# Exploring Transmission of Infectious Diseases on Networks with NetLogo

## Winfried Just, Hannah Callender Highlander and Drew LaMar

This website contains a collection of materials for teaching network-based models of the transmission of infectious diseases, including:

A more extensive website https://qubeshub.org/iontw that provides access to the latest updates of these materials together with sample solutions of exercises, cross-linking to related readings, user feedback, and other contextual information is available at QUBES Hub. QUBES (Quantitative Undergraduate Biology Education and Synthesis) is an NSF funded program and community designed to support faculty and students in the teaching and learning of quantitative biology. The virtual home for QUBES is QUBES Hub https://qubeshub.org, an infrastructure designed for academic collaboration, storage of and discussion around curricular resources, and use of software tools for classroom use.

## Network-based models of disease transmission

Classical compartment-level models of transmission of infectious diseases are based on the uniform mixing assumption, which means that each host is eaqually likely to make contact with each other host of the population during a given time interval. In contrast, network-based models assume that direct transmission of pathogens is possible only between two hosts (humans, animals, or plants) that are connected by the edge of a graph that models the underlying contact network. For many populations of hosts network models appear closer to biological reality than compartment-level models. It is therefore of interest to study how the structure of the contact network might influence the spread of a disease.

A detailed introduction to compartment-level and network models of disease transmission is given in our book chapters

[1] Winfried Just, Hannah Callender, M. Drew LaMar, and Natalia Toporikova (2015); Transmission of infectious diseases: Data, models, and simulations. In Raina Robeva (ed.), Algebraic and Discrete Mathematical Methods for Modern Biology, Academic Press, 193-215.

[2] Winfried Just, Hannah Callender, and M. Drew LaMar (2015); Disease transmission dynamics on networks: Network structure vs. disease dynamics. In: Raina Robeva (ed.), Algebraic and Discrete Mathematical Methods for Modern Biology, Academic Press, 217-235.

A condensed introduction to network models can be found in Network-based models of transmission of infectious diseases: a brief overview at this web site.

The following presentation gives some highlights:

This poster was presented at the University of Portland's Summer Research Symposium on November 9, 2014, in Portland, Oregon. The poster is aimed at readers with little or no background in modeling infectious diseases. The contents provide a brief overview of modeling infectious diseases on networks, including an introduction to some of the basic properties of contact networks. A discussion is provided on some of the capabilities of IONTW for exploring network-based disease transmission models with NetLogo and how simulations and mathematical theory can be used to explore predictions of such models.

## Modules for exploring network-based models of disease transmission

While these modules are a natural continuation of our book chapters, they can also be used independently based on the background material that is posted here.

### Modules

1. A quick tour of IONTW
Level: Undergraduate students of biology or mathematics.
Includes sample solutions for the exercises.

In this module we guide you through some of the capabilities of IONTW. Highlights include the types of networks supported, setting up various types of models of disease transmission, observing the resulting dynamics, and collecting statistics on the outcomes. Along the way, the module also reviews some basic notions of graph theory.

2. Exploring contact patterns between two subpopulations
Level: Undergraduate students of biology or mathematics.
Includes sample solutions for the exercises.

In this module we introduce a construction of generic random graphs for a given degree sequence or degree distribution and explore whether mixing between hosts who belong to different subpopulations is assortative or disassortative.

3. Exploring Erdős-Rényi random graphs with IONTW
Includes sample solutions for the exercises.

In this module we explore in detail the distribution of the sizes of connected components of Erdős-Rényi random graphs and discover the reasons for the similarities and differences between disease transmission on Erdős-Rényi networks and complete graphs that were observed in the explorations of Module 6 of [2].

4. Exploring random regular graphs with IONTW
Level: Undergraduate students of biology or mathematics for Sections 1 and 3; advanced undergraduate and graduate students of mathematics for optional Section 2.
Sample solutions available upon request via https://qubeshub.org/iontw
Requires: Section 2 of Module Exploring contact patterns between two subpopulations and either Subsection 1.1 of Module Exploring Erdős-Rényi random graphs with IONTW or Module 6 of [2]. The optional Section 2 relies on additional material from Module Exploring Erdős-Rényi random graphs with IONTW.

In this module we introduce and explore the structure of random regular graphs. Moreover, we compare the predictions of SIR-models on random regular contact networks with the predictions of corresponding models on Erdős-Rényi networks.

5. Differential equation models of disease transmission
Sample solutions available upon request via https://qubeshub.org/iontw

In this module we explore ODE models of disease transmission and compare some of their predictions with those of agent-based models. Parts of this material will be referenced in later modules.

6. The replacement number
Sample solutions available upon request via https://qubeshub.org/iontw
Requires: Module Exploring random regular graphs with IONTW. An optional subsection requires basic familiarity with differential equation models as covered in Module Differential equation models of disease transmission.

In this module we introduce the important notion of the replacement number, which generalizes the basic reproductive number R0. We investigate how this number behaves near the start of an outbreak in two types of models: The first type is based on the uniform mixing assumption and the second type assumes a contact network that is a random k-regular graph with small k. We also illustrate a method for estimating the value of R0 from epidemiological data.

Sample solutions available upon request via https://qubeshub.org/iontw
Requires: Module The replacement number.
You also need to download the input file degreesFP.txt that will be used in this module.

In this module we introduce the so-called friendship paradox and illustrate how it affects disease transmission on networks that exhibit this phenomenon.

8. Clustering coefficients
Sample solutions available upon request via https://qubeshub.org/iontw
Requires: Some material from Module A quick tour of IONTW is needed for Sections 2 and 3 that form the core of the module. The motivationg example in Section 1 also draws on knowledge of parts of Modules The replacement number, and especially Exploring random regular graphs with IONTW. One optional exercise in the last section refers to the material in Module The friendship paradox.

In this module we introduce several definitions of so-called clustering coefficients. A motivating example shows how these characteristics of the contact network may influence the spread of an infectious disease. In later sections we explore, both with the help of IONTW and theoretically, the behavior of clustering coefficients for various network types.

9. Exploring distances with IONTW
Sample solutions available upon request via https://qubeshub.org/iontw
Requires: Section 2 references some material from Module Exploring Erdős-Rényi random graphs with IONTW and Module Exploring random regular graphs with IONTW.

Section 1 is purely conceptual and invites readers to critically evaluate popular claims based on Stanley Milgram's famous experiment that gave birth to the phrases small-world property and six degrees of separation. In Section 2 we use IONTW to explore distances between nodes in several types of networks. We also propose a definition of the small-world property that is suitable for classes of disconnected graphs.

10. Small-world models
Sample solutions available upon request via https://qubeshub.org/iontw
Requires: Module Clustering coefficients and Module Exploring distances with IONTW.

Small-world networks are classes of networks that have both the small-world property and exhibit strong clustering. Two constructions of such networks are implemented in IONTW. Here we study, both theoretically and with simulation experiments, the structure of these networks and how it influences effectiveness of a certain vaccination strategy.

11. The preferential attachment model
Sample solutions available upon request via https://qubeshub.org/iontw
Requires: This module is fairly self-contained. Subsection 1.1 as well as Sections 2 and 3 require only basic familiarity with network models of disease transmission and IONTW to the extent covered in Module A quick tour of IONTW. Subsections 1.2 and 1.3 reference the construction given in Section 2 of Module Exploring contact patterns between two subpopulations.

Many empirically studied networks have approximately so-called power-law or scale-free degree distributions. In Section 1 we formally define such distributions and explore some of their properties. We also introduce and briefly compare two methods for constructing random networks with approximately power-law degree distributions: generic scale-free networks and the preferential attachment model. In Sections 2 and 3 we explore disease transmission on networks that are obtained from the preferential attachment model and implications for designing effective vaccination strategies.

12. Exploring generic scale-free networks
Sample solutions available upon request via https://qubeshub.org/iontw
Requires: Module The preferential attachment model.

This module is a companion module to Module The preferential attachment model. Here we study in more detail networks that are generic for a given network size and a given exponent of a power-law degree distribution. We explore predicted structural properties of such networks both mathematically and with IONTW.

13. Mathematical models and theorems
Includes sample solutions for the exercises.
Requires: For the most part accessible to any students with solid mathematical preparation who have done some work with IONTW. Prior knowledge of Module Exploring Erdős-Rényi random graphs with IONTW and Module Differential equation models of disease transmission is recommended.

In this module we introduce and compare various types of deterministic and stochastic mathematical models of disease transmission. We then illustrate how one can derive predictions of these models in the form of mathematical theorems.

14. More modules coming soon with your help! We encourage submissions of additional modules or additions to existing ones from students or colleagues who wish to contribute their ideas to this website.

### Research papers using IONTW

1. Claire Seibold and Hannah L. Callender (2016); Modeling epidemics on a regular tree graph. Letters in Biomathematics 3(1), 59--74 Link to paper

## The IONTW simulation tool

IONTW uses the NetLogo programming language, which was developed by Uri Wilensky at the The Center for Connected Learning and Computer-Based Modeling.

IONTW simulates both discrete and continuous time agent-based models of infectious disease dynamics on networks. It can simulate models of type SEIR, SIR, SEI, SI, SEIS, and SIS.
Supported network types include: complete graphs, empty graphs, Erdős-Rényi, nearest-neighbor (1 and 2 dimensions), small world (1 and 2 dimensions), preferential attachment, generic scale-free, spatially clustered, random regular, trees, as well as approximately uniform realizations from custom degree sequences and custom distributions.
The Reference Guide that is included with the software package gives detailed instruction on how to enforce these and other options.

The following presentation gives a more detailed overview of IONTW's capabilities:

This poster was presented at the workshop "Advances in Discrete Networks" at the University of Pittsburgh, December 12-14, 2014.

### Installation of IONTW1.0

We recommend that for the exercises in the book chapters [1] and [2] Version 1.1 of IONTW be used. It is included in the online appendix of [1].

Subsequent updates of the software and related information will be posted at https://qubeshub.org/tools/iontw, where IONTW can also be run directly in your browser.

NetLogo 5.0.5+: You must download and install this program before using IONTW. Note that you do not have to fill out the short form on the next page in order to download the software.
IONTW 1.0 and supporting materials

Online version of Reference Guide to IONTW