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

- Access to and description of our software tool called
**IONTW**, which stands for**I**nfections**O**n**N**e**TW**orks. - A number of modules for active student learning at the level of advanced undergraduate and graduate students of mathematics, biology, or a related field.
- Background readings on 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.

More information about the book and its online supplements can be found at its companion website.

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:

**Poster:** Modeling Infectious Disease through Contact Networks

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.

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.

- How to use these modules
- Network-based models of transmission of infectious diseases: a brief overview
- A brief review of basic probability theory

- 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. - 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.

- Exploring Erdős-Rényi random graphs with
**IONTW**

**Level:**Advanced undergraduate and graduate students of mathematics.

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].

- 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*or Module 6 of [2]. The optional Section 2 relies on additional material from Module**IONTW***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. - Differential equation models of disease transmission

**Level:**Advanced undergraduate and graduate students of mathematics or biology.

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.

- The replacement number

**Level:**Advanced undergraduate and graduate students of mathematics.

Sample solutions available upon request via https://qubeshub.org/iontw

**Requires:**Module*Exploring random regular graphs with*An optional subsection requires basic familiarity with differential equation models as covered in Module**IONTW**.*Differential equation models of disease transmission.*In this module we introduce the important notion of the replacement number, which generalizes the basic reproductive number

*R*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_{0}.*k*-regular graph with small*k.*We also illustrate a method for estimating the value of*R*from epidemiological data._{0} - The friendship paradox

**Level:**Advanced undergraduate and graduate students of mathematics.

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.

- Clustering coefficients

**Level:**Undergraduate and graduate students of mathematics or biology for Sections 1-3, advancd undergraduate and graduate students of mathematics for Section 4.

Sample solutions available upon request via https://qubeshub.org/iontw

**Requires:**Some material from Module*A quick tour of*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**IONTW***The replacement number,*and especially*Exploring random regular graphs with*One optional exercise in the last section refers to the material in Module**IONTW**.*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. - Exploring distances with
**IONTW**

**Level:**Undergraduate students of biology or mathematics for Section 1; advanced undergraduate and graduate students of mathematics for Section 2.

Sample solutions available upon request via https://qubeshub.org/iontw

**Requires:**Section 2 references some material from Module*Exploring Erdős-Rényi random graphs with*and Module**IONTW***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. - Small-world models

**Level:**Advanced undergraduate and graduate students of mathematics.

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. - The preferential attachment model

**Level:**Advanced undergraduate students of biology or mathematics.

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*Subsections 1.2 and 1.3 reference the construction given in Section 2 of Module**IONTW**.*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. - Exploring generic scale-free networks

**Level:**Advanced undergraduate and graduate students of mathematics.

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**. - Mathematical models and theorems

**Level:**Advanced undergraduate and graduate students of mathematics.

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*and Module**IONTW***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.

- 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.

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

**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:

**Poster:** Exploring Disease Transmission On Networks with **IONTW**

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

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.

Online version of Reference Guide to **IONTW**

© 2014 Winfried Just, Hannah Callender Highlander, M. Drew LaMar Last modified March 21, 2017.