Exploring Transmission of Infectious Diseases on Networks with NetLogo
Modules for exploring networkbased models of disease transmission
Modules for exploring networkbased 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.
Background material
 How to use these modules
 Networkbased models of transmission of infectious diseases: a brief overview
 A brief review of basic probability theory
Modules
 A quick tour of IONTW
Level: Undergraduate students of biology or mathematicsIn 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 mathematicsIn 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 ErdősRényi random graphs with IONTW
Level: Advanced undergraduate and graduate students of mathematicsIn this module we explore in detail the distribution of the sizes of connected components of ErdősRényi random graphs and discover the reasons for the similarities and differences between disease transmission on ErdősRényi networks and complete graphs that were observed in the explorations of Module 6 of [1].

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 ErdősRényi random graphs with IONTW or Module 6 of [1]. The optional Section 2 relies on additional material from Module Exploring ErdősRényi random graphs with IONTW.In this module we introduce and explore the structure of random regular graphs. Moreover, we compare the predictions of SIRmodels on random regular contact networks with the predictions of corresponding models on ErdősRényi networks.

Differential equation models of disease transmission
Level: Advanced undergraduate and graduate students of mathematics or biology.In this module we explore ODE models of disease transmission and compare some of their predictions with those of agentbased models. Parts of this material will be referenced in later modules.

The replacement number
Level: Advanced undergraduate and graduate students of mathematics.
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 R_{0}. 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 kregular graph with small k. We also illustrate a method for estimating the value of R_{0} from epidemiological data.

The friendship paradox
Level: Advanced undergraduate and graduate students of mathematics.
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 socalled 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 13, advancd undergraduate and graduate students of mathematics for Section 4.
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 socalled 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.
Requires: Section 2 references some material from Module Exploring ErdősRényi 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 smallworld 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 smallworld property that is suitable for classes of disconnected graphs.

Smallworld models
Level: Advanced undergraduate and graduate students of mathematics.
Requires: Module Clustering coefficients and Module Exploring distances with IONTW.Smallworld networks are classes of networks that have both the smallworld 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.
Requires: This module is fairly selfcontained. 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 socalled powerlaw or scalefree 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 powerlaw degree distributions: generic scalefree 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 scalefree networks
Level: Advanced undergraduate and graduate students of mathematics.
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 powerlaw 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.
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 ErdősRényi 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.

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