Mathematics of Epidemics on Networks
From Exact to Approximate Models
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Article
Open AccessA Functional Approach to Interpreting the Role of the Adjoint Equation in Machine Learning
The connection between numerical methods for solving differential equations and machine learning has been revealed recently. Differential equations have been proposed as continuous analogues of deep neural net...
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Article
Open AccessGlobal bifurcations in a dynamical model of recurrent neural networks
The dynamical behaviour of a continuous time recurrent neural network model with a special weight matrix is studied. The network contains several identical excitatory neurons and a single inhibitory one. This ...
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Article
Open AccessOn Parameter Identifiability in Network-Based Epidemic Models
Modelling epidemics on networks represents an important departure from classical compartmental models which assume random mixing. However, the resulting models are high-dimensional and their analysis is often ...
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Article
Open AccessThe impact of spatial and social structure on an SIR epidemic on a weighted multilayer network
A key factor in the transmission of infectious diseases is the structure of disease transmitting contacts. In the context of the current COVID-19 pandemic and with some data based on the Hungarian population w...
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Article
Open AccessMicro-scale foundation with error quantification for the approximation of dynamics on networks
Epidemics, voting behaviour and cascading failures in power grids are examples of natural, social and technological phenomena that can be modelled as dynamical processes on networks. The study of such importan...
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Article
Analytic Study of Bifurcations of the Pairwise Model for SIS Epidemic Propagation on an Adaptive Network
The pairwise ODE model for SIS epidemic propagation on an adaptive network with link number preserving rewiring is studied. The model, introduced by Gross et al. (Phys Rev Lett 96:208701, 2006), consists of four ...
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Chapter
The Effect of Inhibitory Neurons on a Class of Neural Networks
The understanding of the effect of inhibitory neurons on neural networks’ dynamics is crucial to gain more insight into the biological process. Here we examine the dynamics of a special excitatory-inhibitory n...
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Article
Open AccessEpidemic threshold in pairwise models for clustered networks: closures and fast correlations
The epidemic threshold is probably the most studied quantity in the modelling of epidemics on networks. For a large class of networks and dynamics, it is well studied and understood. However, it is less so for...
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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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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 ...
