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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Explores the emerging subject of epidemic dynamics on complex networks, including theories, methods, and real-world applications
Throughout history epidemic diseases have presented a serious threat to human life, and in recent years the spread of infectious diseases such as dengue, malaria, HIV, and SARS has captured global attention; and in the modern technological age, the proliferation of virus attacks on the Internet highlights the emergent need for knowledge about modeling, analysis, and control in epidemic dynamics on complex networks.
For advancement of techniques, it has become clear that more fundamental knowledge will be needed in mathematical and numerical context about how epidemic dynamical networks can be modelled, analyzed, and controlled. This book explores recent progress in these topics and looks at issues relating to various epidemic systems.
Propagation Dynamics on Complex Networks covers most key topics in the field, and will provide a valuable resource for graduate students and researchers interested in network science and dynamical systems, and related interdisciplinary fields.
Key Features:
- Includes a brief history of mathematical epidemiology and epidemic modeling on complex networks.
- Explores how information, opinion, and rumor spread via the Internet and social networks.
- Presents plausible models for propagation of SARS and avian influenza outbreaks, providing a reality check for otherwise abstract mathematical modeling.
- Considers various infectivity functions, including constant, piecewise-linear, saturated, and nonlinear cases.
Examines information transmission on complex networks, and investigates the difference between information and epidemic spreading.
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Seitenzahl: 474
Veröffentlichungsjahr: 2013
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Table of Contents
Title Page
Copyright
Preface
Summary
Chapter 1: Introduction
1.1 Motivation and background
1.2 A brief history of mathematical epidemiology
1.3 Organization of the book
References
Chapter 2: Various epidemic models on complex networks
2.1 Multiple stage models
2.2 Staged progression models
2.3 Stochastic SIS model
2.4 Models with population mobility
2.5 Models in meta-populations
2.6 Models with effective contacts
2.7 Models with two distinct routes
2.8 Models with competing strains
2.9 Models with competing strains and saturated infectivity
2.10 Models with birth and death of nodes and links
2.11 Models on weighted networks
2.12 Models on directed networks
2.13 Models on colored networks
2.14 Discrete epidemic models
References
Chapter 3: Epidemic threshold analysis
3.1 Threshold analysis by the direct method
3.2 Epidemic spreading efficiency threshold and epidemic threshold
3.3 Epidemic thresholds and basic reproduction numbers
References
Chapter 4: Networked models for SARS and avian influenza
4.1 Network models of real diseases
4.2 Plausible models for propagation of the SARS virus
4.3 Clustering model for SARS transmission: Application to epidemic control and risk assessment
4.4 Small-world and scale-free models for SARS transmission
4.5 Super-spreaders and the rate of transmission
4.6 Scale-free distribution of avian influenza outbreaks
4.7 Stratified model of ordinary influenza
References
Chapter 5: Infectivity functions
5.1 A model with nontrivial infectivity function
5.2 Saturated infectivity
5.3 Nonlinear infectivity for SIS model on scale-free networks
References
Chapter 6: SIS models with an infective medium
6.1 SIS model with an infective medium
6.2 A modified SIS model with an infective medium
6.3 Epidemic models with vectors between two separated networks
6.4 Epidemic transmission on interdependent networks
6.5 Discussions and remarks
References
Chapter 7: Epidemic control and awareness
7.1 SIS model with awareness
7.2 Discrete-time SIS model with awareness
7.3 Spreading dynamics of a disease-awareness SIS model on complex networks
7.4 Remarks and discussions
References
Chapter 8: Adaptive mechanism between dynamics and epidemics
8.1 Adaptive mechanism between dynamical synchronization and epidemic behavior on complex networks
8.2 Interplay between collective behavior and spreading dynamics
References
Chapter 9: Epidemic control and immunization
9.1 SIS model with immunization
9.2 Edge targeted strategy for controlling epidemic spreading on scale-free networks
9.3 Remarks and discussions
References
Chapter 10: Global stability analysis
10.1 Global stability analysis of the modified model with an infective medium
10.2 Global dynamics of the model with vectors between two separated networks
10.3 Global behavior of disease transmission on interdependent networks
10.4 Global behavior of epidemic transmissions
10.5 Global attractivity of a network-based epidemic SIS model
10.6 Global stability of an epidemic model with birth and death and adaptive weights
10.7 Global dynamics of a generalized epidemic model
References
Chapter 11: Information diffusion and pathogen propagation
11.1 Information diffusion and propagation on complex networks
11.2 Interplay between information of disease spreading and epidemic dynamics
11.3 Discussions and remarks
References
Appendix A: Proofs of theorems
A.1 Transition from discrete-time linear system to continuous-time linear system
A.2 Proof of Lemma 6.1
A.3 Proof of Theorem 10.4
A.4 Proof of Theorem 10.3
A.5 Proof of Theorem 10.42
Appendix B: Further proofs of results
B.1 Eigenvalues of the matrix in (6.27)
B.2 The matrix in (6.32)
B.3 Proof of (7.6) in Chapter 7
B.4 The positiveness of : proof of in section 9.1.2
B.5 The relation between and in Section 9.1.3
Index
This edition first published 2014
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Preface
Throughout history, epidemic diseases have been a serious threat to human health and life. In the past few years, many infectious diseases such as dengue, malaria, HIV, and SARS have captured global attention. Many of these, and others, remain a great threat, with potential for new outbreaks—particularly, for example, with a human-transmissible version of the H5N1 avian influenza. Moreover, with the development of globalized transportation, the potential for epidemic transmission has become much greater. Once a disease emerges, it will very likely diffuse globally very rapidly: 2009 H1N1 spread to some 30 countries worldwide in a relatively short period of time leaving more than 800 dead. The continual computer virus attacks on the Internet also illustrate the urgent need for knowledge about modeling, analysis, and control of epidemic dynamics on complex networks. Concerning the advance of techniques, it has become clear that more fundamental knowledge is needed within the context of mathematical and numerical studies on how epidemic dynamical networks can be modeled, analyzed and controlled. The main objective of this book is to present the state-of-the-art and recent progress in the investigation of these important topics and some related issues arising from various epidemic and information systems.
This book covers most emerging topics of epidemic dynamics on complex networks, including models, theories, methods, and global stability analysis. We also extend our discussions to include information propagation dynamics, and address topics such as how information, opinions, and rumors spread in the Internet or social networks. This work has developed from a series of research papers resulting from an on-going collaboration among the three authors and their research groups since 2006.
This is mainly a research monograph and also a textbook that can be used as either a research reference book or for a one-semester introductory course on propagation dynamics and epidemic control on complex networks for upper-division undergraduates and first-year graduates in applied mathematics, engineering, computer science, information science, communication systems, biological and life sciences, applied physics, as well as biomedical and social sciences. It covers most basic topics in the field, and therefore can serve well for self-study of these topics by graduate students and researchers interested in network science and engineering.
Throughout the text we often keep the adjective complex to reflect the historical perspective and to emphasize the nature of the subject, which is in line with the common phrases of complex systems and complex dynamics alike, therefore it should not be seen as redundant.
We would like to take this opportunity to express our gratitude to the editor Ms Ying Liu at China Higher Education Press for her invaluable help and support throughout the writing of this book and the subsequent publication processes. We would also like to thank the editors at Wiley for their timely responses to our book proposal and for all their helpful comments aimed at improving the final product.
We would also like to acknowledge and thank Luonan Chen, Zhen Jin, Xiang Li, Zengrong Liu, Zonghua Liu, Jun-an Lu, Robert MacKay, Chi K. Tse, Binghong Wang, Xiaofan Wang, to mention just a few, and also our research group members, for their kind help and support.
Finally, we would like to thank our postgraduate students for their contributions, helpful discussions and useful suggestions during the writing of this book. Their contributions are too many to be listed individually.
The research was supported jointly by the University Grants Council of Hong Kong (HK UGC GRF PolyU5300/09E and CityU1109/12E), the Australian Research Council Future Fellowship scheme (grant number FT110100896), City University of Hong Kong, the NSFC grant 11072136, the Shanghai University Leading Academic Discipline Project “Complex Systems: Theory, Methods and Technology” (2012–2014) (Project No. A.13-0101-12-004), and a grant of “The First-class Discipline of Universities in Shanghai”. The publication of this book was supported by the China National Publishing Fund for Academic Books in Science and Technology.
Summary
This book evolved from a series of research papers by the three authors and their students published since 2006. It covers the emerging topics of propagation dynamics on complex networks, including models, methods, and stability analysis. Throughout history, epidemic diseases have always been a serious threat to mankind's health and life, and ongoing serious virus attacks on the Internet also illustrate the emergent need for knowledge about modeling, analysis, and control in epidemic dynamics on complex networks. For advance of techniques, it has become clear that more fundamental knowledge will be needed in mathematical and numerical context about how epidemic dynamical networks can be modeled, analyzed, and controlled. The aim of this book is to report the progress made in these topics and some related issues of various epidemic systems. The book will first present a brief history of mathematical epidemiology, and epidemic modeling on complex networks. Then different epidemic models on complex networks, such as staged progression models, models with population mobility, or effective contacts, models on weighted networks, or directed networks, discrete epidemic models, stochastic SIRS epidemic models, and so on, will be discussed. Some threshold analyses by the direct method and by using spectral properties are given. Networked models for SARS and H1N1 are established by setting up plausible models for propagation of the SARS virus and avian influenza outbreaks, which provides a reality-check for the otherwise abstract mathematical models of this text, and it is shown that such models do match well the reality of current emerging diseases. Furthermore, various infectivity functions, including constant, piecewise-linear, saturated, and nonlinear cases, are considered. This book also concentrates on the cases for SIS models with an infective medium, the roles of human awareness in epidemic control, adaptive mechanism between dynamics and epidemics. Methods for epidemic control and different immunization strategies are summarized. Global stability analysis for several networked epidemic models is demonstrated. Finally, information transmission on complex networks and some differences between information and epidemic spreading are investigated.
This book covers most basic topics in the field, and therefore can serve well for self-study of the subjects by graduate students and researchers interested in network science and dynamical systems, and related interdisciplinary fields.
Chapter 1
Introduction
In this chapter we provide a brief introduction to the remainder of the book.
The uninitiated may require a broader background to the topic of complex networks. Rather than overburden out current presentation, we refer interested readers to some good introductory books and papers [1–14] for more background information on complex networks and network science.
1.1 Motivation and background
Throughout history, infectious diseases have always been a serious threat to human health and life. It is therefore of great practical significance to study epidemic transmission and then to take effective measures to prevent and control them. Toward this end, much research has fallen within the field of epidemiology, which uses mathematical modeling as an analytical approach. Traditionally, epidemic models were based on uniformly mixing populations, which are unable to characterize epidemic propagation in large-scale social contact networks with disparate heterogeneity. However, the fact that most population-based epidemics spread through physical interactions raises contact networks as a basic tool for mathematical description of contagion dynamics. In the last decade, spurred by the availability of real data and the maturation of network theory, there has been a burst of research on network-based epidemic transmission [15–26].
Beyond ordinary infection diseases, recurring computer virus attacks (as well as computer worms and other malware vectors) on the Internet also illustrate the urgent need for knowledge about modeling, analysis and control of epidemic dynamics on complex networks.
The World Health Organization (WHO) announced in 2012 [27] that some time in the next couple of years Guinea worm will become only the second known disease, after smallpox, to be completely eradicated. The disease has been known to afflict humans for thousands of years. Unlike other diseases, the campaign against Guinea worm has focused not on developing a cure, but on educating people about how the disease spreads and how infestation can be prevented.
While Guinea worm may be almost eradicated, people worry that several other infectious diseases are re-emerging [28]: Tuberculosis: poor-quality diagnoses, treatment, and medicines contributed to the rise of 8.7 million new cases in 2011, particularly in Eastern Europe, India, China, and parts of Africa; Leprosy: 219 000 new cases were reported last year, mostly in Africa and Asia; and, Bubonic plague: the same Black Death that wiped out millions in Europe has cropped up in the United States, and between 1000 and 2000 cases of plague are still reported worldwide each year.
Information spread can also appear to propagate like a virus. In 2011, in the wake of the Fukushima nuclear disaster, rumors spread throughout China that iodized table salt could be used to help prevent radiation sickness. The subsequent rumors and panic-buying lead to a shortage of salt in both China and neighboring territories. Organized, or coherent, spread of rumors combined with lack of judgement on the part of public news agencies, led to official information sources appearing to lose credibility. Conversely, institutionalized and individual cyber-attacks have gained recent prominence. Naturally, network structure and propagation dynamics become key features in controlling and understanding such mechanisms.
Epidemics on networks is a rapidly expanding field of considerable contemporary interest to researchers in a broad spectrum of areas including applied mathematics, probability, physics, biology, and so on. There is a need for a book at an introductory research level that gives a balanced overview of the current state-of-the-art in this area. Concerning the advance of techniques, it has become clear that more fundamental knowledge is needed within the context of mathematical and numerical studies on how epidemic dynamical networks can be modeled, analyzed, and controlled. This book, based on existing research, aims to address this need. We discuss in detail different epidemic models on complex networks and a variety of applicable mathematical techniques. Using mean-field approximation we provide a detailed analysis of epidemic dynamics, the theory of complex networks, and qualitative theory and stability methods of ordinary differential equations. The current volume serves to present recent progress in the investigation of these important topics and some related topics.
1.2 A brief history of mathematical epidemiology
In this section we give a brief, largely descriptive, history of mathematical epidemiology. Many undergraduate texts provide extensive coverage of the details.
Epidemiological modeling is a large subject in mathematical biology, a single short section about its brief history is of course insufficient to give a complete picture of the field. So we here refer the readers to a book and a review article [29, 30] for more details.
1.2.1 Compartmental modeling
The recorded earliest mathematical epidemic model dates from the eighteenth century. In 1760, by using ordinary differential equations, Daniel Bernoulli studied smallpox vaccination, and gave the Bernoulli equations [31]. Bernoulli's results showed that universal inoculation against smallpox could increase life expectancy.
Later in 1889, En'ko built the chain-binomial model for measles and scarlet fever. To understand the recurring epidemics of measles, in 1906, Hamer gave a discrete mathematical model, and presented the mass-action principle [32, 33]. In 1911, Ronald Ross established and studied the malaria transmission model, and gave the standard incidence ratio and the basic reproduction number (sometimes called the basic reproductive number, basic reproductive rate, basic reproductive ratio, and denoted as ). In epidemiology, the basic reproduction number of an infection is the number of cases that one case generates on average over the course of its infectious period. The roots of the concept can be traced through the work of Alfred Lotka, Ronald Ross, and others, but its first modern application in epidemiology was by George MacDonald in 1952, who constructed population models of the spread of malaria. In 1926, by studying the spreading patterns of the Black Death in 1665–1666 and the plague in 1906, A. G. McKendrick and W. O. Kermack formulated a simple deterministic model that was the modern mathematical epidemic model–the SIR compartmental model, which was successful in predicting the behavior of outbreaks in many recorded epidemics. Based on this model, they presented the threshold theory to determine eventual endemic or disease-free status of a disease. In 1949, Bartlett's measles model [34] was built.
In a compartmental model of infectious disease, individuals are divided into several classes, for example, the compartments: susceptible (S), latent (E), infected (I), vaccinated (V), and/or removed (recovered) (R). The E status is also used to represent the stage when individuals have been exposed to a disease and are therefore infected, but not yet infectious. Depending on the propagation process, we can build various compartmental models by combining these different classes (or creating new ones). Examples of such include SI, SIS, SIR, SIRS, SEI, SEIS, SEIR, SEIRS, SIV, and so on. The sequence in which classes are listed typically corresponds to the infection pathway. In a compartmental SIS model, say, each individual can be in two discrete states, either susceptible to or infected by the virus particle, and susceptible individuals (S) may become infected (I) owing to contact wth other infected individuals, and infected individuals also may recover to susceptible state (S), with a certain recovery rate. Apart from percolation models [22], this book will discuss most such models.
As George E. P. Box said, “Essentially, all models are wrong, but some are useful.” Certainly, all the models we include here are wrong as they are mean-field approximations for the spreading of real epidemic diseases. Nonetheless, this is a useful approximation and many of these models have helped people to plan effective actions against various serious epidemic diseases.
After building a model, we need then to study it by qualitative, analytical, experimental (including numerical), and theoretical methods. Research methods for modern epidemic dynamics models can be summarized briefly as follows.
For a deterministic compartmental model, that is built based on uniform mixture hypothesis, we may study it by either theory or methods [35, 36] developed in ordinary differential equations, partial differential equations, delay differential equations, impulsive equations, and difference equations. The trends in these research areas are for higher model dimension and deeper and more refined analysis.
For a stochastic model, we may apply stochastic processes and stochastic dynamical methods. In contrast to the above, the trends with these models are toward specific diseases and toward deterministic and stochastic mixed models.
1.2.2 Epidemic modeling on complex networks
Many epidemic systems can be represented as a graph or network, where nodes stand for individuals and a link connects a pair of nodes—indicating interaction between individuals.
Patterns of this type can quickly become very complex and it is usually not sufficient to describe the connectivity between two nodes as uniform or homogenous. Heterogenous contact rates reflect that the node degree , the number of contacts with other individuals for a given individual, are not uniform. Instead, such heterogeneity can be represented by the degree distribution . Real networks underlying disease transmission have been represented not only by conventional graphs such as lattices, regular trees, or classical random graphs, but also by complex networks, such as the WS (Watts–Strongatz) small-world networks [14] or the BA (Barabási–Albert) scale-free networks [3].
Traditional epidemic models are useful for uniformly mixing populations with homogenous contacts. However, these are unable to characterize epidemic propagation in large-scale social contact networks with distinct heterogeneities. As is well known, all models are inaccurate simplifications of nature. By taking the heterogeneity into account, complex-network modeling of epidemics provides a somewhat more accurate viewpoint. A compartmental model based on uniform mixing can be viewed as a networked model with Delta degree distribution, an approximation to the Poissonian or power-law degree distribution. Conversely, if we take the degree distribution as the Delta distribution, a networked model will then become a uniformly mixing compartmental model.
Networked epidemic models are typically considered with networked mean-field theory, which was pioneered by two physicists, Pastor-Satorras and Vespignan [23, 37, 38], although some earlier results were already reported in a mathematical textbook [29]. The basic idea is, according to traditional compartment models, to classify all nodes on the network by disease states and, based on this, subdivide nodes according to their degrees, such that nodes with the same degree belong to one class, which has the same dynamics patterns. The core here is the dynamical behavior for the nodes with the same degree can be represented by the average behavior. Based on such contact networks, many epidemic models, such as SI [39], SIS [23] and SIR [40], have been investigated.
To understand the mechanism of disease spreading and other similar processes, such as rumors spreading [20], networks of movie actor collaboration [3, 13] and science collaboration [41], WWW [42, 43], and the Internet [44], it is of great significance to inspect the effect of complex networks' features.
In China alone, some early research on networked epidemic transmission models was carried out by many researchers [4, 45–57]. Some results on propagation and immunization of infection on general networks with both homogenous and heterogenous components, and influence of dynamical condensation on epidemic spreading in scale-free networks [49, 52, 53, 58, 59, 60], global stability analysis of networked epidemic models [61–67] are obtained, to mention only a few.
After building a mathematical model, we may then apply the following cycle: run algorithms to compute with the model; analyze errors where results differ from data; create modifications of the mathematical model; (develop pure mathematics theory that is perhaps increasingly irrelevant); analyze improved model, and so on… In this book, however, we will concentrate on theoretical analysis of the models we build. In Chapter 4 we consider the problem of comparing these models with the real data.
Apart from the difficulty caused by very high dimension in networked epidemic models, some other problems for these models are that they did not properly take the population dynamics into account. These factors include the impact of birth, death, and migration on the network topology and the spreading patterns of diseases [23, 29, 30, 37, 38, 68–70]. Finally, networked models based on pair approximation [71] seem a further step to make networked models more accurate.
1.3 Organization of the book
This book consists of 11 chapters. Chapter 1 gives an introduction, motivation, and background for this work. In this chapter we present a brief history of mathematical epidemiology, and epidemic modeling on complex networks. Chapter 2 discusses different epidemic models on complex networks, such as staged progression models, models with population mobility, or effective contacts, models on weighted networks, or directed networks, discrete epidemic models, and stochastic SIRS epidemic models. Chapter 3 details some threshold analyses by the direct method and by using spectral properties. Chapter 4 analyzes networked models for SARS and H1N1, and sets up plausible models for propagation of the SARS virus and avian influenza outbreaks. This provides a reality-check for the otherwise abstract mathematical models of this text. We show that such models do match well the reality of current emerging diseases. Chapter 5 deals with various infectivity functions, including constant, piecewise-linear, saturated, and nonlinear cases. Chapter 6 concentrates on the case for SIS models with an infective medium; Chapter 7 discusses the roles of human awareness in epidemic control; Chapter 8 reveals adaptive mechanism between dynamics and epidemics; and Chapter 9 summarizes methods for epidemic control and different immunization strategies. Finally, Chapter 10 demonstrates global stability analysis; and Chapter 11 investigates information diffusion and pathogen propagation on complex networks, and discusses some differences between information and epidemic spreading.
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Chapter 2
Various epidemic models on complex networks
The dynamical behavior of the SIS (susceptible-infected-susceptible) model and the SIR (susceptible-infected-recovered/removed) model, widely considered as the conventional way to describe the fundamental mechanism of diseases, has been widely studied on regular networks and complex networks [1–16]. In this chapter we introduce several networked epidemic models based on these [17–20].
There is a whole body of research about computational epidemiology (see [21] and references therein). In this chapter (and for most of this book), we will not discuss computational approaches in detail. Nonetheless, we briefly mention such models here as one of the primary modern methods to understand complicated disease dynamics.
2.1 Multiple stage models
Dynamical behavior of disease spreading has long been an important topic for mathematical research. The SIS and the SIR are omnipresent and convenient mathematical models to describe the fundamental mechanism of diseases [22]. For the SIS epidemic model, each individual can exist in two states: S (susceptible) and I (infected). Over time (at each time step, in a discrete model), the susceptible individual that is connected to an infected neighbor will be infected with rate . Meanwhile, the infected individuals may be recovered and become susceptible at a rate . For the SIR model, once an infected individual becomes R (recovered), then that individual will not be infected again.
For certain types of networks, where most nodes have similar degrees—that is, the degree distributions have small fluctuations exhibiting a normal distribution, for example, random networks, regular networks, small-world networks [23]—we call them homogenous networks. In contrast to homogenous networks, those networks with large fluctuations in degree distributions are called heterogenous networks, such as scale-free networks [24]. In a scale-free network, the probability that any node has links to other nodes is distributed according to a power-law (i.e., the degree distribution exhibits extremely large fluctuations).
Researchers have mainly studied the dynamics of epidemics on homogenous networks, and many remarkable results have been obtained. Many real complex systems have been shown to be scale-free networks, such as the WWW (World Wide Web), the Internet, and so on. Moreover, many epidemic diseases occur in communities, which also exhibit characteristics consistent with a scale-free network, for example, human sexual contacts show scale-free characteristics [25]. Recently, the spread of epidemic diseases on scale-free networks has been studied by many researchers, and [26] provides a review of recent advances. The striking result is that for SIS and SIR models the epidemic threshold is null for sufficiently large scale-free networks [7–9].
Different diseases have many different mechanisms, and the SIS and SIR models cannot adequately represent all kinds of diseases. Consequently, for different diseases, the corresponding dynamical models should be established. For instance, the susceptible individuals should be classified into different cases because of their different immunities; similarly, so should the infected individuals because of their infectivities. Sometimes, the disease will progress through several distinct stages.
Individuals can be coarsely classified into three states, S, I, and R. To better explore the mechanism of epidemic spreading on complex networks, in this section we suppose that the S and I states can be subdivided into subclasses according to their different immunities, different infectivities, and so on [27]. That is, our models can describe , and . To make the models more reasonable (particularly for slowly acting diseases), we also consider the birth and death of individuals. By using the method as in [4], we suppose that all individuals are distributed on the network, and each node of the network is empty or occupied by at most one individual. For computation and simulation, the numbers 0,1,2,3 denote that a node has no individual (is empty), a healthy (susceptible) individual, an infected individual, and a recovered individual, respectively. Each node can change its state with a certain rate. An empty node can give birth to a healthy (susceptible) individual at the rate . The susceptible individual can be infected at a rate that is proportional to the number of infected individuals in the neighborhood or die at certain rate . The infected individual can be cured at certain rate or die at certain rate . If an individual dies, that node will become empty once again.
2.1.1 Multiple susceptible individuals
Now, we consider the susceptible individuals with several different cases according to their ages or immunities. To consider the heterogeneity of complex networks, denote the density of the susceptible individuals with degree and also belong to the -th class, and and denote the density of the infected individuals and the recovered individuals with degree , respectively. Then the mean-field equations can be written as:
2.1
where is the density of empty nodes that will give birth to nodes with degree , and , , are the birth rates, infectivity rates, and the natural death rates for the -th class susceptible individuals, respectively, , are the natural death rate and the rate from for infected individuals, and is the natural death rate of recovered individuals. For uncorrelated networks, can be written as [7, 8]
2.2
where .
2.1.2 Multiple infected individuals
Here, we suppose that the infected individuals are classified into several different cases according to their infectivity rates or natural death rates. Let , , denote the -th infected individual with degree . The corresponding mean-field equations are given by
2.3
Here, the new infected individuals will come into the -th infectivity individuals with probability , so . Other parameters are similar to those in Subsection 2.1.1, and
2.4
2.1.3 Multiple-staged infected individuals
In the above section, we considered that the infected individuals may have several cases. However, as was discussed in [28], each case of infected individuals can also develop in several stages. Hence, we introduce the Multiple-Staged infected individuals models in this section. Let , , , denote the -th infected individual which is in the j-th stage.
To simplify the computation, we do not consider the natural death rate for , , , but suppose that they only go into state with certain rates (the method for the Multiple-Staged infected individuals models with natural death is the same, but that is somewhat more complicated).
The dynamic equations are
2.5
Here, the individuals' degree is given as the superscripts to differentiate from the subscripts . The infectivity rates for on susceptible individuals are , and are the rates of the transformation , , , and are the rates of the transformation . Here, we suppose that each can infect susceptible individuals, and new infected individuals will come into the -th infectivity individuals with probability , so we also have . Similarly, are given by
2.2 Staged progression models
Because of the different mechanisms of different diseases, the SIS and SIR models can only faithfully be applied to model a limited range of actual diseases (albeit, highly effectively). Often, for real diseases, the infected individuals may experience several distinct stages [3], for example, individuals who are infected by HIV-AIDS may pass through several stages: being highly infectious in the first few weeks after becoming infected, then having low infectivity for many years, and becoming gradually more infectious as their immune systems break down, eventually they progress to full blown AIDS. Moreover, such staged progression models may be applied to situations where the behavior of infected individuals changes with time, and so does their infectivity. In view of all the above facts, an alternative staged progression model is introduced in this section [28], and the epidemic spread for the staged progression model on both homogenous and heterogenous networks is discussed.
2.2.1 Simple-staged progression model
We now consider the basic staged progression model. In Section 2.2.2 we treat the homogenous case, and in Section 2.2.3 we consider the heterogenous case. In the following section, we introduce birth and death to this basic model.
2.2.2 Staged progression model on homogenous networks
We assume that the individuals can exist in two states: susceptible (S) and infected (I), where the infected individuals are subdivided into subgroups with different infection stages such that infected susceptible individuals enter the first subgroup and then gradually progress from this subgroup to subgroup . Let be the infection rate when susceptible individuals acquire infection from an infected neighbor belonging to subgroup , for , and be the average percentage of infected individuals transiting from subgroup to subgroup , for , and let be the rate at which infected become susceptible individuals again [3].
On a homogenous network, we suppose that every node has the same degree , where is the average number of the nearest neighbors of one node. We denote by and the densities of the susceptible population and the infected population in subgroup at time step , respectively. Consequently:
2.6
with the population unchanged. In what follows, and throughout the rest of this book, we will drop the parenthetical dependence on time [and just write or to mean to , respectively] whenever our meaning is clear.
At first, a susceptible individual may become infected through contact with its infected neighbors, then all the infected individuals will enter the first stage and pass through different stages with different rates. Finally, some of the infected individuals may recover and then become susceptible again. So the evolution equations of densities can be expressed as follows:
2.7
2.2.3 Staged progression model on heterogenous networks
In the previous subsection, we discussed the epidemic threshold of the staged progression models on homogenous networks. However, many real-world networks show heterogenous properties. For instance, scientific-collaboration networks, Internet, and the World Wide Web are all observed to be heterogenous networks. Therefore, we will study the staged progression models on this type of networks.
Here, denotes the density of the susceptible individuals with degree at step , and denotes the density of the infected individuals with degree and belongs to subgroup at step . We also have
2.8
for all time . Similar to the homogenous case, the mean-field equations for heterogenous networks are
2.9
where , denotes the probability that a link emanates from a susceptible node with degree to an infected individual in subgroup . So , where is the probability that a node with degree points to a node with degree . For uncorrelated networks, , then we have
2.2.4 Staged progression model with birth and death
In [4], Liu and coworkers analyzed the spread of diseases with birth and death, where they supposed that individuals are distributed on a network, and each node of the network is empty or occupied by at most one individual. They used the numbers 0,1,2 to denote that the node has no individual, a healthy (susceptible) individual, and an infected individual, respectively. Each node can change its state with a certain rate. An empty node can give birth to a healthy (susceptible) individual at the rate . The susceptible individual can be infected at a rate proportional to the number of infected individuals in the neighborhood or die at certain rate . The infected individual can be cured at certain rate or die at certain rate . If an individual dies, that node will become an empty node again.
Here, we discuss the staged progression model with birth and death. To be consistent with the above section, the symbols used in above section will have the same meaning in this section. In addition, we use and to denote the birth rate and death rate of susceptible individual respectively, and , to stand for the death rate of individuals who are infected and belong to subgroup .
2.2.5 Staged progression model with birth and death on homogenous networks
On homogenous networks, the dynamical equations of staged progression model with birth and death are
2.10
2.2.6 Staged progression model with birth and death on heterogenous networks
For the staged progression model with birth and death on heterogenous networks, we have the following dynamical equations:
2.11
2.3 Stochastic SIS model
The research in [7–9] shows that both of the above-mentioned kinds of heterogenous contact rates can decrease the epidemic threshold for SIS models.
On the other hand, the critical parameter, that is, the infection rate, is always related to susceptibility and infectiousness of individuals [29–33]. Such individual-based infection rates are sometimes ascribed to heterogenous social or sexual contact rates as specified by . In [34], Olinky and Stone analyzed a new SIS model. They studied the role of disease transmission by introducing degree correlated transmission rate and admission rate , where is the probability that an infected node would actually transmit an infection through a link connected to a susceptible node, and is the probability that a susceptible node would actually admit an infection through a link connected to an infected node. Since the infection rate is not constant, it may be different for different infectious links. If we denote by the infection rate by the link between node and node , then according to the meanings of , , we have
2.12
Thus, heterogenous infection rates must change with epidemic propagation or systemic evolution. Moreover, the change of infection rate must impact the epidemic behaviors in turn. Through theoretical and simulating analysis, Olinky and Stone [34] found that the epidemic threshold may not vanish in scale-free networks.
Therefore, both heterogenous infection rates and heterogenous contact rates have different influences on the epidemic spreading. In this section, we focus on the following two problems:
We hope that the results presented in this section [35] will give some insight into the spread of real diseases (human diseases or computer viruses).
In the real world, an epidemic always occurs on a finite network [9], even though the size of the network may be very large. Hence, we consider disease transmission in a finite population where susceptibility and infectiousness of nodes depend on the node-connectivity. Here, the epidemic disease model is built on a Barabási–Alberts (BA) scale-free network [10, 24, 36, 37].
We analyze the epidemic spread by the SIS mechanism [2]. According to [6], a physically plausible case has the transmission rate of node given by . Here, denotes the degree of node . Similar to [34], we select the following general forms:
2.13
These forms can include more general cases induced by monopoly correlated to node-degree. Different diseases may correspond to different values of parameters in .
Based on the assumption (2.13), a discrete-time stochastic epidemic transmission process is determined (for convenience, we denote the process by ), and the dynamics are specified by the following transition probabilities of each single node: at node , , with the probability ; , with the probability .
Here, denotes that infected nodes are in the neighborhood set of node .
2.3.1 A general concept: Epidemic spreading efficiency
If an infected node links with a susceptible node, we call them an infection pair. We denote the set of all infection pairs by . At each time, the infection rate of each infection pair can be computed according to (2.12). For a finite population, one can compute the arithmetic mean value of all values of in the whole population at time . This means that the average of infection rates is actually a function of time, denoted by , which can be computed by the following formula:
2.14
The numerator represents the total number of effective infection rates at time step . The denominator accounts for the number of infection pairs. If we make the complementary definition of , then at other cases except the cases referred in (2.12). The matrix reflects the distribution of infection rate over the whole network at time step , and the quantity is just the mean value of the distribution.
At time step , we denote by a set composed of all susceptible nodes. For each , the probability from susceptible state to infectious state is equal to . Hence, when , the transition rate is just . The number of infected nodes at time [denoted by ] is therefore (similar to the method in [38]). We have
2.15
Therefore, (2.14) can be changed into
2.16
From (2.16), it is clear that reflects the efficiency of epidemic transmission on the network, as it is actually the number of nodes infected along one link. Simple simulation can show that the quantity is a function of time, and oscillates with a very small amplitude. In addition the time-varying curve oscillates around a common mean value. We make numerical simulations to check other parameter cases and find a similar phenomenon, and even periodic behavior, for some parameters.
Based on the small-amplitude oscillation of this kind, we define an average quantity:
2.17
where is the duration of transmission simulations. The time average of this kind can not only capture the main feature in a large time span but also keep the function of . Therefore, we call it epidemic spreading efficiency (ESE). When , we obtain that . In this case, the quantity ESE refers as an extension of the classical infection rate.
2.4 Models with population mobility
Many good results about epidemic diseases on networks have been obtained. However, most assumed that a node is an individual; as a result, the deeper structures of networks were neglected, such as the mobility of individuals between different cities was ignored. Most recently, Colizza et al. [39] studied the behavior of two basic types of reaction-diffusion processes ( and ), where they supposed that a node of the network can be occupied by any number of individuals and the individuals can diffuse along the link between nodes. The two basic reaction-diffusion processes can be used to model the spreading of epidemic diseases with SIS model [39]. In epidemic terminology, a node can be viewed as a city, that is, all people have the same degree if they live in the same city (the node with degree ), and the diffusion of particles among different nodes can be considered as the travel of people among different cities. They supposed that the infection may happen inside a city. However, the infection may also happen in different cities by other media, for example, for Avian Influenza, different places' poultry can be infected by migratory birds even though the domestic poultry has little or no mobility.
We suppose that the infection can also happen in different cities, and study the effect of this kind of epidemic spreading on the epidemic threshold [40]. This can be done by introducing a probability of spreading of the infection to the neighboring nodes without the need of diffusion of infected particles. In fact, as we will show, this mechanism is in part equivalent to the diffusion of individual particles.
We discuss two cases of infection taking place in different cities. Firstly, the infection rate is proportional to [16, 41]; secondly, the infection rate is proportional to [7, 8].
In this section, the spreading of epidemic diseases happens on an uncorrelated network, that is, the conditional probability that a link departing from a node of degree points to a node of degree is independent of , that is, , where .
2.4.1 Epidemic spreading without mobility of individuals
To determine the effect of mobility of individuals, we first assume that mobility is zero. In this case, the dynamical equations are
2.18
where is the rate for infected individuals becoming susceptible again, and is the conditional probability that a node with degree is connected to a node with degree . The parameters and are the epidemic rates inside the same city and between different cities, respectively. The term is just as the first type process in [39], and the term stands for the density of that is infected by other cities' infected individuals. Here, we should note that the total density is not changed because there is no mobility of individuals among different cities (this case is different from the discussion in the following sections where we consider the mobility; as a result, the individual's degree may change), so we can let for all .
In the following sections, we take into account the mobility of individuals in different cities, so the individuals' degrees may change, that is, the total density is not an invariant, but the average density is.
2.4.2 Spreading of epidemic diseases among different cities
Similar to paper [39], we denote the size of the network as , and and are the numbers of susceptible and infective individuals, respectively, so the total number of individuals in the network is and is the average density of people. Because the number of individuals on each node is a random non-negative integer, we set and as the numbers of and on node . To take into account the heterogenous quality of networks we have to explicitly consider the presence of nodes with very different degree . A convenient representation of the system is therefore provided by the following quantities:
where is the number of nodes with degree and the sums run over all nodes having degree equal to .
Just as in [39], we also assume that the mobility of people is unitary time rate 1 along one of the links departing from the node in which they are at a given time. This implies that at each time step an individual occupying in the node with degree will travel to another city with probability (with rate in continuous time).
Thus, the dynamics of epidemic spreading can be described as follows:
2.19
We now explain the right-hand side terms of the first equation of (2.19). The first term is obtained by considering that at each time step the infected people live in a city of degree move to other cities with unitary rate, and the positive term contributing to the infected individual density is obtained by summing the contribution of all individuals moving to the city of degree from their neighbors of degree , including the new infected individuals generated by the term , . The right-hand side terms of the second equation of(2.19) are similar.
2.4.3 Epidemic spreading within and between cities
In this section, we assume that the epidemic disease not only occurs within individual cities but also between connected cities. Moreover, we also consider two types of epidemic spreading inside each city. In the case of type , we consider that each individuals may be infected by all the individuals in the same cities, and in this case the epidemic rate is when the spreading of the epidemic disease happen in the same cities. This case is discussed in the following subsection. In the case of type , we consider that each individual has a finite number of contacts with others, and in this case the epidemic rate has to be rescaled by the total number of individuals in city , that is,
