Mathematics of Epidemics on Networks: From Exact to Approximate Models

Preface
About the Authors
Contents
1 Introduction to networks and diseases
	1.1 Mathematical modelling of epidemics: the basics
		1.1.1 Deterministic epidemic models: compartmentalSIS & SIR
		1.1.2 Stochastic epidemic models: compartmental SIS & SIR
		1.1.3 Linking stochastic and deterministic compartmentalmodels
	1.2 Networks
		1.2.1 Basic tools for representing networks
		1.2.2 Characterising networks
		1.2.3 Network-generating algorithms
	1.3 Disease spread on networks: the main topic of the book
		1.3.1 Stochastic simulation
		1.3.2 Mathematical modelling
		1.3.3 Topics covered and not covered in the book
2 Exact propagation models on networks: top down
	2.1 An introductory example
	2.2 Continuous-time Markov chains
	2.3 Master equations for arbitrary networks
		2.3.1 State space and transition rates for arbitrary dynamics
		2.3.2 State space and transition rates for binary dynamics
		2.3.3 Master equations for binary dynamics
		2.3.4 Master equations for SIR dynamics
	2.4 Lumping
		2.4.1 Partition of the state space
		2.4.2 A motivating example
		2.4.3 Lumping of linear systems
		2.4.4 The use of graph symmetries to lump a binary dynamic network model
	2.5 Applications of lumping
		2.5.1 Lumping for some small networks
		2.5.2 Lumping for some classes of networks of arbitrary size
	2.6 Conclusions and outlook
3 Propagation models on networks: bottom-up
	3.1 Illustrative examples
		3.1.1 Closures: a succinct overview
	3.2 General bottom-up model
		3.2.1 SIS dynamics
		3.2.2 SIR dynamics
	3.3 Differential inequalities and a comparison theorem for ODEs
	3.4 Closures at the level of pairs and triples for SIS dynamics
		3.4.1 The individual-based model
		3.4.2 The individual-based model overestimates the true probability of nodes being infected
		3.4.3 Steady states of the individual-based model
		3.4.4 Stability of the steady states of the individual-basedmodel
		3.4.5 Relation between the exact, individual-based and classic compartmental SIS model
		3.4.6 Closures at the level of triples for SIS dynamics and numerical examples
	3.5 Closures at the level of pairs for SIR dynamics
		3.5.1 The individual-based SIR model
		3.5.2 The final epidemic size based on the individual-based SIR model
		3.5.3 SIS: an upper bound on SIR
	3.6 General closures at higher levels for SIR dynamics for networks with and without loops
		3.6.1 The relationship between closures and structural network properties
		3.6.2 Feasibility of generalised closure and examples
		3.6.3 Exact, numerically computable model on tree graphs
		3.6.4 Numerical examples of the performance of closed systems for SIR disease
	3.7 Conclusions and outlook
4 Mean-field approximations for homogeneous networks
	4.1 Exact, unclosed models
		4.1.1 The variables of mean-field models: population levelcounts
		4.1.2 Exact differential equations for the singles and pairs
	4.2 Closures at the pair and triple level and the resulting models
		4.2.1 Closures
		4.2.2 Closed systems
		4.2.3 Clustered pairwise model
	4.3 Analysis of the closed systems
		4.3.1 SIS homogeneous mean-field equations at the single level
		4.3.2 SIR homogeneous mean-field equations at the single level
		4.3.3 SIS homogeneous pairwise equations
		4.3.4 SIR homogeneous pairwise equations
	4.4 Basic reproductive ratio
	4.5 Comparison of mean-field models to simulation and exact master equations
		4.5.1 Comparison to exact master equations, dependence on system size and infection parameters
		4.5.2 Comparison to simulation, dependence on networkstructure
	4.6 Derivation of mean-field models from master equations
		4.6.1 Derivation of the homogeneous mean-field model at the single level for the SIS epidemic
		4.6.2 Mean-field models for arbitrary dynamics
	4.7 Detailed analytical study of pairwise models
		4.7.1 Homogeneous SIS pairwise model
		4.7.2 Homogeneous SIR pairwise model
	4.8 Conclusions and outlook
5 Mean-field approximations for heterogeneous networks
	5.1 Exact, unclosed models
	5.2 Closures at the pair and triple level and the resulting models
		5.2.1 Closures
		5.2.2 Closed systems
		5.2.3 Compact pairwise model
		5.2.4 Super-compact pairwise model
	5.3 Analysis of the closed systems
		5.3.1 SIS heterogeneous mean-field model at the single level
		5.3.2 SIR heterogeneous mean-field model at the single level
		5.3.3 SIS compact pairwise model
	5.4 Comparison of models to simulation
	5.5 Detailed analytical study of the mean-field models
		5.5.1 Heterogeneous SIS mean-field model at the single level
		5.5.2 Heterogeneous SIR mean-field model at the single level
		5.5.3 Compact pairwise SIS model
	5.6 Effective degree models
		5.6.1 Basic effective degree model
		5.6.2 Compact effective degree model
	5.7 Conclusions and outlook
6 Percolation-based approaches for disease modelling
	6.1 Typical SIR outbreaks
		6.1.1 Dependence on network size
		6.1.2 Dependence on transmission probability
		6.1.3 Epidemic definitions for finite networks
		6.1.4 Epidemic definition in the infinite network limit
	6.2 Epidemic probability in Configuration Model networks
		6.2.1 Discrete-time Markovian model
		6.2.2 Continuous-time Markovian model
		6.2.3 Non-Markovian model
	6.3 Percolation and SIR disease
		6.3.1 Properties of bond percolation
		6.3.2 Bond percolation and discrete-time SIR disease
		6.3.3 Directed percolation and continuous time SIR disease
		6.3.4 Non-Markovian disease spread
	6.4 Epidemic size in Configuration Model networks
		6.4.1 Discrete-time Markovian model
		6.4.2 Continuous-time Markovian model
		6.4.3 Non-Markovian model
	6.5 Edge-based compartmental modelling: epidemic dynamics
		6.5.1 Epidemic size with many initially infected nodes
		6.5.2 Predicting time evolution of the epidemic
	6.6 SIS disease
	6.7 Conclusions and outlook
7 Hierarchies of SIR models
	7.1 The models and their assumptions
		7.1.1 Mean-field models
		7.1.2 Improved models
	7.2 Hierarchy
		7.2.1 Non-equivalence of PW, ED and EBCM models
	7.3 Equivalence of compact ED, compact PW and EBCM models
		7.3.1 Equivalence of compact ED and EBCM models
		7.3.2 Equivalence of compact PW and EBCM model
	7.4 Limiting approximations
		7.4.1 Reduction of EBCM model to HetMF model
		7.4.2 Reduction of HetMF model to HomMF model
	7.5 Conclusions and outlook
8 Dynamic and adaptive networks
	8.1 Link-conserving rewiring models
		8.1.1 Pairwise model
		8.1.2 Effective degree model
	8.2 Random link activation and deletion models
		8.2.1 Pairwise model
		8.2.2 Effective degree model
	8.3 Link-status-dependent link activation and deletion models
		8.3.1 Oscillating epidemics in a dynamic network model
		8.3.2 Bifurcation analysis of the pairwise model
		8.3.3 Simulation-based bifurcation analysis and trackingof the oscillatory cycle
	8.4 Link deactivation and activation on fixed networks
	8.5 EBCM-based approach
		8.5.1 SIR disease in dynamic degree-preserving networks
	8.6 Conclusions and outlook
9 Non-Markovian epidemics
	9.1 Pairwise model with multiple stages of infection
		9.1.1 Extended compartmental SIR model
		9.1.2 Pairwise model
	9.2 Pairwise model for epidemics with non-Markovian recoverytimes
		9.2.1 Derivation of the standard compartmental and pairwisemodel
		9.2.2 Dynamics in time and final epidemic size
		9.2.3 Pairwise model for a general recovery process
	9.3 EBCM approach to non-Markovian dynamics
	9.4 Conclusions and outlook
10 PDE limits for large networks
	10.1 Model, methods and motivation
		10.1.1 Aims and methods of investigation
		10.1.2 Motivation for the model: binary network dynamics
	10.2 General results for one-step processes
		10.2.1 Differential equations for the moments and for the PGF
		10.2.2 Fokker–Planck equation
	10.3 One-step processes with linear coefficients
		10.3.1 Differential equation for the first moment
		10.3.2 Probability-generating function of the distribution
		10.3.3 Fokker–Planck equation
		10.3.4 Fokker-Planck equation for constant coefficients:random walk
	10.4 Mean-field equations for the moments
		10.4.1 The case of quadratic coefficients
		10.4.2 Polynomial coefficients
		10.4.3 Upper and lower bounds for the expected value
		10.4.4 Higher order closure based on an a priori distribution
	10.5 PDE approximation for the distribution
		10.5.1 The PGF for polynomial coefficients
		10.5.2 First order PDE approximation
		10.5.3 Fokker–Planck equation
	10.6 The accuracy of the mean-field and Fokker-Planckapproximations
		10.6.1 Relation of the Fokker–Planck and mean-field equations
		10.6.2 Mean-field equation
		10.6.3 Fokker–Planck equation
	10.7 Conclusions and outlook
11 Disease spread in networks with large-scale structure
	11.1 A sample social network
		11.1.1 SIR epidemics
		11.1.2 SIS epidemics
	11.2 Small-world networks
		11.2.1 SIR epidemics
		11.2.2 SIS epidemics
	11.3 Preferential attachment networks
		11.3.1 SIR epidemics
		11.3.2 SIS epidemics
	11.4 Conclusions and outlook
Appendix A Stochastic simulation of epidemics
	A.1 Efficient simulations
		A.1.1 Gillespie algorithm
		A.1.2 Event-driven algorithm
	A.2 Time shifting of simulation results
	References
Index