Summary
Using a website simulation tool, students build on their understanding of random processes on networks to interact with the graph of a social network of individuals and simulate the spread of a disease. They decide which two individuals on the network are the best to vaccinate in an attempt to minimize the number of people infected and "curb the epidemic." Since the results are random, they run multiple simulations and compute the average number of infected individuals before analyzing the results and assessing the effectiveness of their vaccination strategies.Engineering Connection
Simulations of real systems are used throughout science and engineering to test hypotheses, understand the nature of particular problems, and generate effective solutions. For example, biomolecular engineers use computers to extensively simulate complex reaction networks in order to form and test hypotheses about how interactions of certain molecules (such as proteins) in our cells lead to disease (such as cancer). Public health professionals also use computer simulations to provide solutions to the challenge of distributing the smallest possible number of vaccines in order to minimize the number of people falling ill to infectious diseases. For example, when the flu arrives in a highly populated area, distributing a limited number of vaccines in an appropriate manner is key to minimizing distribution costs and alleviating potential vaccine shortages.
Learning Objectives
After this activity, students should be able to:
 Use a website applet to simulate the spread of a disease on a social network of interacting individuals.
 Determine a vaccination strategy that minimizes the number of people infected by the disease.
 Compute an appropriate quantity for evaluating the effectiveness of a vaccination strategy.
More Curriculum Like This
Building on their understanding of graphs, students are introduced to random processes on networks. They walk through an illustrative example to see how a random process can be used to represent the spread of an infectious disease, such as the flu, on a social network of students.
Students learn about complex networks and how to use graphs to represent them. An illustrative example shows how a random process can be used to represent the spread of an infectious disease, such as the flu, on a social network of students, and demonstrates how scientists and engineers use mathemat...
Students learn and apply concepts and methods of graph theory to analyze data for different relationships such as friendships and physical proximity. They are asked about relationships between people and how those relationships can be illustrated.
Educational Standards
Each TeachEngineering lesson or activity is correlated to one or more K12 science,
technology, engineering or math (STEM) educational standards.
All 100,000+ K12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN),
a project of D2L (www.achievementstandards.org).
In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics;
within type by subtype, then by grade, etc.
Each TeachEngineering lesson or activity is correlated to one or more K12 science, technology, engineering or math (STEM) educational standards.
All 100,000+ K12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).
In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.
NGSS: Next Generation Science Standards  Science

Use a computer simulation to model the impact of proposed solutions to a complex realworld problem with numerous criteria and constraints on interactions within and between systems relevant to the problem.
(Grades 9  12 )
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Do you agree with this alignment?
This Performance Expectation focuses on the following Three Dimensional Learning aspects of NGSS:Science & Engineering Practices Disciplinary Core Ideas Crosscutting Concepts Use mathematical models and/or computer simulations to predict the effects of a design solution on systems and/or the interactions between systems. Alignment agreement:
Both physical models and computers can be used in various ways to aid in the engineering design process. Computers are useful for a variety of purposes, such as running simulations to test different ways of solving a problem or to see which one is most efficient or economical; and in making a persuasive presentation to a client about how a given design will meet his or her needs. Alignment agreement:
Models (e.g., physical, mathematical, computer models) can be used to simulate systems and interactions—including energy, matter, and information flows—within and between systems at different scales. Alignment agreement:
Common Core State Standards  Math

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
(Grades 9  12 )
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International Technology and Engineering Educators Association  Technology

Systems, which are the building blocks of technology, are embedded within larger technological, social, and environmental systems.
(Grades 9  12 )
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State Standards
Florida  Math

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
(Grades
9 
12 )
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Do you agree with this alignment?
Florida  Science

Explain how scientific knowledge and reasoning provide an empiricallybased perspective to inform society's decision making.
(Grades
9 
12 )
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National Council of Teachers of Mathematics  Math

use representations to model and interpret physical, social, and mathematical phenomena
(Grades
PreK 
12 )
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Materials List
Each student needs:
 computer with internet access
 Curb the Epidemic Worksheet
 pen or pencil
 (optional) calculator
Introduction/Motivation
(Begin by asking students a few questions about modeling complex networks for infection control and the SIR model, as described in the Assessment section.)
Public policy makers must often make difficult choices when deciding how to use limited resources. Often, no single "obvious" choice exists, and decisions are usually guided by computer simulations.
Today, you will be presented with a social network of individuals and asked to choose which two individuals should get vaccinated for the best public benefit. Then, you will test your choices by running simulations and analyzing the results.
Vocabulary/Definitions
infectious: A student capable of spreading a disease.
probability: A number (between 0 and 1) that tells us how probable the occurrence of an event is, with 0 meaning that the event cannot happen and 1 meaning that the event always happens.
process: A variable that changes with time.
random process : A variable that changes with time, but cannot be completely predicted.
resistant: A student who is immune to future infections of the same disease.
simulation: A calculation of a process using computers.
SIR model: A mathematical model of disease spreading over social networks.
susceptible: A student capable of becoming infected by a disease.
Procedure
Background
In this activity, students simulate how the flu spreads on a social network. To do this, they use the freely accessible interactive application, Spread of Disease on a Social Network, available at this Johns Hopkins University website: http://www.cis.jhu.edu/~goutsias/teachingApplet/webapp.html. A few pointers:
 The interactive applet of a graph represents a social network of 20 individuals (nodes) who interact with each other in a manner determined by the edges (connecting lines).
 Users can examine the interactive graph by clicking and dragging on nodes to see in detail how nodes are connected to each other. The location of the nodes in the graph is arbitrary; all that matters is the existence of the nodes and the edges between the nodes.
 The node color represents the state of that node with respect to infection: susceptible individuals are blue nodes, infectious individuals are red and resistant individuals are green.
 The node labeled "1" starts off as infectious (red); all other nodes start off as susceptible (blue).
 Users enter numbers into the two text fields below the graph to choose which two nodes to vaccinate (that is, make resistant). After examining the node relationships in the graph, students decide on two nodes (that is, fill in the textboxes with their choice of numbers; the default values are 2 and 3).
 Users simulate a single time step by pressing the "Take Timestep" button, which enables visualization of how a disease spreads over a social network. In each time step, infection spreads with probability 0.5, and infected individuals recover with probability 0.25.
 The disease is eradicated when no nodes are red (that is, infectious), meaning no one can become infected anymore.
 Refreshing the web page resets the simulation.
Refer to the associated Processes on Complex Networks lesson for background information on random processes on networks and the SIR (susceptible, infectious, resistant) model.
Before the Activity
 Make copies of the Curb the Epidemic Worksheet.
 Set up enough computers with internet access, either one student per machine, or small groups of students per machine.
With the Students
 Have students each sit at a computer (or divide the class into small groups, each at a computer).
 Hand out the worksheet.
 Direct students to open up the Spread of Disease on a Social Network simulation website at http://www.cis.jhu.edu/~goutsias/teachingApplet/webapp.html.
 Introduce the activity challenge to students: This website simulation tool enables you to interact with the graph of a social network and simulate the spread of a disease. Your challenge: Minimize the number of people who get infected with the flu by examining the social network and mindfully choosing which two of the 20 individuals are the best ones to get vaccinated. After you have run multiple simulations, you'll analyze the results and assess the effectiveness of your vaccination strategies.
 Familiarize students with how the interactive applet works. Point out the simulation's main features and variables. Give them a minute to play with it and reset by refreshing the page.
 Before students start the simulations, pose the following problem to the class: When running a simulation, you need to keep track of the number of infected individuals (red nodes) at each time step in order to record the total number of infected individuals at the end of each simulation. This can be tedious and prone to errors. Does anyone see a quick calculation that can be performed at the end of each simulation to find the total number of individuals infected during the simulation? (Answer: At the end of each simulation only susceptible individuals (blue nodes) and resistant individuals (green nodes) will remain. You know that two individuals are resistant due to vaccination (the ones you chose), while the rest are resistant because they were infected at some point during the simulation. Therefore, you can simply count the number of resistant individuals (all green nodes) at the end of a simulation and subtract 2 to obtain the total number of individuals infected during the simulation.)
 Direct students to run their simulations, using the worksheet to record data and answer questions. They first must decide which two nodes are best to vaccinate in order to reduce the number of individuals infected by the disease. Each student should use his/her same choice of vaccinated individuals to run 10 separate applet simulations, starting from the beginning until the disease is completely eradicated. After each simulation, record in the worksheet table how many nodes became infected during the course of that simulation (that is, once the flu virus runs its course).
 After all students have completed the simulations, ask them to compute the average number of nodes infected in their simulations and record this number on the worksheet.
 As a class, call on the three students with the lowest averages to explain why they chose to vaccinate the nodes they did. (Common successful strategies include: choosing nodes directly connected to the initially infectious individual, choosing nodes with a high degree [that is, with many edges from other nodes], choosing nodes that "block off" other nodes from ever becoming infected.)
 For extra credit (or as time permits), have students investigate which vaccination choices are the least effective. Students may enjoy attempting to maximize the size of the epidemic.
 Have students turn in their worksheets for grading.
 Conclude by leading a class discussion to compare results and conclusions.
Worksheets and Attachments
Assessment
PreActivity Assessment
Opening Questions: Before starting the activity, ask students a few questions to review what they learned in the Processes on Complex Networks lesson.
 What do the terms susceptible, infected, and resistant mean when we are talking about modeling an epidemic?
 What is the SIR model of disease spreading and what is it used for?
 How does the SIR model relate to graphs and complex networks?
 What does it mean to simulate the SIR model?
 What is the purpose of vaccination and how can it be incorporated into the SIR model?
ActivityEmbedded Assessment
Focus: Monitor the students and observe their level of engagement in the simulation activity.
PostActivity Assessment
Worksheet: As students run their simulations, have them complete the Curb the Epidemic Worksheet to record their data and answer questions, including an extra credit run. Review their answers to gauge their mastery of the subject matter.
Analytical Discussion: Ask students to share the strategies they used to choose which two individuals to vaccinate to best meet the challenge (the fewest infected nodes in order to minimize the epidemic). As time permits, have them rerun the activity, making improvements to their approach using successful strategies learned from other students and the class discussion, or applying the lessons learned to the extra credit challenge to maximize the epidemic.
Additional Multimedia Support
This activity uses the freely accessible interactive application, Spread of Disease on a Social Network, available at the Complex Systems Science Laboratory in the Whitaker Biomedical Engineering Institute at The Johns Hopkins University website: http://www.cis.jhu.edu/~goutsias/teachingApplet/webapp.html
Related to the topic, show students a fourminute video by Penn State University researchers called Science Cast: How Easily Do Diseases Spread through a Closed Group of People? on YouTube: https://www.youtube.com/watch?v=5rWKlN_nz5Y
Contributors
Garrett Jenkinson and John Goutsias, The Johns Hopkins University, Baltimore, MD; Debbie Jenkinson and Susan Frennesson, The Pine School, Stuart, FLCopyright
© 2013 by Regents of the University of Colorado; original © 2012 The Johns Hopkins UniversitySupporting Program
Complex Systems Science Laboratory, Whitaker Biomedical Engineering Institute, The Johns Hopkins UniversityAcknowledgements
The generous support of the National Science Foundation, Directorate for Computer and Information Science and Engineering (CISE), Division of Computing and Communication Foundations (CCF), is gratefully acknowledged.
Last modified: August 23, 2017
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