# 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

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

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

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 .

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
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 . 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

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
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 models based on the uniform mixing assumption and in models that assume 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.

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
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 Module 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
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
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
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
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