Mathematics of Epidemics on Networks
From Exact to Approximate Models
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Article
Approximating Quasi-Stationary Behaviour in Network-Based SIS Dynamics
Deterministic approximations to stochastic Susceptible–Infectious–Susceptible models typically predict a stable endemic steady-state when above threshold. This can be hard to relate to the underlying stochasti...
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Article
Open AccessThe Impact of Contact Structure and Mixing on Control Measures and Disease-Induced Herd Immunity in Epidemic Models: A Mean-Field Model Perspective
The contact structure of a population plays an important role in transmission of infection. Many ‘structured models’ capture aspects of the contact pattern through an underlying network or a mixing matrix. An ...
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Article
Open AccessA monotonic relationship between the variability of the infectious period and final size in pairwise epidemic modelling
For a recently derived pairwise model of network epidemics with non-Markovian recovery, we prove that under some mild technical conditions on the distribution of the infectious periods, smaller variance in the...
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Chapter and Conference Paper
Fast Variables Determine the Epidemic Threshold in the Pairwise Model with an Improved Closure
Pairwise models are widely used to model epidemic spread on networks. This includes the modelling of susceptible-infected-removed (SIR) epidemics on regular networks and extensions to SIS dynamics and contact ...
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Article
Open AccessEdge-Based Compartmental Modelling of an SIR Epidemic on a Dual-Layer Static–Dynamic Multiplex Network with Tunable Clustering
The duration, type and structure of connections between individuals in real-world populations play a crucial role in how diseases invade and spread. Here, we incorporate the aforementioned heterogeneities into...
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Article
Open AccessMean-field models for non-Markovian epidemics on networks
This paper introduces a novel extension of the edge-based compartmental model to epidemics where the transmission and recovery processes are driven by general independent probability distributions. Edge-based ...
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Book
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Chapter
Disease spread in networks with large-scale structure
This book has developed analytic models of disease spread on networks. All of our tractable models require closure assumptions. The closure process assumes that we can explain the dynamics at the network scale...
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Chapter
Introduction to networks and diseases
Mathematical models are caricatures of real systems that aim to capture the fundamental mechanisms of some process in order to explain observations or predict outcomes. No model — no matter how complicated — i...
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Chapter
Propagation models on networks: bottom-up
In this chapter, we present a different approach to deriving exact models. In Chapter 2, we began with equations for every possible state of the system and then aggregated them into a simpler form. Here, we be...
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Chapter
Mean-field approximations for heterogeneous networks
Section 4.5 showed that the homogeneous mean-field approximations cannot capture the system behaviour for networks with heterogeneous degree distributions. The heterogeneity in degree can significantly affect ...
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Chapter
Hierarchies of SIR models
This chapter focuses on the relationships between the continuous-time SIR models we have previously derived and identifying conditions under which they are appropriate. Unless otherwise noted, the models discu...
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Chapter
Non-Markovian epidemics
Early studies of non-Markovian epidemics focused on SIR dynamics on fully connected networks, or homogeneously mixing populations, with the infection process being Markovian but with the infectious period take...
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Chapter
Exact propagation models on networks: top down
Chapter 1 introduced SIS and SIR diseases and some weaknesses of compartmental models that can be remedied by considering networks. In this chapter, we begin our n...
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Chapter
Mean-field approximations for homogeneous networks
As seen in Chapters 2 and 3, because of the high-dimensionality of exact mathematical models describing spreading processes on networks, the models are often neither tractable nor numerically solvable for n...
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Chapter
PDE limits for large networks
In previous chapters, it was shown that dynamics on networks can be described by continuous-time Markov chains, where probabilities of states are determined by master equations. While limiting mean-field ODE m...
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Chapter
Percolation-based approaches for disease modelling
The methods introduced thus far are applicable to both SIS and SIR diseases. This chapter focuses primarily on SIR disease. Once a node u becomes infected with an SIR disease, no other node affects the timing of ...
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Chapter
Dynamic and adaptive networks
An important feature of many real-world networks is the transient nature of some interactions. Thus far, our models have explicitly assumed that the network is static. That is, we assume that the rate of partn...
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Chapter
Map** Out Emerging Network Structures in Dynamic Network Models Coupled with Epidemics
We consider the susceptible – infected – susceptible (SIS) epidemic on a dynamic network model with addition and deletion of links depending on node status. We analyse the resulting pairwise model using classi...
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Article
Open AccessOptimizing Hybrid Spreading in Metapopulations
Epidemic spreading phenomena are ubiquitous in nature and society. Examples include the spreading of diseases, information and computer viruses. Epidemics can spread by local spreading, where infected nodes can o...
