identifying and monitoring individuals who were exposed to an infected individual to stop chains of transmission early.

• Vaccination – moving individuals directly into the “Recovered” (immune) category, lowering the proportion of susceptible individuals and reducing outbreak potential.

• Herd immunity – achieved when enough of the population is immune (through vaccination or prior infection) that disease transmission can no longer sustain itself.

A key concept that links all these strategies is the basic reproduction number (R₀), which represents the average number of new infections caused by a single infected individual in a fully susceptible population. R₀ can be calculated as 

• If R₀ > 1, the disease can spread and cause an outbreak.

• If R₀ = 1, transmission will be stable in the population

• If R₀ < 1, the outbreak will fade out

Interventions like vaccination and quarantine work by reducing the effective reproduction number until it falls below 1.

This concept can be illustrated with a real world example. In 1971, African swine fever spread to Cuba, threatening both food security and the economy (Simeón-Negrín & Frías-Lepoureau, 2002). Because there was no vaccine and the virus spread rapidly among pigs, officials resorted to culling (burning and burying) nearly half a million pigs. This drastic step reduced the number of susceptible hosts and ultimately helped contain the outbreak. While devastating, it demonstrates the same principle modeled in the SIR framework: reducing the susceptible pool can halt transmission.

A related concept in the herd immunity threshold (HIT), or the proportion of the population that must be immune (through vaccines or prior infection) to prevents sustained disease spread. HIT is equal to one minus the reciprocal of the basic reproduction number:

For example, if R₀ = 3, then:

This means that about two thirds of the population must be immune to stop the outbreak.

In Part 1 of this lab, you will simulate an outbreak in a small sample population to build intuition about transmission, recovery, and vaccination. In Part 2, you’ll apply these ideas to a case study of CDV in wild foxes, a virus that has caused major population crashes in carnivores worldwide. You will test how contact rates (driven by population density) and vaccination strategies change the course of an epidemic and explore the concept of herd immunity.

While the models used here are simplified, they highlight fundamental principles of disease ecology. By the end of the lab, you should be able to connect disease dynamics to ecological theory, understand how density and immunity influence outbreaks, and appreciate the challenges of managing disease in wildlife populations.

Part 2: Canine Distemper in Wild Foxes

Canine distemper virus (CDV) is a highly contagious disease that spreads among carnivores like foxes, raccoons, and wolves. Infected animals show symptoms such as fever, neurological distress, and often death. CDV has caused major population crashes in wildlife. Vaccination through oral baits is one method scientists have tried in endangered populations.

You will use the  BioInteractive Outbreak Simulator to model a fox population experiencing a CDV outbreak, and test how different population densities and vaccination levels affect the outcome. To begin we will simulate a baseline scenario of high-density population in the absence of vaccination.

Simulation 1: Baseline

1. Click on the SIR Model Advanced tab and read through the differences between the basic and advanced model.

2. Scroll down and enter the following parameters into the simulator:

   • Total days = 100

   • Transmission rate = 100

   • Recovery rate = 10

   • Initial susceptible individuals = 999

   • Initial infected = 1

   • Initial removed = 0

3. Run the simulation. Download the graph for submission, and record the following data:

   • Day of peak infection

   • Peak number of infectious individuals

   • Total removed at the end of the simulation

   • Number of susceptible individuals at the end of the simulation

Simulation 2: Effect of population density/transmission rate

In ecology, population density refers to how many organisms live in a given area. Density influences how often individuals in a population come into contact,  which in turn affects disease transmission. For example, in a low-density fox population, we would expect less interaction and slower disease spread; in a high-density population, we would expect crowding and fast spread. Simulate each of these scenarios with the parameter values below (leave other parameter values the same as in Simulation 1 above).

4. Low-density scenario: Transmission rate = 20

   • Download the graph for submission and recored the following data

     • Day of peak infection

     • Peak number of infectious individuals

5. High-density scenario: Transmission rate = 150

   • Download the graph for submission and recored the following data

   • Day of peak infection

   • Peak number of infectious individuals

6. Describe how the shapes of the curves differ between the two scenarios. Why?

Simulation 3: Effect of vaccination

Vaccines are not just for humans. Vaccination programs can also be deployed in wildlife populations to keep diseases in check. Vaccination effectively removes a portion of individuals from the susceptible pool, without them ever becoming infected. Accordingly, we will adjust the Initial Removed parameter to simulate vaccination in our model. 

Successful vaccination should enable populations to achieve herd immunity faster, with fewer fatalities.

7.  Calculate the Herd Immunity Threshold (HIT) when the transmission rate is 100% and the recovery rate 10%.

Simulate each of the scenarios below with the listed parameter values (For all simulations, Transmission rate = 100% and Recovery rate = 10%).

8. Low vaccination (20% scenario): Initial Susceptible Individuals = 799, Initial Removed Individuals = 200

   • Download the graph for submission and recored the following information

     • Did an outbreak occur?

     • Day of peak infection?

     • Day HIT was passed?

9. Medium vaccination (50% scenario): Initial Susceptible Individuals = 499, Initial Removed individuals = 500

   • Download the graph for submission and recored the following information

     • Did an outbreak occur?

     • Day of peak infection?

     • Day HIT was passd?

10. High vaccination (80% scenario): Initial Susceptible Individuals = 199, Initial Removed individuals = 800

    • Download the graph for submission and recored the following information

      • Did an outbreak occur?

      • Day of peak infection?

      • Day HIT was passed?

Reflection Questions

11. Canine distemper has a high fatality rate. How would the outcomes vary for a less lethal disease, like canine coronavirus or seasonal flu?

12. The simple SIR model used here does not distinguish between individuals that are removed from the susceptible population because they attained immunity (through vaccination or exposure to the disease) and individuals that were removed because they died. How could we modify the model to account for this difference? What additional parameter(s) would we need to add?

13. Based on your simulation, what population factors (like density or immunity levels) have the biggest impact on preventing an outbreak?

14. What are some challenges of vaccinating wild animals in the real world? How do you expect those challenges to affect disease control?

15. If vaccines were not available, what recommendations would you make as wildlife ecologist trying to manage CDV in an endangered fox population?