Presentations included in a minisymposium at the Eighth International Symposium on BEER, Illinois... more Presentations included in a minisymposium at the Eighth International Symposium on BEER, Illinois State University, Normal, IL
In this module1 we introduce the important notion of the replacement number, which generalizes th... more In this module1 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 kregular graph with small k. We also illustrate a method for estimating the value of R0 from epidemiological data.
Press New to initialize a next-generation SIR-model on a network GReg(200, 4) with one index case... more Press New to initialize a next-generation SIR-model on a network GReg(200, 4) with one index case in an otherwise susceptible population. Press Metrics to verify that R0 = 2. By the result of our module The replacement number at this website we should have Rst t ≈ 1.5 > 1 for sufficiently small positive t. Thus we would expect to see a significant proportion of major outbreaks in addition to some minor ones. You may want to run a few exploratory simulations to check whether this is what you will see in the World window and the Disease Prevalence plot.
This short activity introduces a discussion of reproducibility in scientific research and ways th... more This short activity introduces a discussion of reproducibility in scientific research and ways that we might address these ideas with undergraduates. There is an opportunity to use a prototype version of Serenity to explore BIRDD data.
Collection of presentations that were part of a minisymposium at the Seventh International Sympos... more Collection of presentations that were part of a minisymposium at the Seventh International Symposium on Biomathematics and Ecology: Education and Research in 2014.
Why do your friends have more friends than you do? The question may sound offensive. We don’t eve... more Why do your friends have more friends than you do? The question may sound offensive. We don’t even know you. How can we assume than you have fewer friends than your friends have on average? Because most people do. This so-called friendship paradox has first been described and studied in [1]. It does seem counterintuitive: If we are talking about the average number of friends of average friends of an average person, shouldn’t this average out to the average number of friends of an average person? Enough loose talk about averages that makes the average person’s head spin. Let’s steady our thoughts with some solid mathematical definitions. Consider a graph G that represents friendships between persons numbered 1, . . . , N . The degree ki of node i represents the number of i’s friends. The “average” number of friends of a randomly chosen person can be most naturally interpreted as the mean degree 〈k〉 that is given by
where qk is the probability that a randomly chosen node has degree k, and cγ and γ are positive c... more where qk is the probability that a randomly chosen node has degree k, and cγ and γ are positive constants. As this formula makes sense only for k > 0, we will tacitly assume that the graph contains no isolated nodes, that is, q0 = 0. Graphs with a power-law distribution of degrees are often called scale-free networks. This phrase needs to be handled with care. Since qk > 0 for all k > 1, Equation (1) could be literally true only if there were infinitely many nodes in the graph.2 For graphs with finitely many nodes, (1) can be satisfied only approximately. If this is the case, we will write that the graph is approximately a scale-free network. A number of constructions of approximately scale-free networks have been proposed in the literature. The preferential attachment model of Barabási and Albert [2] was studied in some detail in our module [4]. Here we focus on some properties that are common to all types of approximately scale-free networks and on properties of generic s...
This module has two parts. The first part is purely conceptual and invites readers to critically ... more This module has two parts. The first part 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 the second part we use IONTW to explore distances between nodes in several types of networks. We also explore various possible formal definitions of the small-world property and propose one that is suitable for classes of disconnected networks. The parts are somewhat independent of each other, but we recommend that the reader work through both parts in the order given here. The exercises within each part should definitely not be attempted out of sequence.
The main purpose of this document is to give a brief but mathematically rigorous description of t... more The main purpose of this document is to give a brief but mathematically rigorous description of the network-based models of transmission of infectious diseases that are studied on this web site1. Readers will be able to find a much more detailed development of this material in our book chapters [5, 6]. We also briefly describe how the network-based models that are defined here are related to compartment-level models that are used in most on the literature on mathematical epidemiology.
What is an Educational Gateway? The Quantitative Undergraduate Biology Education & Synthesis (QUB... more What is an Educational Gateway? The Quantitative Undergraduate Biology Education & Synthesis (QUBES; qubeshub.org) project has adopted a “scientific gateways” model [1] to accelerate undergraduate biology education reform. As such, QUBES provides an accessible and integrated cyberinfrastructure that makes it possible to coordinate and streamline the work of a diverse and distributed community of biology educators. The QUBES services include an online professional development model (faculty mentoring networks – FMNs), an open educational resources publication and versioning platform, diverse types of community hosting, workshop support, and access to cloud-based computational resources. The integration of these functionalities within a single gateway provides important opportunities for both individual faculty and education projects to engage with the professional community and amplify their scholarship. We argue that professional participation through a scientific gateway reflects a...
Why do your friends have more friends than you do? The question may sound offensive. We don’t eve... more Why do your friends have more friends than you do? The question may sound offensive. We don’t even know you. How can we assume than you have fewer friends than your friends have on average? Because most people do. This so-called friendship paradox has first been described and studied in [1]. It does seem counterintuitive: If we are talking about the average number of friends of average friends of an average person, shouldn’t this average out to the average number of friends of an average person? Enough loose talk about averages that makes the average person’s head spin. Let’s steady our thoughts with some solid mathematical definitions. Consider a graph G that represents friendships between persons numbered 1, . . . , N . The degree ki of node i represents the number of i’s friends. The “average” number of friends of a randomly chosen person can be most naturally interpreted as the mean degree 〈k〉 that is given by
After clicking New you will see a picture of the complete graph K5 in the World window. In this g... more After clicking New you will see a picture of the complete graph K5 in the World window. In this graph each node i has degree ki = 4; it is a 4-regular graph. More generally, for every N the complete graph KN with N nodes is N − 1-regular. Now choose network-type → Empty Graph and click New again. You will see the empty graph K5. Each node in an empty graph has degree 0. Empty graphs are 0-regular. For a third example, choose network-type → Nearest-neighbor 1 num-nodes: 10 d: 2
As we explained in the brief overview of network-based models of transmission of infectious disea... more As we explained in the brief overview of network-based models of transmission of infectious diseases at this web site, for most populations of hosts the actual contact network is not known, and we want to model it as a random graph. There are various constructions of such random graphs. They give us networks that usually share some, but not all properties of real contact networks. The most basic of these constructions gives Erdős-Rényi random graphs, named after the two Hungarian mathematicians who first systematically explored these graphs in the seminal paper [1]. These graphs serve as a benchmark against which all other constructions of random networks can be compared.
Many empirically studied networks have approximately so-called power-law or scale-free degree dis... more 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.
Poster on the use, adaptation, and reuse of Open Educational Resources within the QUBES community... more Poster on the use, adaptation, and reuse of Open Educational Resources within the QUBES community at OpenEd 2018
Presentations included in a minisymposium at the Eighth International Symposium on BEER, Illinois... more Presentations included in a minisymposium at the Eighth International Symposium on BEER, Illinois State University, Normal, IL
In this module1 we introduce the important notion of the replacement number, which generalizes th... more In this module1 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 kregular graph with small k. We also illustrate a method for estimating the value of R0 from epidemiological data.
Press New to initialize a next-generation SIR-model on a network GReg(200, 4) with one index case... more Press New to initialize a next-generation SIR-model on a network GReg(200, 4) with one index case in an otherwise susceptible population. Press Metrics to verify that R0 = 2. By the result of our module The replacement number at this website we should have Rst t ≈ 1.5 > 1 for sufficiently small positive t. Thus we would expect to see a significant proportion of major outbreaks in addition to some minor ones. You may want to run a few exploratory simulations to check whether this is what you will see in the World window and the Disease Prevalence plot.
This short activity introduces a discussion of reproducibility in scientific research and ways th... more This short activity introduces a discussion of reproducibility in scientific research and ways that we might address these ideas with undergraduates. There is an opportunity to use a prototype version of Serenity to explore BIRDD data.
Collection of presentations that were part of a minisymposium at the Seventh International Sympos... more Collection of presentations that were part of a minisymposium at the Seventh International Symposium on Biomathematics and Ecology: Education and Research in 2014.
Why do your friends have more friends than you do? The question may sound offensive. We don’t eve... more Why do your friends have more friends than you do? The question may sound offensive. We don’t even know you. How can we assume than you have fewer friends than your friends have on average? Because most people do. This so-called friendship paradox has first been described and studied in [1]. It does seem counterintuitive: If we are talking about the average number of friends of average friends of an average person, shouldn’t this average out to the average number of friends of an average person? Enough loose talk about averages that makes the average person’s head spin. Let’s steady our thoughts with some solid mathematical definitions. Consider a graph G that represents friendships between persons numbered 1, . . . , N . The degree ki of node i represents the number of i’s friends. The “average” number of friends of a randomly chosen person can be most naturally interpreted as the mean degree 〈k〉 that is given by
where qk is the probability that a randomly chosen node has degree k, and cγ and γ are positive c... more where qk is the probability that a randomly chosen node has degree k, and cγ and γ are positive constants. As this formula makes sense only for k > 0, we will tacitly assume that the graph contains no isolated nodes, that is, q0 = 0. Graphs with a power-law distribution of degrees are often called scale-free networks. This phrase needs to be handled with care. Since qk > 0 for all k > 1, Equation (1) could be literally true only if there were infinitely many nodes in the graph.2 For graphs with finitely many nodes, (1) can be satisfied only approximately. If this is the case, we will write that the graph is approximately a scale-free network. A number of constructions of approximately scale-free networks have been proposed in the literature. The preferential attachment model of Barabási and Albert [2] was studied in some detail in our module [4]. Here we focus on some properties that are common to all types of approximately scale-free networks and on properties of generic s...
This module has two parts. The first part is purely conceptual and invites readers to critically ... more This module has two parts. The first part 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 the second part we use IONTW to explore distances between nodes in several types of networks. We also explore various possible formal definitions of the small-world property and propose one that is suitable for classes of disconnected networks. The parts are somewhat independent of each other, but we recommend that the reader work through both parts in the order given here. The exercises within each part should definitely not be attempted out of sequence.
The main purpose of this document is to give a brief but mathematically rigorous description of t... more The main purpose of this document is to give a brief but mathematically rigorous description of the network-based models of transmission of infectious diseases that are studied on this web site1. Readers will be able to find a much more detailed development of this material in our book chapters [5, 6]. We also briefly describe how the network-based models that are defined here are related to compartment-level models that are used in most on the literature on mathematical epidemiology.
What is an Educational Gateway? The Quantitative Undergraduate Biology Education & Synthesis (QUB... more What is an Educational Gateway? The Quantitative Undergraduate Biology Education & Synthesis (QUBES; qubeshub.org) project has adopted a “scientific gateways” model [1] to accelerate undergraduate biology education reform. As such, QUBES provides an accessible and integrated cyberinfrastructure that makes it possible to coordinate and streamline the work of a diverse and distributed community of biology educators. The QUBES services include an online professional development model (faculty mentoring networks – FMNs), an open educational resources publication and versioning platform, diverse types of community hosting, workshop support, and access to cloud-based computational resources. The integration of these functionalities within a single gateway provides important opportunities for both individual faculty and education projects to engage with the professional community and amplify their scholarship. We argue that professional participation through a scientific gateway reflects a...
Why do your friends have more friends than you do? The question may sound offensive. We don’t eve... more Why do your friends have more friends than you do? The question may sound offensive. We don’t even know you. How can we assume than you have fewer friends than your friends have on average? Because most people do. This so-called friendship paradox has first been described and studied in [1]. It does seem counterintuitive: If we are talking about the average number of friends of average friends of an average person, shouldn’t this average out to the average number of friends of an average person? Enough loose talk about averages that makes the average person’s head spin. Let’s steady our thoughts with some solid mathematical definitions. Consider a graph G that represents friendships between persons numbered 1, . . . , N . The degree ki of node i represents the number of i’s friends. The “average” number of friends of a randomly chosen person can be most naturally interpreted as the mean degree 〈k〉 that is given by
After clicking New you will see a picture of the complete graph K5 in the World window. In this g... more After clicking New you will see a picture of the complete graph K5 in the World window. In this graph each node i has degree ki = 4; it is a 4-regular graph. More generally, for every N the complete graph KN with N nodes is N − 1-regular. Now choose network-type → Empty Graph and click New again. You will see the empty graph K5. Each node in an empty graph has degree 0. Empty graphs are 0-regular. For a third example, choose network-type → Nearest-neighbor 1 num-nodes: 10 d: 2
As we explained in the brief overview of network-based models of transmission of infectious disea... more As we explained in the brief overview of network-based models of transmission of infectious diseases at this web site, for most populations of hosts the actual contact network is not known, and we want to model it as a random graph. There are various constructions of such random graphs. They give us networks that usually share some, but not all properties of real contact networks. The most basic of these constructions gives Erdős-Rényi random graphs, named after the two Hungarian mathematicians who first systematically explored these graphs in the seminal paper [1]. These graphs serve as a benchmark against which all other constructions of random networks can be compared.
Many empirically studied networks have approximately so-called power-law or scale-free degree dis... more 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.
Poster on the use, adaptation, and reuse of Open Educational Resources within the QUBES community... more Poster on the use, adaptation, and reuse of Open Educational Resources within the QUBES community at OpenEd 2018
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