A survey on modelling of infectious disease spread and control on social contact networks
aa r X i v : . [ phy s i c s . s o c - ph ] F e b A survey on modelling of infectious disease spread andcontrol on social contact networks
Md ShahzamalMacquarie University, Sydney, [email protected] KhanUniversity of Queensland, Brisbane, [email protected]
Abstract
Infectious diseases are a significant threat to human society which was over sightedbefore the incidence of COVID-19, although according to the report of the WorldHealth Organisation (WHO) about 4.2 million people die annually due to infectiousdisease. Due to recent COVID-19 pandemic, more than 2 million people died dur-ing 2020 and 96.2 million people got affected by this devastating disease. Recentresearch shows that applying individual interactions and movements data couldhelp managing the pandemic though modelling the spread of infectious diseaseson social contact networks. Infectious disease spreading can be explained with thetheories and methods of diffusion processes where a dynamic phenomena evolves onnetworked systems. In the modelling of diffusion process, it is assumed that conta-gious items spread out in the networked system through the inter-node interactions.This resembles spreading of infectious virus, e.g. spread of COVID-19, within apopulation through individual social interactions. The evolution behaviours of thediffusion process are strongly influenced by the characteristics of the underlyingsystem and the mechanism of the diffusion process itself. Thus, spreading of in-fectious disease can be explained how people interact with each other and by thecharacteristics of the disease itself. This paper presenters the relevant theories andmethodologies of diffusion process that can be used to model the spread of infectiousdiseases.
Keywords: diffusion; disease spread; contact networks, human movements, and covid-19 Introduction
Infectious diseases are a significant threat to human society which was over sighted beforethe incidence of COVID-19, although according to the report of the World Health Organ-isation (WHO) about 4.2 million people die annually due to infectious disease. Due toCOVID-19, more than 2 million people died during 2020 and 96.2 million people affectedby this disease. The infectious diseases can also spread through animal contact networks,insect contact networks and even with plant contact networks [1, 2, 3]. Therefore, it isvery important to understand the theories and methods for modelling spread of infectiousdisease and its control. The popular approach to model infectious disease spread is usingtheories and techniques of diffusion process on contact networks. The availability of highcomputational power allows researchers to simulate and analyse dynamics of infectiousdisease spread using the models of diffusion. Therefore, building simulation tools basedon the diffusion modelling theories can help authorities to make policies and strategiesto manage spreading of infectious diseases within a population.A diffusion processes is defined as a dynamic phenomena on a networked system,connecting a set of nodes, that starts from a node or a set of nodes and spread over thenetworked system through inter-node interactions. In the diffusion process, contagiousitems (infectious particles, a piece of information, innovation and a specific behaviouretc.) initially grow on one or more nodes of the networks and then spread throughneighbouring nodes over the network. The interactions among nodes are responsiblefor transmitting contagious items from one node to the other nodes. The interactionscan happen through physical contact such as being in a common location, touch thesame object within a time frame and transferring of objects from one node to othernodes through medium such as air. In infectious diseases scenarios, infected individualsinteract with susceptible individuals and disease is transmitted to susceptible individuals.Thus, the nodes of a networked system underlying of a diffusion process are representedby individuals and the inter-node interactions causing transmission of infectious particlesare called contacts, edges, or links in the study of disease spreading on social contactnetworks [4, 5, 6].Modelling of diffusion dynamics on networked system or social contact networks isrequired to understand what factors shape the diffusion behaviours. In reality, the factorsgoverning the behaviours of diffusion process vary depending where it happens. However,the spreading of contagious items, e.g. infectious particles, over a contact network isoften treated as the coupling of results of three factors, namely individual interactions,characteristics of contagious items, and environments. These three factors make strongcontributions to diffusion phenomena on a contact network. The roles and influence ofthese factors depend on the context of diffusion processes.Researches on diffusion modelling identified interactions between individuals are the2ain drivers for spreading contagious items on contact networks. The individual inter-action patterns provide pathways to spread contagious items on contact networks. Forexample, how many susceptible individuals an infected individual meets during their in-fectious period determine the spreading speed of the infectious disease. If an infectedindividual is connected to many other individuals, there is a high probability to transmitdisease to others by him. On the other hand, if an infected individual has no contact withother individuals, the disease is not transmitted. Thus, the final size of the epidemic,total number of infection caused during an outbreak of an infectious disease, depends onthe contact patterns distribution in the population. There are several interaction prop-erties such as how contact happens, contact frequency and contact duration etc. at theindividual levels and contact degree distribution, clustering coefficient etc. at the net-work level that are studied to understand and model diffusion processes on social contactnetworks.The contagious items can be infectious particles for spreading infectious diseases. Thecontagious items themselves play strong roles in their spread in a population and is iden-tified as the critical factors to spread disease in the literature. The internal spreadingpotentiality of contagious items varies based on its characteristics. For example, the in-fectiousness of contagious particles is defined by the disease types and highly infectiousdiseases usually spread faster in a population. The spreading potential is also affected bythe process of infectious particles generation. The impacts of contagious items also varydepending on the recipient individual behaviours. For example, the impacts of infectiousparticles varies according to the susceptibility of the individuals. Therefore, the charac-teristics of contagious items are often considered in the modelling of diffusion processes.In case of modelling infectious disease spreading, the characteristics of infectious itemsvaries widely due to the various nature of disease.The spreading of contagious items is often influenced by the environment. The envi-ronments represent the characteristics of the space where diffusion processes occur. Forexample, the impacts of infectious particles are determined by weather conditions such astemperature and humidity etc. The infectious particles generally lose their infectiousnessover time and may also depend on the weather conditions. The underlying medium wherecontagious items spread can also be heterogeneous. For example, a disease can spreadthrough multiple platforms such as proximity contact networks, transportation contactnetworks, and air-travel contact networks. Thus, diffusion modelling is also required toconsider the heterogeneity of diffusion medium.To model the diffusion on contact networks, these factors should be included to cap-ture realistic diffusion behaviours. The impacts of contagious properties and environmenthave been studied for a long time in the literature [7]. However, analysing contact struc-ture to understand the spreading of contagious items is comparatively new. The recentexploration of data on individual interactions have fuelled research on unravelling contact3atterns affecting contagious spreading [8, 9, 10, 11].A spark has been seen on modelling of COVID-19 spread and control using socialcontact networks properties [12, 13, 14, 15]. The authors of [12] shows that there is sig-nificant variation in the number of connections and social contacts each individual makein a social interaction networks. Individuals with more social contacts are more likelyto attract and spread infection. These individuals are likely the drivers of the epidemic,so-called superspreaders. When many superspreaders are immune, it becomes more dif-ficult for the disease to spread, as the connectedness of the social network dramaticallydecreases. Authors of [16] also show that properly taking some relevant network featuresinto account, linear growth can be naturally explained. Further, the effect of nonpharma-ceutical interventions (NPIs), like national lockdowns, can be modeled with a remarkabledegree of precision without fitting or fine-tuning of parameters.This paper analyse the approaches to integrate individual contact patterns with dif-fusion modelling and discuss how these approaches are used in modelling of infectiousdiseases.
Diffusion modelling is an intensively researched area due to its wide applications. As thearea of diffusion is diverse, the models developed are extremely varied in their approaches.Broadly speaking, the models developed can be divided into two groups based on theirpurposes: i) explanatory models, and ii) predictive models [17, 18, 19, 20]. The explana-tory models are usually developed to understand the factors affecting diffusion dynamicson a contact network. This often allows one to answer the questions such as which nodesare influential, what is the underlying reason for the way diffusion occurs, and what isappropriate diffusion controlling strategy? On the other hand, predictive models usuallypredict the spreading intensity and the final number of individuals received contagiousitems based on certain factors. It is often the case that the explanatory models find thekey influential factors and apply them for developing predictive models. This paper fo-cuses on understanding the diffusing approaches for explanatory models. There has beena range of approaches for explanatory models of diffusion. This section discusses someof them that are widely used in different fields of diffusion ranging from disease spread-ing on individual contact networks to information spreading on online social networks(OSN) [19, 20].
Compartment epidemic models are frequently applied to study diffusion in many appli-cations such as information diffusion, innovation diffusion and computer virus spread-4ng [21, 22, 23]. The fundamental concept of these approaches is to divide the systeminto several compartments or partitions. Each compartment represents a set of individ-uals having a specific status. Then, the dynamics of the diffusion are determined by theflows between these compartments. Widely used compartments are Susceptible havingindividuals who are exposed to the contagious items, Infected having individuals whohave adopted the contagious items and started forwarding items (infecting) others, andRecovered having individuals who were infected but now recovered. This compartmentmodel is called SIR propagation model and the dynamics of the compartments are givenby dSdt = − βIS, dIdt = βIS − τ I, dRdt = − τ I (1)where S , I and R are the fractions of the population in the Susceptible, Infected and Re-covered compartments respectively, β is the transmission rate from Susceptible to Infectedcompartment and τ is the transmission rate from Infected to Recovered compartment.Thus, the dynamics of the system is given by S ( t ) + I ( t ) + R ( t ) = 1. In the diffusionmodelling, β is given as β = γβ , where γ is the number of potential contacts on averageindividuals has with others through which contagious items transmit with a rate β . Bychanging the value of γ , τ and β one, therefore, can study the diffusion dynamics ofdiffusion processes.The compartment model can be analysed easily mathematically in the simple case. Itrequires no more details than needed to reproduce and explain observed behaviours. Itreduces data collection cost and computational cost. Clearly, it can be applied in manysituations where high precision is not necessary. However, the assumption of havinghomogeneous interaction γ between individuals is not realistic. Many researchers havepointed out that the interaction between individuals is clearly heterogeneous as individu-als do not have the same level of contact with all its neighbours [24, 25, 26, 25]. Moreover,the constant rate of transmission probability β and constant recovery rate τ are not re-alistic in many diffusion processes. This is because individuals have different contactintensity with infected individuals and heterogeneous susceptibilities to the contagiousitems [27, 28, 29].Heterogeneity is often integrated into the compartment model by dividing the maincompartments (such as S, I, R) into sub-compartments. These sub-divisions can be con-structed based on the age, risk behaviours, or spatial diversities of individuals. Then, thetransmission probability can be divided into k sub-classes and the model can be param-eterised by means of a k × k transmission rate matrix instead of a constant transmissionrate β [30, 31]. For example, some disease spread models split individuals spatially (dividepopulation with different regions) and assign heterogeneity for infection risks [32, 33, 34].These approaches are called meta-population models. Similarly, different infectious pe-riods can be implemented in the model by dividing population into k sub-classes which5esolve the limitations of the constant rate of recovery [27].The integration of heterogeneity with sub-compartmentalisation relax some of themost unrealistic assumptions of basic compartmental models. However, many limitationsof the compartment model still prevail and new issues arise by doing sub-compartmentalisation.The analysis in [35] shows that individual’s interaction is still random and transient inthese models. Hence, individuals in the divided sub-populations behave homogeneously.In case of meta-population models, dividing a population into the various spatial groupscan create asynchronous between these groups, but for time t → ∞ they become ho-mogeneous [36]. Therefore, the real steady state heterogeneity cannot be captured bymeta-population models. Thus, high partitioning in compartmental models lose the sim-plicity. It also requires more data to fit the model and increases the data collectioncost. To overcome the limitations of the compartment models and capture realistic contactpatterns, network science is adopted for modelling diffusion processes on the networkedsystems [37, 38]. The network based diffusion modelling is also empowered by the graphtheory where contact networks are often generated by graph models. In addition, graphtheory is applied to study the characteristics of contact networks. The core entities inthe network-based modelling are nodes, representing individuals, and links connectingone node to other nodes in the network, which represent interactions between individ-uals. A contact network can be represented by a graph where vertices correspond tonodes and edges to links. The network models provide a range of flexibility for assigningnodes various attributes and defining links with a range of properties. A wide range ofefficient network based models has been developed in the literature for studying diffusionprocesses [37, 38, 39]. For generating contact networks, a fundamental aspect is to buildnetwork structure called the network topology based upon which nodes interact witheach other. There are four types of network structure namely regular lattice networks,random networks, small-world networks and scale-free networks that are frequently usedto study diffusion processes [40, 41, 42, 43]. In addition, data-driven network models arealso derived from real-world data and they may assume some properties of the theoreticalmodels.Regular lattices are the simplest representation of contact network structures wherenodes are only connected to their nearest neighbour nodes in a lattice with a regularfashion [42, 43]. The regular lattice networks assume large path lengths, i.e. the averagedistance between two nodes is very high and the clustering coefficient is very high as well.Therefore, these are not realistic [44, 45]. The random network models improve regularlattice model where nodes contact with each other in a random fashion and each pair of6odes has an equal probability to be connected. Furthermore, the average path lengthsin the random networks match with many real-world networks with appropriate con-tact probabilities [46]. However, the clustering coefficient is too low for these networks.Recently an approach is introduced called the small-world network model based on sixdegrees of separation phenomena which states that if you choose any two individuals any-where on Earth, you will find a path of only six acquaintances on average between them.In a small-world network, most of the nodes are not neighbours of each other, but theneighbours of a given node are likely to be neighbours of each other. The Watts-Strogatzmodel [47] generates such networks where the existing links of a regular lattice are re-wired with a defined probability. The generated networks assume high local clusteringco-efficient and short path length [48]. The latest approach to generate contact structureis scale-free networks developed by the Barabasi-Albert model [49] where node’s degreefollows a power-law distribution. In the Barabasi-Albert model, the scale-free networksare self-organised with growing and preferential attachment processes. The research foundthat many real-world systems have a power-law degree distribution [50, 51].The above structured contact networks can be analysed mathematically and numer-ically. The authors of [52] has presented surveys on the methods of designing contactnetworks applying real contact data. These networks are, however, static in nature wherenode attributes and link properties are not changed during the observation period. Thesecontact networks are often represented with adjacency matrices of binary values. Thisallows the use of algebra to calculate various network properties and corelate it with dif-fusion dynamics unfolded on it. While having some strong benefits over compartmentalmodels, these network models still have several shortcomings, namely that in these mod-els, the quality of contacts is overlooked. For example, the duration of contacts affectsthe transmission probability, and frequency of contact etc. There have, however, beensome models to overcome these limitations with weighted contacts [53, 54, 55]. However,the weighted contact networks do not capture burstiness of the contact which is foundto have an impact on spreading dynamics [56]. The other crucial temporal factor is thatthe contact sequences among individual are completely missing in these network models.For studying diffusion processes with realistic contacts, there have been several ap-proaches to make the contact network dynamic as well [39, 57, 58]. The dynamic networksassume the links are transient in status, i.e., links appear and disappear. However, therelationship between two linked nodes is often permanent. In the dynamic contact net-work models, the above static network models can be implemented as the underlyingstructure (capturing permanent social relationship among individuals) and an additionalmechanism is added on top of that to maintain the link dynamic. The dynamic contactnetwork models are often difficult to analyse with exact mathematical solutions. Thus,an approximation is often used to characterise the system dynamics [59]. There are noanalytical solutions for many dynamic contact network models and such models are only7sed for simulations to explore diffusion dynamics for wide scenarios of developed mod-els. These classes of contact networks are often efficient tools to validate the simulationsresults of data-driven individual-level diffusion models.
The other trend of diffusion modelling on contact networks is to apply individual-basedmodels [60, 61, 62, 63]. In these models, all operations are executed at the individual-leveland thus the integration of many realistic contact properties becomes easier. The otherfundamental concept is that individual-based models are implemented upon a communityof targeted individuals and that are situated in an environment. In these models, everyindividual plays its role and interacts with its respective environment. Thus, infectiousitems are received by an individual according to its behaviour and surrounding conditions,and it transmits contagious items to other individuals by regenerating it. However, it isnot so easy to define the boundaries of the model class based on individual comparedto compartmental models or network models as the assumptions in the individual basedmodels varies largely.The system dynamics in the individual-based model are generated with all individualactions happened simultaneously within the respective simulation environment of theindividual [64]. The respective environment depends on the modelling approaches andit may include parts or sometimes all of the other individuals. Thus, all individuals areaffected by the state of neighbours in their simulation environment at the same time.Individual response to a specific environment can be deterministic or can be stochasticevents. The reactions process to a simulation environment is often implemented witha set of rules (e.g. IF-THEN operations). For disease simulation, such a rule can beIF the individual is susceptible and if there was a contact with an infected individual,THEN switch the status from susceptible to infectious with a certain probability P .Some individual-based models implement the process of adaptation, learning or evolution.These models are called agent-based models, which is a subset of individual-based models,and have simulated intelligence [65, 66, 67, 68].The most significant advantage of individual-based models is that they allow for theinclusion of natural mechanisms for every desired aspect of the model to be as realisticas possible. They can offer characterisation even at the link level and environmentalconditioning including complex biological mechanisms [1, 69]. The current exploration ofdata on social interactions leverages the benefit of this class of model as they easily allowmodelling of individual interactions over time. The complexity of implementing higher-level architecture such as clustering and community structure are reduced as the networkformation mechanism is implemented at a lower level. The contact networks createdby dynamic contact network models can also be simulated with individual-level models.8owever, the individual based models become difficult with detailed information andrequire more effort to analyse sensitivity. To achieve stable insights, repeated simulationsare conducted with a high number of parameter combinations. Therefore, the individualbased model requires more computing resources and computation time. These modelsoften cannot be analysed mathematically due to their stochastic nature and large numberof parameters. Contact networks are widely used tools to study diffusion on social contact networks.This allows to characterise the diffusion behaviours and simulate the spreading dynam-ics. These contact networks can mimic the social interaction and reveal the influence ofindividual interactions. Contact networks are often generated by graph models. It is dis-cussed in the previous section, dynamic contact networks are realistic to model diffusionprocesses. However, the development of dynamic graph models for generating dynamiccontact networks is still at an early stage compared to the static models. There have beenlimited number of approaches to develop dynamic graph model. This section presents abrief details of current dynamic graph modelling approaches which consist of representingcontact networks and links generation. Here, dynamic contact graphs/networks repre-sent temporal or time-varying graphs/networks where edges between a pair of nodes aredynamic as their availability for transmission are not permanent.
The evolution of a dynamic contact graph can be captured in many ways. The evolutionin the graph can occur due to changes in the status of nodes and status of links. Thelinks in the static graphs represent a relationship between a pair of nodes and is createdthere is at least one interaction during observation period [70, 71]. In dynamic graphs,however, the links are often differentiated from contacts (links and contacts have differentmeaning) [39, 72, 58]. The contacts indicate interactions between a linked pair of nodesoccurring at certain times during an observation period. Dynamic graph modelling isrequired to incorporate timing information of these contacts with link dynamics. Thedevelopment of a dynamic graph model often depends on how the graph is represented.The dynamic contact graphs can be represented in the following ways.
Contact sequences:
Many real world interaction data sets comes with the entriescontaining identities of interacted nodes and the time when the interaction happened,even with other some meta information such as gender and locations. The interactiontime can be a time stamp or a time interval sequence. For examples, works of [73, 62]have collected interactions between two individuals using RFID and wearable sensors.9his representation is a straightforward and practical format computationally. However,analysing diffusion processes on the graphs with this format would be difficult as theydo not count some properties such as contact duration. It is also difficult to visualise thecontact graphs and hence representing it to audience.
Multi-layer graphs:
The dynamic graph can be visualised well if it is representedwith as a sequence of static graphs. In this method, the observation time is dividedinto discrete time steps and a static graph is constructed for each time step [74, 75].Thus, the dynamic graph becomes a multi-layer graph with each sequence of the staticgraph as a single layer. This allows one to understand and analyse the dynamic graphusing static graph theories and then combine the results for the sequence of times toobtain overall results. This method is applicable where the time resolution is high (orcontinuous) compared to dynamic process on the studied graph. For studying infectiousdisease, this method has a limitation as the disease cannot be transmitted over a pathduring a sequence of the graph and thus analysing multi-layer graphs cannot capture realdynamics. The time-lines representation of contacts is one of the extended approacheswhere nodes are placed in one axis and times in another axis. The advantage with thisrepresentation is that the time-respecting paths (sequences of contacts of increasing times)between nodes are easy to identify as these are all paths that do not turn backwards in thetime dimension. The structure of time-respecting paths can be represented as a binarymatrix as it is in a static graph with an adjacency matrix. Thus, the dynamic graph can beexpressed as a binary tensor. The limitation is that the corresponding adjacency tensor,as a data structure, takes a lot of memory and requires high computational overhead toprocess such graph [59, 76].
Dynamic links graph:
In this representation, temporal variations are capturedwith only one dynamic graph where nodes and links change their status over time. Theunderlying graph is a static graph with the fixed links among nodes. In fact, the un-derlying graph captures the fixed topology of the dynamic graph and can be treated asthe foot-print of nodes [77, 78]. Then, the static graph structure evolves over time wherecontacts can appear and disappear. This means a time dimension is added with thestatic network. This approach is considered for the class of graphs where the targetedresearch question is to understand how the structure has evolved and how it affects thediffusion process unfolded on the graph. The dynamic graphs are typically data-orientedwhere the focus of study is on a data set, its structure, and how something behaves onit e.g. how disease spreading would behave on the graph. In addition, these observationsmay vary with used data sets and generalisation of results are often difficult. However,it is used widely due to its flexibility to implement and capture properties of real contactnetworks.
Time-node graphs:
The recent trend of dynamic graph modelling is to extend theconcept of node into temporal node i.e at each time step the same node is considered as10 different node. Then, the graph is built among the temporal nodes. This approach iscalled the static expansion of a temporal graph [79]. This type of graph can be practicalsince it is straightforward to apply static graph methods also over the time dimension.Eventually one usually needs to map the time nodes back to the original nodes. Thisrequires high computational power which is available in the current technology. However,the applicability is limited by the size of the networks.
A general representation of the dynamic contact graph can be described as follows. Con-sider a dynamic contact graph G T that is built with a set of nodes Z , a set of relationships L between these nodes (links, contacts), and a labelling sets Y which represents any prop-erty such as links weights, set of node attributes; that is, L ⊆ Z × Z × Y . The relationsbetween nodes are assumed to take place over a time span Γ ⊆ T denoting the lifetimeof the system. The temporal domain T is generally assumed to be N + for discrete-timesystems or R + for continuous-time systems. The dynamics of the system can be subse-quently described by a dynamic contact graph, G T = ( Z, L, Γ , φ, ψ ), where φ : L × Γ → { , } φ is called presence function, indicates whether a given link is available at a given time.The status of node can also be varied over time where they can be active or inactive tocreate links at a certain time. Thus, the model can be extended by adding a node statusfunction ψ : Z × Γ → { , } where the activation function ψ of nodes depend on a given time. Given a G T =( Z, L, Γ , φ, ψ ), the graph G = ( Z, L ) is called underlying graph of G T . This static graph G should be seen as a sort of footprint of G T , which flattens the time dimension andindicates only the pairs of nodes that have relations at some time in Γ. The connectivityof G = ( Z, L ) does not imply that G T is connected at a given time instant with theconnectivity of G . Within this general graph definition, various contact networks canbe generated by varying details in links and nodes definition. Thus, graph models areoften accompanied with a network generation method. This network generation methodhandles incorporating details to links and nodes while graph defines the relationships atthe abstract level. Some network generation methods that have been designed under theframeworks of various graph models are presented here. Static graphs with link dynamics:
The simplest method of generating a dynamicgraph is first to generate a static graph with links using a static graph model and thendefine a sequence of contacts for each link generated. To avoid complexity, contact11eneration process is often kept independent of the network position of the links in thisapproach. The authors of [77] have applied the following procedures to generate adynamic graph.i) a static graph is constructed from a multigraph that is generated using theconfiguration model [80] and deleting the duplicate links and self-linksii) an active interval is generated for each link when contacts can occur i.e.nodes are present in the graph. The duration of active intervals is gener-ated using a truncated power-law. For a link, the active interval starts at auniformly random starting time within a sampling time frame (observationperiod)iii) a sequence of contact times is generated following an inter-event timedistribution over the observation period. This also generates burstiness in thecontact patterns.iv) finally the contact time sequence is wrapped with the corresponding activeinterval of each link. The contact sequence times (step 2) which are within theactive time interval are taken and other are deleted for a link. The wrappingis done for all links in the graph and a dynamic contact graph is obtainedThe authors of [78] used a similar method to generate contact graph. These methods eas-ily integrate contact dynamics and burstiness to the topology of a static graph. However,the inter-event times are not influenced by the topology structure and node properties.
Temporal exponential random graphs:
The exponential random graph model(ERGM) [81] is widely used to generate static graphs in the study of social dynamics.The ERGM model parameters reflect the importance and weight of selected topologicalelements and sub-graphs such as triangles and stars. The ERGM model generates aclass of graphs and the model parameter inferred from an empirical data capture thecorresponding graphs. A similar modelling framework is used by the authors [82, 83] togenerate temporal exponential random graphs (TERGM). Thus, the temporal dynamicsof links are influenced by the network topology in TERGM. In TERGM, a set of statesof the nodes observed over a time window for an ongoing dynamical process is applied toestimate the model parameters. Thus, the probability of making a contact between a pairof nodes is bounded with the time window of applied data set for modelling. The temporalexponential random graph models are not node-oriented model (nodes connectivity is notbuild at the node level, but connectivity is defined based on the network topology) thatmakes possible to change the network with its basic building block such as links. Thus, itis difficult to achieve a good fit unless the successive networks are close to each other [84].
Coordinated temporal graph models:
The above models cannot implement nodeand link level operations such as which neighbour nodes a host node contacts frequently,12ow recent changes in contact sequences affect the future contacts, whether the contactcreation and deletion follow any specific social mechanism. There have been several worksto incorporate these characteristics of dynamic contact graphs [85, 86, 87, 71]. The workof [71] has developed a dynamic graph model for face-to-face interactions. This is aspatio-temporal graph implemented with a two-dimensional random walk. In this model,the propensity of walking closer to a node is proportional to the attractiveness assignedto it. Therefore, the more attracted a walker is to its neighbour nodes, the slower itswalk becomes. A similar approach is implemented in the dynamic graph model developedby [88]. This model for online setting and their assumption is that some individuals aremuch more central in a temporal graph than they are in an aggregated static graph.Thus, random communication partners are assigned to a node by a basal rate and apositive feedback mechanism. The authors have applied stochastic point processes tomodel dynamic contact graphs. In this model, a node creates and breaks links accordingto a Bernoulli process with memory. The probability of an event between two nodesincreases with the number of events recently occured between them.
Activity-driven graph modelling:
A comparatively simple dynamic graph mod-elling approaches is proposed by [57]. This model is called activity driven network mod-elling (ADN). They adapt the graph sequence framework of dynamic graph modellingand generate a simple graph G T at (it is a discrete time system) time t . The graphgeneration procedures are as follows:i) node i is assigned an activity potential a i . This is usually done with apower-law distribution. The activity potential is assigned to all nodes in thegraphii) the graph generation process goes through increasing a time counter to t and assume that G T is empty, i.e. all N nodes have no link and contactmemory from previous time stepiii) activate node i with a probability a i ∆ t . If node i is activated, it is con-nected with other m randomly chosen distinct nodes. Repeat the step 2 andstep 3The distinguishing characteristics of this model are that the activity of the nodesgoverns the link creation. In contrast, the previous models are connectivity driven wherethe network’s topology is at the core of the model formation. The ADN overcomes thetimescale separation assumption and explicitly accounting for the concurrent evolutionof the interactions in a graph and the dynamic process evolving on it [89]. The studies of[90, 91] show that many important aspects of the system dynamics can be characterisedusing a heterogeneous mean-field approach. Interestingly, it is found that some systemproperties are directly related to the activity potential of nodes. The diffusion dynamics13re quite different from that of aggregated static networks [91]. However, the basic ADNmodel has several limitations such as nodes contact with a fixed number of links duringeach activation, the contacts are not repetitive, and social structure among individual isnot maintained. When nodes are active in the basic activity driven network, they randomly create con-nections with other nodes. In other words, at any time, a node may connect with anyother nodes in the graphs. That means this method does not apply any prior knowledgeof social or geographic relationships that could alter the selection of one link over an-other. However, this assumption severely challenges the feasibility of the ADN modellingwhen it is implemented for real graphs. There have been great efforts to incorporaterealistic features with the basic ADN. An individual can interact with other individualsarbitrarily or by choice. Thus, some links tend to be persistent in time and such linksare generated in the household, at the office and with close friends. This property isintegrated introducing memory effects in the link formation. Therefore, social graphs canhave two types of links. The first class describes strong ties that identify time repeatedand frequent interactions among specific couples of nodes. The second class characterisesweak ties among agents that are activated only occasionally. It is natural to assume thatstrong ties are the first to appear in the system, while weak ties are incrementally addedto the contact set of each node. This approach is studied in the work of [92] where itis assumed that a node will connect to a new node with a probability P ( n + 1) = ηη + n ,where n is the current contact set sizes of the node and η is the tendency to broaden theircontact set sizes. Therefore, the probability of contacting with a node from previouslycontacted nodes is 1 − P ( n ). This method of repeating with the old contacted nodes andextending contact set size is called the reinforcement process. In the above process, everynode has the same tendency to extend the contact graph. In reality, however, individualhave heterogeneous tendency to extend the contact set size. This issue is addressed bythe work of [93] where they proposed to assign a heterogeneous value of η . For a node i , the probability of contacting a new node is given by p ( n i ) = (cid:18) n i η i (cid:19) − α i where α i is the reinforcement of node i and η i is the characteristic number indicating thesize of contact set size before reinforcement start. The value of η i is often assigned withpower-law and the distribution of resultant contact set size will be a power-law.In the social graphs, individuals have a tendency to make a close social circle andmake a community. Therefore, the underlying social structure of a dynamic contact14raph also should maintain community structure. The community structure emerges ina graph by creating triadic closure when making a new connections [94]. One typicalmechanism to make triadic closure is to use common neighbours (CN) indices, where twonodes i and j are going to interact if their neighbour nodes set has substantially overlap.This means that the probability of these two nodes interacting is proportional to thenumber of common neighbours. In other words, triadic closure can be created if thehost node chose a new neighbour from its neighbour’s neighbour [94, 95]. The randomnew neighbour selection mechanism of basic activity driven graph can not emerge thecommunity saturate. The work of [72] has upgraded the basic ADN integrating a triadicclosure creation mechanism. If a host node i has not contacted any node yet, it randomlypicks another node from the entire graph j and creates a link. Otherwise, the host nodetries to make a new link with the triadic closure mechanism. As the first step, it selectsrandomly one neighbour node j from his contact set with a probability. If node j is notselected or has no other neighbours node except node i , node i looks for another randomnew node and creates a link. If node j is elected and has neighbour nodes, then it selectsa random neighbour node k from the neighbour set of j . Then node i from a link with k and create triadic closure.The basic activity-driven graph model assigns heterogeneous potentiality to nodes.This generates a heterogeneous distribution of interactions. However, the research onsocial interaction shows that individual’s interactions have bursty nature, i.e the inter-event time of activation of nodes in ADN is required to be heterogeneous [96, 56]. Thisbursty activity has a strong influence on graph evolution and diffusion unfolded on it.The inter-event time t i is directly connected with the activity of node i and can definedas a i =
An infectious disease is transmitted from an infected individual to susceptible individ-uals via transferring organisms/microbes capable of causing infection [99, 100]. Theseorganisms/microbes are called pathogens. In this paper, contagious items or infectiousparticles refer to these pathogens. The infectious items enter the body of susceptibleindividuals and deposit on mucus membranes of body parts such as mouth, nose, throat,and lungs where they can cause an infection. Therefore, for an infectious disease topersist within a population, relevant contagious items are required to be transmittedcontinuously to new bodies. The contagious items are transmitted through two mech-anisms: 1) direct transmission and 2) indirect transmission [8, 101, 6, 58, 102]. Directtransmission occurs through individual-to-individual interactions transferring contagiousitems without any intermediate transmission medium between these two individuals. Theexample includes physical touches (such as shaking hands, kissing etc.) and contact ofblood and body fluids. Direct transmission is found in infectious disease such as commoncolds, sexually transmitted diseases etc. For many infectious diseases [103, 104, 99, 100],infected individuals generate particles containing infectious microbes by their respiratoryactivities like talking, laughing, coughing or sneezing. These particles are scattered intothe environment of the proximity of the infected individuals. The infectious particlesthen deposit onto objects or surfaces and survive long enough time to transfer to othersusceptible individuals who subsequently touch the objects. This creates the indirecttransmission of diseases where intermediate medium or objects are required to transmitinfectious particles. Examples of diseases with the indirect transmission are Coronavirus,Rhinovirus, and Influenza etc.The ways indirect transmission occur are not the same in all cases and can be classifiedinto different modes which are based on the roles of the intermediate medium and theproperties of the infectious particles when transmitting through the intermediate medium.The respiratory activities of infected individuals generate droplets containing infectiousparticles and the sizes of the droplets often define the mode of transmission. Dropletwhose size is comparatively large, often assumes to be greater than 5 µ m, are transmittedthrough the air to nearby susceptible individuals. This mode of indirect transmission iscalled droplet transmission. However, the droplet whose size is small, often assumes tobe less than 5 µ m, evaporates quickly and becomes droplet nuclei. These droplet nucleiare suspended in the air for a long time and can travel large distances. Thus, they cantransmit to susceptible individuals with a long time delay after their generation, even tosusceptible individuals who are far away up to 100m from the source infected individual.16his mode of indirect transmission is called airborne transmission. Indirect transmissionof infectious particles can also happen through vectors (mosquitoes, flies and mites etc.)that carry infectious particle from an infected individual to susceptible individuals withdelay and at substantial distances. In this situation, the contagious items present in theblood or skin of an infected individual are ingested by vectors. Then, it is developedin the vectors itself. Susceptible individuals are usually infected through the bite of aninfectious vector, though other ways of entry are possible. Examples of vector-bornedisease transmissions are yellow fever, malaria, plague and dengue etc. [103, 104, 99,105]. Another indirect mode of transmission is to spread disease through contaminatedobjects spatially. Examples of such diseases includes water-borne diseases, food-bornediseases [106, 107].It has been observed that the spreading of some infectious diseases is dominated byonly direct transmission and can be modelled by creating direct transmission links forco-presence interaction between infected and susceptible individuals. However, the in-fectious diseases that spread based on indirect transmission or have additional indirecttransmission along with direct transmission cannot be modelled by creating only directtransmission links for co-presence interactions. This thesis considers airborne infectiousdisease spreading as a case study for understanding and modelling the impacts of indirecttransmission links. For airborne diseases, infected individuals generate droplets contain-ing infectious particles through various respiratory activities. The authors of [108] havefound that an infected individual generates on average 75,000 particles/cough but it canbe up to 500,000 particles/cough. They have also found that 60% of these particles canreach the alveolar region of lungs if the particles are inhaled by another individual. It isfound that the cough frequency of an infected individual is on average 18/hr [109]. Thus,an infected individual deposits about 1 . × particles during a one hour stay at alocation. Up to 50% of these particles evaporate and become droplet nuclei (airborneparticles) which are suspended in the air for a longer time [110]. Airborne particles arealso added to the environment by breathing, talking and laughing. There have been awide range of studies to understand the viral load of airborne particles. The studies showthat most of the influenza virus is contained in the droplets whose sizes are < µ m. Theworks of [111, 112, 111] have found that up to 75% virus is contained within dropletswith sizes < µ m. The exhaled breath of an influenza patient can generate on average0.5 plaque-forming units (PFU) for influenza viruses [113]. The study of [112] foundthat a cough can generate up to 77 PFU virus. Inhalation of 0.7 - 3.5 PFU of influenzais sufficient to cause infection in 50% of susceptible individuals [114]. Therefore, it canbe concluded that the generated airborne particles have sufficient viral load to causeinfection if they are inhaled.The impact of airborne transmission is different to the other model of indirect trans-mission as airborne particles can travel spatially while large droplets settle nearby. The17iterature indicates the various range of travel distances for airborne particles. The traveldistances depend on the weather conditions and air-flows. The authors of [115] showthat airborne particles can travel up to 100m in the direction of air-flow. The traveldistance can also be interpreted from the analysis of SARS outbreak occurred in theAmoy Gardens Hong Kong in the year 2003. The study of [116] has revealed that theinfection had reached the Block-E which was at 60m distance from the Block-B whereinfection had started, although there was no indication of physical interaction among theresidences of these buildings. Thus, it was concluded that the infection particles travelledto the Block-E through airborne transmission. A number of studies have also shown thatit is also possible to disperse airborne particles between flats in a building [117, 118] andbetween wards in a hospital [119]. The experiment of [118] shows that airborne particlescan also travel from one building to the nearby buildings. The travel distance of airborneparticles is extended in the open area. The authors of [120] have studied the presenceof influenza A virus around pigs farms by collecting air samples at different distancesfrom the farms. They have noticed a significant amount of RNA copies of the virus atthe distance of 1.5Km from the pig farms that had influenza A infected pigs. There-fore, the airborne indirect transmission mode of infectious disease has strong potential tospread diseases. The airborne infectious diseases spreading is an important applicationof diffusion process with indirect transmissions. An interaction (e.g. being in the same location) between infected and susceptible indi-viduals poses an infection risk for the susceptible individual. Infection risk assessmentcan be divided into two steps: determining the intake dose of infectious particles andfinding the corresponding infection probability [121]. The infectious particles that reachthe target infection site are called the intake dose. The intake dose is estimated basedon the exposure level to the infectious particles, the pulmonary ventilation rate of sus-ceptible individuals, the exposure time interval, and the respiratory deposition of theinfectious particles. Then, the infection probability is calculated by a mathematical for-mula. Two approaches are applied to determine if an infection occurs: deterministic andstochastic. The first approach assumes that each individual has an inherent resistanceup to a dose of infectious particles. Thus, a susceptible individual contracts the diseasewhen a target infection site is exposed to a dose equivalent to or exceeding the thresholddose. In the stochastic approach, any amount of intake dose causes disease with a certainprobability. The infectious particles are usually randomly distributed in the suspensionmedium. Thus, the estimated exposure level and intake dose of airborne particles arealways expected values rather than exact values. Therefore, the stochastic models areappropriate for studying airborne disease spread. Models that are frequently used for18ssessing infection risk for airborne diseases are now discussed.A wide range of models has been developed for the spread of airborne disease. Theserange from simple models that are easy to apply to complex models that require greaterdetail of the disease spreading process. Unfortunately, these details is not always availablefor many diseases. In the literature, the Wells-Riley model or its modification are widelyused to estimate infection risks [121, 122]. The Wells-Riley equation is given as P I = 1 − exp (cid:18) − IgptQ (cid:19) (2)where P I is the probability of causing infection to a susceptible individual for the intakedose E = Igpt/Q , I is the number of infected individuals at the interaction room, p is thebreathing rate of the susceptible individual (L/s), g is the average quanta generation rate(quanta/s), t is the exposure time interval, and Q is the room ventilation rate (L/s). The P I is, in fact, the ratio between the number of infections caused for E and the susceptibleindividuals. This model is based on the concept of quanta which is the number of dropletnuclei required to cause infection for 63% of all exposed susceptible individuals. The ratio P I provides the reproduction number of the studied diseases which is frequently used todetermine disease spreading dynamics for the large population. The model parameterquanta generation rate ϑ is required to be estimated from the real outbreak cases. Thisis very difficult for many diseases as it requires data from real outbreak scenarios. Themodel is also limited due to its assumption that particles are homogeneously distributedin the air, and that every particle reaches to the target infection site. It does not considerthe duration of particle generation.There have been several modifications to overcome these limitations. The authors of[122] incorporated the effect of respiratory protection system that may filter the inhaledinfectious particles by multiplying a fraction term with the intake dose as E = IgptθQ ,where θ is the fraction of infectious particles reached to a target infection site. Airdisinfection and particle filtration are used in the many buildings that reduce the effectiveinfectious particles to cause infection. These factors are included for the Wells-Rileyequation in the work of [123]. However, collecting such data is difficult and expensive forlarge scale simulation. The assumption about the homogeneity of particle distribution inthe interaction area is addressed by [124]. They considered the time-weighted averagepathogen concentration in the room air to incorporate the non-steady-state conditions inthe Wells-Reily equation. This model is given by P I = 1 − exp (cid:18) − pIg ( Qt + e − ϕt − V Q (cid:19) (3)where Q is the air change rate or disinfection rate, ϕ is the particle accumulation rateand V is the volume of interaction area. In spite of these improvements, the Wells-Reily19odel still requires the total exposure during an outbreak to find the quanta generationrate and that is not possible for many diseases.Rudnick and Milton [125] developed a model where the exhaled air volume fraction isused to estimate the number of quanta that the susceptible individuals are exposed to: P I = 1 − exp (cid:18) − g ¯ ωItN (cid:19) (4)where ¯ ω is the average volume fraction of room air that is exhaled breath and N is thetotal number of people in the premises. To find the quanta generation rate based on the ¯ ω ,one requires a knowledge of carbon dioxide concentration in the room. These models stillfollow the well-mixed assumption of particle concentration. Some works [118, 126] addressthis problem by experimenting the dispersion of tracer gas and integrating impacts withmodel.In the models discussed above, the quanta generation rates are not well understoodfor many diseases. However, the infectious particles generation rates, their formation,pathogen loads and their survivable time etc. are now becoming available. The authorsof [127] first introduce a dose response model based on the infectious particles concentra-tion instead of quanta. The model is P I = 1 − exp (cid:18) − IgθptQ (cid:19) (5)where g is the number of infectious particles released per infected per unit time and θ isthe fraction of infectious particles reaches the target site. In this equation, the quantageneration rate g is replaced by θg . The authors defined the source strength g with coughfrequency, pathogen concentration in the respiratory fluids and the volume of expiatorydroplets introduced into the air in a cough. This model also based on the homogeneity.Recently, the authors of [128] have also introduced a model based on the infectiousparticle concentration considering non-steady-state conditions as P I = 1 − exp (cid:18) − I ( g − r ) θptQ (cid:20) − VQT (cid:16) − e − QτV (cid:17)(cid:21)(cid:19) (6)where g is the particles generation rate, r is the mortality rate of the generated particles, θ is the deposition fraction of the inhaled particles, τ is the duration of particle generation,and t is the duration susceptible individuals breath in infectious particles.In the above equations, the temporal variation in the particle concentration is cap-tured using a non-steady-state model. However, this model assumes that all infectedindividuals arrive at the same time and this may not happen in reality. The variationsin the arrival time of infected individuals also introduce the fluctuations in the particlesconcentration. The current models also do not capture the exposure that susceptible in-20ividuals receive after infected individuals leave the interaction locations. Thus, it wouldbe more appropriate to find exposure level due to contact with each infected individualand sum them up to find total exposure. Therefore, the arrival and departure of eachinfected individual can be tracked independently and hence the exposure during indirectinteractions. This also allows one to assign a random value of Q to each contact tocapture heterogeneous particle concentrations at different locations.The disease transmission probability can also be calculated for both the direct andindirect contacts as follows [8]. This model resolves the problems mentioned in theprevious section. This model define a contact called same place different time (SPDT)that has both direct and indirect infection transmission probability. This accounts whenan infected individual and susceptible individuals have been to a place. If a node in thesusceptible compartment receives a SPDT link from a node in the infectious compartment,the former is subject to exposure E l of infectious pathogens for both direct and indirecttransmission links according to the following equation E l = gpV r h r ( t i − t ′ s ) + e rt l (cid:16) e − rt i − e − rt ′ l (cid:17)i + gpV r (cid:16) e − rt ′ l − e − rt ′ s (cid:17) e rt s (7)where g is the particle generation rate of infected individual, p is the pulmonary rate ofsusceptible individual, V is the volume of the interaction area, r is the particles removalrates from the interaction area, t s is the arrival time of the infected individual, t l is theleaving time of the infected individual, t ′ s is the arrival time of susceptible individualsand t ′ l is the leaving time of susceptible individuals from the interaction location and t i isgiven as follows: t i = t ′ l when the SPDT link has only a direct component, t i = t l if theSPDT link has both direct and indirect components, and t i = t ′ s otherwise. If t s < t ′ s , t s is set to t ′ s for calculating an appropriate exposure [8]. If a susceptible individual receives m SPDT links from infected individuals during an observation period, the total exposure E is E = m X k =0 E kl (8)where E kl is the received exposure for k th link. The probability of infection for causingdisease can be determined by the dose-response relationship defined as P I = 1 − e − σE (9)where σ is the infectiousness of the virus that causes infection [99].21 Diffusion control
Controlling diffusion dynamics on individual contact networks has a wide range of appli-cations ranging from mitigating the spread of infectious diseases to marketing products.The methods developed for controlling diffusion depends on the context and applications.For example, controlling diffusion for marketing a products focuses on maximising thespreading of items to the largest proportion of populations [129, 130] while diffusion con-trolling for infectious disease focuses on minimising the number of infections reducingthe number of individuals received spreading items [131]. However, the key task in allcontrolling methods is to find a set of individuals and change their behaviours to alterspreading rates. These individuals often have high spreading potential and are calledsuper-spreaders. The size of the set should be minimal to reduce vaccination cost as wellas achieve the control goals.Most of the efforts of developing a control strategy are put on finding the optimalset of individuals. Accordingly, researchers search for the contact properties of individ-uals and their behaviours relevant to the spreading of contagious items [132, 133, 134].Individual’s preference and exposure intensity to contagious items, personal status andtheir surrounding environment often define the spreading potential. Thus, the influen-tial individuals are often searched based on the individual’s behaviours [135, 136]. Thenetwork properties such as the number of connection of individuals to others are alsokey factors to determine one’s spreading potential. Understanding personal behavioursand modelling is a complex process. In addition, research focuses on understanding theimpacts of contact properties on diffusion dynamics.Various measures of network properties are applied to find influential individuals forcontrolling diffusion of contagious items. The widely used measures to find the importantindividuals are degree centrality, betweenness centrality, k-core score and PageRank cen-trality etc [132]. Individuals contact degrees defined the number of connections to otherindividuals is frequently used as topological measures of influence. In the social contactnetworks with broad degree distribution, it is observed that individuals with high degreedetermine the diffusion dynamics [137]. However, degree based methods sometimes un-derestimate the low degree individuals that can be influential through connecting highdegree individuals. The page ranking algorithm developed to rank the content in theWorld Wide Web is also adapted to find the pivotal individuals in social contact net-works. The ranking mechanism of PageRank is simple and straight forward [138] wherethe importance of page is measured by counting the number and quality of links to thatpage. The PageRank algorithm is only applicable to directed networks. Betweennesscentrality is also a good candidate in many applications as it is a measure of the num-22er of shortest paths passing through one individual [139]. Thus, it is most likely thatindividuals having high betweenness centrality will play key roles in shaping the diffu-sion dynamics on the networks. It is efficient but requires high computational resources.Moreover, it is applicable for undirected networks. The K-core score present the positionsof individuals in the social networks with the k-index obtained by iteratively removingk-degree nodes [140]. These networks measures are based on static networks and maynot provide optimal performance in dynamic contact networks.The one way of finding influential nodes in dynamic contact networks is to use tem-poral versions of traditional centrality metrics [133, 134]. In this work, time respectingpaths, paths are created based on the time order of links availability, are used to calculatethe betweenness centrality and closeness centrality. Eigenvector centrality is also modi-fied for dynamic networks [141]. These methods require complete information regardingcontact networks. The random walk is applied in [142] for measuring temporal central-ity. This does not require global information. However, all these algorithms require hugecomputational resources. The temporal centrality measurement is in the early stage andis not still feasible for applying in large social contact networks.Applicability of these methods depends on the application scenarios. For example,betweenness centrality can be applied to find the influential individuals in online socialcontact networks as the contact information often is available [132, 143].
It is difficult to apply the above approaches for controlling disease spreading as it is quitedifficult to collect contact information of a population. The vaccination strategies arerequired to be developed based on the contact information that can be obtained locally.The key task of a vaccination strategy is to choose a set of individuals based on thelocal contact information. The simplest way of selecting a set of individuals is to chooserandomly from the population and is called random vaccination [144, 145, 146]. Thisapproach does not consider the disease spreading behaviours of the chosen individuals.Therefore, the information collection cost is minimal. However, it requires a large set ofindividuals to be vaccinated for achieving hard immunity to disease. Thus, the researchis directed to select the individuals who have strong disease spreading potentials [147].These methods are called targeted vaccination. In the targeted vaccination, the numberof individuals to be vaccinated is often small and the effectiveness of strategies is substan-tially high if an appropriate set of individuals are chosen. Therefore, the infection costcan be substantially lower in a targeted vaccination strategy with the reasonable cost ofinformation collection and vaccination cost.There has been a wide range of vaccination strategies using obtainable contact infor-mation [148, 149, 150, 151]. All these methods do not develop vaccination strategies based23n local contact information. Sometimes global information is also used and methods aredeveloped to find the global metrics with locally obtainable contact data. There have beenseveral other methods that depend on the movement behaviours of individuals insteadof collecting information on interactions between individuals. Vaccination strategies arealso varied based on the implementation scenarios. There are two specific vaccination sce-narios: preventive vaccination (pre-outbreak) and reactive vaccination (post-outbreak).The vaccination strategies that apply local contact information is now first discussed.Then, the implementation of vaccination strategies is discussed. As examining vaccina-tion strategies in the real-world scenarios are expensive and difficult, empirical contactnetworks or synthetic contact networks are frequently applied to test and validate devel-oped vaccination strategies. The discussion includes the network model based vaccinationstrategies.The authors of [145, 149] present an elegant way of implementing vaccination us-ing local contact information called acquaintance vaccination (AV). According to thisstrategy, a randomly picked node is asked to name a neighbour node to be vaccinated.Therefore, no knowledge of the node degrees and any other global information of net-work are required. In fact, it selects the node that has a large number of connection toother nodes. Its efficiency greatly exceeds that of random vaccination. The acquaintancemethod is also improved in few other ways. Instead of vaccinating random acquaintance,it is more effective to vaccinate the acquaintance who has more frequent contact [148].That means the selected individual should be asked to name friends who contact fre-quently. This method substantially improves the efficiency of AV strategy. The works of[152, 153, 154] have shown that if neighbour nodes with many connections are vaccinatedthen the performances are improved significantly.The acquaintance based strategies still require a large number of individuals to bevaccinated to achieve control goal. Thus, there have been a fair amount of works tosearch for other contact properties that can be obtained locally. The analysis of socialcontact networks shows that individuals are connected to various communities [155, 156].These properties of the contact network are exploited by several works where the conceptof the bridge nodes is introduced as these nodes provide the pathways for a diseaseto propagate from one community to another community. Therefore, vaccinating suchnodes will be a more effective strategy than of selecting random acquaintance. However, itrequires the searching methods that use only the local contact information. The searchingalgorithm developed by the work of [157] can find the bridge nodes using stochasticsearching methods that need only local structural information. They found that thedeveloped strategy based on these bridge nodes is more efficient than random strategies.A similar approach is applied in [158] where bridge hub nodes, nodes bridging betweentwo communities, are chosen for vaccination.The above vaccination strategies have been developed based on static network prop-24rties. However, the real-world social contact networks are dynamic which has a strongimpact on disease spreading and hence designing a vaccination strategy. The study [73]shows that the contact rates between a pair of nodes are broadly distributed. Therefore,the selection of an acquaintance in AV strategy is not sufficient to find the appropri-ate nodes to be vaccinated. For example, the infection risk for being in a contact withan infected individual is relevant to the contact duration. Moreover, there is a higherrisk if one susceptible interacts frequently with the infected individual. The works of[159, 160, 161] consider this information in neighbour selecting instead of selecting ran-dom neighbours. The authors of [159] use the most recent contact for vaccination andthey also apply weight to capture the contact rates with the neighbouring nodes. Thesesvaccination strategies outperform acquaintance vaccination.The collection of contact information is often difficult. Thus, vaccination with detailedlocal contact information may be infeasible in real-world scenarios and lose the benefitof using the local contact information. There have been several other vaccination modelsbased on individual movement behaviours where individuals contact properties are notconsider explicitly [150]. Beyond the contact properties, the work of [162] consider theindividuals who travel long distance for vaccination. The similar approach is taken by[163] where individuals who visit many locations are vaccinated.The above vaccination strategies shows that node ranking is conducted based on thecontact information about neighbouring nodes. However, in the diffusion processes withindirect interactions, it is difficult to identify the neighbours contacted through indirectinteractions. Thus, there is a need to understand the efficiency of strategies with indirectinteraction and find the best strategy. The authors of [102] addressed this issues to buildvaccination strategies with the indirect links.
For modelling disease spread within a society, a proper infection risk assessment modeland a proper contact network are required. This paper analysed a wide range of networksmodels and infection risk models. There are simple infection risk assessment models thatcapture practical situation through simple model parameters. The complex model cancapture the temporal dynamics of individual interactions and estimate more accurateinfection risk. But, it require granular level contact information which may not availableall time. In addition, it is observed that many factors affect the spreading dynamics ofcontagious items on contact networks. However, it is clear that interaction pattern ofindividuals is one of the key factors in driving diffusion processes on contact networks.There have, therefore, been a wide range of efforts to understand and integrate theimpacts of interaction patterns with diffusion modelling [25, 24, 26, 164]. There are,however, still some critical factors to be addressed in constructing proper diffusion models25hat capture realistic contact patterns. In addition, current opportunities for gatheringindividual-level contact data have attracted the researchers to deep dive further in thisfield by looking at contact patterns at the granular level [165, 62, 5, 166, 101]. Themodels presented in this paper would provide a guideline to model infectious disease likeCOVID-19. [1] Matt J Keeling and Pejman Rohani.
Modeling infectious diseases in humans andanimals . Princeton University Press, 2008.[2] Thomas O Richardson and Thomas E Gorochowski. Beyond contact-based trans-mission networks: the role of spatial coincidence.
Journal of The Royal SocietyInterface , 12(111), 2015.[3] Mike J Jeger, Marco Pautasso, Ottmar Holdenrieder, and Mike W Shaw. Mod-elling disease spread and control in networks: implications for plant sciences.
NewPhytologist , 174(2):279–297, 2007.[4] Robert Moss, Roslyn I Hickson, Jodie McVernon, James M McCaw, Krishna Hort,Jim Black, John R Madden, Nhi H Tran, Emma S McBryde, and Nicholas Geard.Model-informed risk assessment and decision making for an emerging infectiousdisease in the asia-pacific region.
PLoS neglected tropical diseases , 10(9), 2016.[5] Chunlin Huang, Xingwu Liu, Shiwei Sun, Shuai Cheng Li, Minghua Deng,Guangxue He, Haicang Zhang, Chao Wang, Yang Zhou, Yanlin Zhao, et al. In-sights into the transmission of respiratory infectious diseases through empiricalhuman contact networks.
Scientific Reports , 6, 2016.[6] Md Shahzamal, Raja Jurdak, Reza Arablouei, Minkyoung Kim, Kanchana Thi-lakarathna, and Bernard Mans. Airborne disease propagation on large scale socialcontact networks. In
Proceedings of the 2nd Int. Workshop on Social Sensing , pages35–40. ACM, 2017.[7] Minkyoung Kim, Dean Paini, and Raja Jurdak. Real-world diffusion dynamicsbased on point process approaches: A review.
Artificial Intelligence Review , pages1–30, 2018. 268] Md Shahzamal, Raja Jurdak, Bernard Mans, and Frank de Hoog. Indirect inter-actions influence contact network structure and diffusion dynamics. arXiv preprintarXiv:1906.02405 , 2019.[9] Marta C Gonz´alez and Albert-L´aszl´o Barab´asi. Complex networks: From data tomodels.
Nature Physics , 3(4):224, 2007.[10] Marta C Gonzalez, Cesar A Hidalgo, and Albert-Laszlo Barabasi. Understandingindividual human mobility patterns. nature , 453(7196):779, 2008.[11] Marc Barth´elemy, Alain Barrat, Romualdo Pastor-Satorras, and Alessandro Vespig-nani. Dynamical patterns of epidemic outbreaks in complex heterogeneous net-works.
Journal of theoretical biology , 235(2):275–288, 2005.[12] Patrick Bryant and Arne Elofsson. Modelling the dispersion of sars-cov-2 on adynamic network graph. medRxiv , 2020.[13] Marino Gatto, Enrico Bertuzzo, Lorenzo Mari, Stefano Miccoli, Luca Carraro, Re-nato Casagrandi, and Andrea Rinaldo. Spread and dynamics of the covid-19 epi-demic in italy: Effects of emergency containment measures.
Proceedings of theNational Academy of Sciences , 117(19):10484–10491, 2020.[14] Sheryl L Chang, Nathan Harding, Cameron Zachreson, Oliver M Cliff, and MikhailProkopenko. Modelling transmission and control of the covid-19 pandemic in aus-tralia.
Nature communications , 11(1):1–13, 2020.[15] Theresa Kuchler, Dominic Russel, and Johannes Stroebel. The geographic spread ofcovid-19 correlates with the structure of social networks as measured by facebook.
Journal of Urban Economics , page 103314, 2020.[16] Stefan Thurner, Peter Klimek, and Rudolf Hanel. A network-based explanation ofwhy most covid-19 infection curves are linear.
Proceedings of the National Academyof Sciences , 117(37):22684–22689, 2020.[17] Nigel Meade and Towhidul Islam. Modelling and forecasting the diffusion ofinnovation–a 25-year review.
International Journal of forecasting , 22(3):519–545,2006.[18] Adrien Guille, Hakim Hacid, Cecile Favre, and Djamel A Zighed. Informationdiffusion in online social networks: A survey.
ACM Sigmod Record , 42(2):17–28,2013.[19] Romualdo Pastor-Satorras, Claudio Castellano, Piet Van Mieghem, and AlessandroVespignani. Epidemic processes in complex networks.
Reviews of modern physics ,87(3):925, 2015. 2720] A. Barrat, Marc B., and A. V.
Dynamical processes on complex networks . Cam-bridge university press, 2008.[21] Roy M Anderson and Robert M May.
Infectious diseases of humans: dynamics andcontrol . Oxford university press, 1992.[22] Kenneth J Himmelstein. Compartmental models and their application. by keithgodfrey. academic press inc., 24–28 oval road, london nwi 7dx, england. 1983. 293pp. 15.5 × . Journal of Pharmaceutical Sciences , 73(7):1018,1984.[23] Daniel Gruhl, Ramanathan Guha, David Liben-Nowell, and Andrew Tomkins. In-formation diffusion through blogspace. In
Proceedings of the 13th internationalconference on World Wide Web , pages 491–501. ACM, 2004.[24] RT Mikolajczyk, MK Akmatov, S Rastin, and Mirjam Kretzschmar. Social contactsof school children and the transmission of respiratory-spread pathogens.
Epidemi-ology & Infection , 136(6):813–822, 2008.[25] Jo¨el Mossong, Niel Hens, Mark Jit, Philippe Beutels, Kari Auranen, Rafael Mikola-jczyk, Marco Massari, Stefania Salmaso, Gianpaolo Scalia Tomba, Jacco Wallinga,et al. Social contacts and mixing patterns relevant to the spread of infectiousdiseases.
PLoS medicine , 5(3):e74, 2008.[26] Niel Hens, Nele Goeyvaerts, Marc Aerts, Ziv Shkedy, Pierre Van Damme, andPhilippe Beutels. Mining social mixing patterns for infectious disease models basedon a two-day population survey in belgium.
BMC infectious diseases , 9(1):5, 2009.[27] Helen J Wearing, Pejman Rohani, and Matt J Keeling. Appropriate models for themanagement of infectious diseases.
PLoS medicine , 2(7), 2005.[28] Timo Smieszek. A mechanistic model of infection: why duration and intensity ofcontacts should be included in models of disease spread.
Theoretical Biology andMedical Modelling , 6(1):25, 2009.[29] Timo Smieszek, Lena Fiebig, and Roland W Scholz. Models of epidemics: whencontact repetition and clustering should be included.
Theoretical biology and medicalmodelling , 6(1):11, 2009.[30] Patrick Rhomberg. On the parallelization of network diffusion models. 2017.[31] James A Yorke, Herbert W Hethcote, and Annett Nold. Dynamics and control ofthe transmission of gonorrhea.
Sexually transmitted diseases , 5(2):51–56, 1978.2832] Alun L Lloyd and Robert M May. Spatial heterogeneity in epidemic models.
Journalof theoretical biology , 179(1):1–11, 1996.[33] Michael Pitcher, Ruth Bowness, Simon Dobson, and Stephen Gillespie. A network-based metapopulation model to simulate a pulmonary tuberculosis infection. In
BOOK OF ABSTRACTS , page 61, 2017.[34] Sophie Meakin, Mike Tildesley, Emma Davis, and Matt Keeling. A metapopulationmodel for the 2018 ebola outbreak in equateur province in the democratic republicof the congo. bioRxiv , 2018.[35] James S Koopman. Mass-action and system analysis of infection transmission.
Ecological paradigms lost: routes of theory change , pages 143–155, 2005.[36] Matt J Keeling. Extensions to mass-action mixing.
Ecological paradigms lost: routesof theory change. Elsevier, Burlington, MA , pages 107–142, 2005.[37] Mark EJ Newman. Spread of epidemic disease on networks.
Physical review E ,66(1), 2002.[38] Matt J Keeling and Ken TD Eames. Networks and epidemic models.
Journal ofthe Royal Society Interface , 2(4):295–307, 2005.[39] P. Holme. Modern temporal network theory: a colloquium.
The European PhysicalJournal B , 88(9):234, 2015.[40] Hamed Seyed-Allaei, Ginestra Bianconi, and Matteo Marsili. Scale-free networkswith an exponent less than two.
Physical Review E , 73(4):046113, 2006.[41] Xiao Fan Wang and Guanrong Chen. Complex networks: small-world, scale-freeand beyond.
IEEE circuits and systems magazine , 3(1):6–20, 2003.[42] Rainer Hegselmann. Modeling social dynamics by cellular automata.
Computermodeling of social processes , pages 37–64, 1998.[43] Rainer Hegselmann and Andreas Flache. Understanding complex social dynamics:A plea for cellular automata based modelling.
Journal of Artificial Societies andSocial Simulation , 1(3):1, 1998.[44] Parongama Sen, Subinay Dasgupta, Arnab Chatterjee, PA Sreeram, G Mukherjee,and SS Manna. Small-world properties of the indian railway network.
PhysicalReview E , 67(3), 2003.[45] Tao Zhou, Gang Yan, and Bing-Hong Wang. Maximal planar networks with largeclustering coefficient and power-law degree distribution.
Physical Review E , 71(4),2005. 2946] Andrew Barbour and Denis Mollison. Epidemics and random graphs. In
Stochasticprocesses in epidemic theory , pages 86–89. Springer, 1990.[47] Duncan J Watts and Steven H Strogatz. Collective dynamics of ‘small-world’networks. nature , 393(6684):440, 1998.[48] Mark EJ Newman and Duncan J Watts. Renormalization group analysis of thesmall-world network model.
Physics Letters A , 263(4-6):341–346, 1999.[49] R´eka Albert and Albert-L´aszl´o Barab´asi. Statistical mechanics of complex net-works.
Reviews of modern physics , 74(1):47, 2002.[50] Lada A Adamic, Rajan M Lukose, Amit R Puniyani, and Bernardo A Huberman.Search in power-law networks.
Physical review E , 64(4), 2001.[51] Aaron Clauset, Cosma Rohilla Shalizi, and Mark EJ Newman. Power-law distribu-tions in empirical data.
SIAM review , 51(4):661–703, 2009.[52] Martina Morris.
Network epidemiology: A handbook for survey design and datacollection . Oxford University Press on Demand, 2004.[53] Jukka-Pekka Onnela, Jari Saram¨aki, J¨orkki Hyv¨onen, G´abor Szab´o, M ArgolloDe Menezes, Kimmo Kaski, Albert-L´aszl´o Barab´asi, and J´anos Kert´esz. Analysisof a large-scale weighted network of one-to-one human communication.
New journalof physics , 9(6):179, 2007.[54] Christel Kamp, Mathieu Moslonka-Lefebvre, and Samuel Alizon. Epidemic spreadon weighted networks.
PLoS computational biology , 9(12), 2013.[55] Xiangwei Chu, Jihong Guan, Zhongzhi Zhang, and Shuigeng Zhou. Epidemicspreading in weighted scale-free networks with community structure.
Journal ofStatistical Mechanics: Theory and Experiment , 2009(07):P07043, 2009.[56] Renaud Lambiotte, Lionel Tabourier, and Jean-Charles Delvenne. Burstiness andspreading on temporal networks.
The European Physical Journal B , 86(7):320,2013.[57] N. Perra, B. Gon¸calves, R. Pastor-Satorras, and A. Vespignani. Activity drivenmodeling of time varying networks.
Scientific reports , 2, 2012.[58] Md Shahzamal, Raja Jurdak, Bernard Mans, and Frank de Hoog. A graph modelwith indirect co-location links. arXiv preprint arXiv:1806.03386 , 2018.[59] Eugenio Valdano, Luca Ferreri, Chiara Poletto, and Vittoria Colizza. Analyticalcomputation of the epidemic threshold on temporal networks.
Physical Review X ,5(2), 2015. 3060] S. Bansal, Bryan T G., and Lauren A. M. When individual behaviour matters:homogeneous and network models in epidemiology.
Journal of the Royal SocietyInterface , 4(16):879–891, 2007.[61] Simon Cauchemez, Alain-Jacques Valleron, Pierre-Yves Boelle, Antoine Flahault,and Neil M Ferguson. Estimating the impact of school closure on influenza trans-mission from sentinel data.
Nature , 452(7188):750, 2008.[62] Juliette Stehl´e, Nicolas Voirin, Alain Barrat, Ciro Cattuto, Vittoria Colizza,Lorenzo Isella, Corinne R´egis, Jean-Fran¸cois Pinton, Nagham Khanafer, WouterVan den Broeck, et al. Simulation of an seir infectious disease model on the dy-namic contact network of conference attendees.
BMC medicine , 9(1):87, 2011.[63] Damon JA Toth, Molly Leecaster, Warren BP Pettey, Adi V Gundlapalli,Hongjiang Gao, Jeanette J Rainey, Amra Uzicanin, and Matthew H Samore. Therole of heterogeneity in contact timing and duration in network models of influenzaspread in schools.
Journal of The Royal Society Interface , 12(108):20150279, 2015.[64] John H Holland. Studying complex adaptive systems.
Journal of systems scienceand complexity , 19(1):1–8, 2006.[65] Elmar Kiesling, Markus G¨unther, Christian Stummer, and Lea M Wakolbinger.Agent-based simulation of innovation diffusion: a review.
Central European Journalof Operations Research , 20(2):183–230, 2012.[66] Kirsten A Copren and NL Geard. An individual based model examining the emer-gence of cooperative recognition in a social insect.
Sociobiology , 46(2):349–361,2005.[67] Zhijing Xu, Kathryn Glass, Colleen L Lau, Nicholas Geard, Patricia Graves, andArchie Clements. A synthetic population for modelling the dynamics of infectiousdisease transmission in american samoa.
Scientific reports , 7(1), 2017.[68] Christian Stummer, Elmar Kiesling, Markus G¨unther, and Rudolf Vetschera. In-novation diffusion of repeat purchase products in a competitive market: an agent-based simulation approach.
European Journal of Operational Research , 245(1):157–167, 2015.[69] Anna Machens, Francesco Gesualdo, Caterina Rizzo, Alberto E Tozzi, Alain Barrat,and Ciro Cattuto. An infectious disease model on empirical networks of humancontact: bridging the gap between dynamic network data and contact matrices.
BMC infectious diseases , 13(1):185, 2013.3170] Y. Zhang, Jing C., Shu-Min Z., Qi Z., and X. L. Modelling temporal networks ofhuman face-to-face contacts with public activity and individual reachability.
TheEuropean Physical Journal B , 89(2):26, 2016.[71] Michele Starnini, Andrea Baronchelli, and Romualdo Pastor-Satorras. Model-ing human dynamics of face-to-face interaction networks.
Physical review letters ,110(16):168701, 2013.[72] G. Laurent, Jari S.¨aki, and M´arton Karsai. From calls to communities: a modelfor time-varying social networks.
The European Physical Journal B , 88, 2015.[73] Rossana Mastrandrea, Julie Fournet, and Alain Barrat. Contact patterns in a highschool: a comparison between data collected using wearable sensors, contact diariesand friendship surveys.
PloS one , 10(9):e0136497, 2015.[74] Kyu-Min Lee, Byungjoon Min, and Kwang-Il Goh. Towards real-world complexity:an introduction to multiplex networks.
The European Physical Journal B , 88(2):48,2015.[75] Mikko Kivel¨a, Alex Arenas, Marc Barthelemy, James P Gleeson, Yamir Moreno,and Mason A Porter. Multilayer networks.
Journal of complex networks , 2(3):203–271, 2014.[76] Benjamin Bach, Emmanuel Pietriga, and Jean-Daniel Fekete. Visualizing densedynamic networks with matrix cubes. In
IEEE Conference on Information Visual-ization [Poster] , 2013.[77] Petter Holme. Epidemiologically optimal static networks from temporal networkdata.
PLoS computational biology , 9(7):e1003142, 2013.[78] Marta Sarzynska, Elizabeth A Leicht, Gerardo Chowell, and Mason A Porter. Nullmodels for community detection in spatially embedded, temporal networks.
Journalof Complex Networks , 4(3):363–406, 2015.[79] Othon Michail. An introduction to temporal graphs: An algorithmic perspective.
Internet Mathematics , 12(4):239–280, 2016.[80] Alice Albano, Jean-Loup Guillaume, S´ebastien Heymann, and B´en´edicte Le Grand.A matter of time-intrinsic or extrinsic-for diffusion in evolving complex networks.In
Proceedings of the 2013 IEEE/ACM International Conference on Advances inSocial Networks Analysis and Mining , pages 202–206. ACM, 2013.[81] Peng Wang, Ken Sharpe, Garry L Robins, and Philippa E Pattison. Exponentialrandom graph (p ∗ ) models for affiliation networks. Social Networks , 31(1):12–25,2009. 3282] Steve Hanneke, Wenjie Fu, Eric P Xing, et al. Discrete temporal models of socialnetworks.
Electronic Journal of Statistics , 4:585–605, 2010.[83] Pavel N Krivitsky. Modeling of dynamic networks based on egocentric data withdurational information.[84] Tom AB Snijders. Statistical models for social networks.
Annual review of sociology ,37:131–153, 2011.[85] Yoon-Sik Cho, Aram Galstyan, P Jeffrey Brantingham, and George Tita. Latentself-exciting point process model for spatial-temporal networks. arXiv preprintarXiv:1302.2671 , 2013.[86] Naoki Masuda, Taro Takaguchi, Nobuo Sato, and Kazuo Yano. Self-exciting pointprocess modeling of conversation event sequences. In
Temporal Networks , pages245–264. Springer, 2013.[87] Christoph Stadtfeld, James Hollway, and Per Block. Dynamic network actor mod-els: Investigating coordination ties through time.
Sociological Methodology , 47(1):1–40, 2017.[88] Alexander V Mantzaris and Desmond J Higham. A model for dynamic communi-cators.
European Journal of Applied Mathematics , 23(6):659–668, 2012.[89] Alessandro Rizzo and Maurizio Porfiri. Innovation diffusion on time-varying activitydriven networks.
The European Physical Journal B , 89(1):20, 2016.[90] Sergio G´omez, Jes´us G´omez-Gardenes, Yamir Moreno, and Alex Arenas. Non-perturbative heterogeneous mean-field approach to epidemic spreading in complexnetworks.
Physical Review E , 84(3), 2011.[91] Kaiyuan Sun, Andrea Baronchelli, and Nicola Perra. Contrasting effects of strongties on sir and sis processes in temporal networks.
The European Physical JournalB , 88(12):326, 2015.[92] M´arton Karsai, Nicola Perra, and Alessandro Vespignani. Time varying networksand the weakness of strong ties.
Scientific reports , 4, 2014.[93] Enrico Ubaldi, Alessandro Vezzani, M´arton Karsai, Nicola Perra, and RaffaellaBurioni. Burstiness and tie activation strategies in time-varying social networks.
Scientific Reports , 7, 2017.[94] Ginestra Bianconi, Richard K Darst, Jacopo Iacovacci, and Santo Fortunato. Tri-adic closure as a basic generating mechanism of communities in complex networks.
Physical Review E , 90(4), 2014. 3395] Simone Daminelli, Josephine Maria Thomas, Claudio Dur´an, and Carlo VittorioCannistraci. Common neighbours and the local-community-paradigm for topologi-cal link prediction in bipartite networks.
New Journal of Physics , 17(11), 2015.[96] Juliette Stehl´e, Alain Barrat, and Ginestra Bianconi. Dynamical and bursty inter-actions in social networks.
Physical review E , 81(3), 2010.[97] Enrico Ubaldi, Alessandro Vezzani, M´arton Karsai, Nicola Perra, and RaffaellaBurioni. Burstiness and tie reinforcement in time varying social networks. arXivpreprint arXiv:1607.08910 , 2016.[98] I. Pozzana, Kaiyuan S., and N. P. Epidemic spreading on activity-driven networkswith attractiveness.
Physical Review E , 96(4), 2017.[99] Aaron Fernstrom and Michael Goldblatt. Aerobiology and its role in the transmis-sion of infectious diseases.
Journal of pathogens , 2013, 2013.[100] Stephanie A Boone and Charles P Gerba. Significance of fomites in the spreadof respiratory and enteric viral disease.
Applied and Environmental Microbiology ,73(6):1687–1696, 2007.[101] Md Shahzamal, Raja Jurdak, Bernard Mans, Ahmad El Shoghri, and FrankDe Hoog. Impact of indirect contacts in emerging infectious disease on social net-works. In
Pacific-Asia Conference on Knowledge Discovery and Data Mining , pages53–65. Springer, 2018.[102] Md Shahzamal, Bernard Mans, Frank de Hoog, Dean Paini, and Raja Jurdak.Vaccination strategies on dynamic networks with indirect transmission links andlimited contact information.
Plos one , 15(11):e0241612, 2020.[103] Erik Rottier and Margaret Ince.
Controlling and preventing disease: the role of wa-ter and environmental sanitation interventions . WEDC, Loughborough University,2003.[104] Gabrielle Brankston, Leah Gitterman, Zahir Hirji, Camille Lemieux, and MichaelGardam. Transmission of influenza a in human beings.
The Lancet infectiousdiseases , 7(4):257–265, 2007.[105] S. T Stoddard, Amy C M., Gonzalo M V., Valerie P. S., Tadeusz J K., Uriel K.,John P E., and Thomas W S. The role of human movement in the transmission ofvector-borne pathogens.
PLoS neglected tropical diseases , 3(7), 2009.[106] Marnie L Brennan, R Kemp, and Robert M Christley. Direct and indirect contactsbetween cattle farms in north-west england.
Preventive veterinary medicine , 84(3-4):242–260, 2008. 34107] Martin Lange, Stephanie Kramer-Schadt, and Hans-Hermann Thulke. Relevanceof indirect transmission for wildlife disease surveillance.
Frontiers in veterinaryscience , 3:110, 2016.[108] William G Lindsley, Terri A Pearce, Judith B Hudnall, Kristina A Davis, Stephen MDavis, Melanie A Fisher, Rashida Khakoo, Jan E Palmer, Karen E Clark, IsmailCelik, et al. Quantity and size distribution of cough-generated aerosol particlesproduced by influenza patients during and after illness.
Journal of occupationaland environmental hygiene , 9(7):443–449, 2012.[109] Robert G Loudon and Linda C Brown. Cough frequency in patients with respiratorydisease 1, 2.
American Review of Respiratory Disease , 96(6):1137–1143, 1967.[110] Richard James Thomas. Particle size and pathogenicity in the respiratory tract.
Virulence , 4(8):847–858, 2013.[111] WG Lindsley, FM Blachere, RE Thewlis, A Vishnu, KA Davis, et al. Measurementsof airborne influenza virus in aerosol particles from human. 2010.[112] William G Lindsley, John D Noti, Francoise M Blachere, Robert E Thewlis,Stephen B Martin, Sreekumar Othumpangat, Bahar Noorbakhsh, William T Gold-smith, Abhishek Vishnu, Jan E Palmer, et al. Viable influenza a virus in airborneparticles from human coughs.
Journal of occupational and environmental hygiene ,12(2):107–113, 2015.[113] Nikolai Nikitin, Ekaterina Petrova, Ekaterina Trifonova, and Olga Karpova. In-fluenza virus aerosols in the air and their infectiousness.
Advances in virology ,2014, 2014.[114] Gerone PJ Alford RH, Kasel JA and Knight V. Human influenza resultingfrom aerosol inhalation.
Proceedings of the Society for Experimental Biology andMedicine , 122(3):800–804, 1966.[115] Zhuyang Han, Wenguo Weng, Quanyi Huang, and Shaobo Zhong. A risk estimationmethod for airborne infectious diseases based on aerosol transmission in indoorenvironment. In
Proceedings of the World Congress on Engineering , volume 2,2014.[116] Ignatius TS Yu, Yuguo Li, Tze Wai Wong, Wilson Tam, Andy T Chan, Joseph HWLee, Dennis YC Leung, and Tommy Ho. Evidence of airborne transmission ofthe severe acute respiratory syndrome virus.
New England Journal of Medicine ,350(17):1731–1739, 2004. 35117] Jiachen Mao and Naiping Gao. The airborne transmission of infection between flatsin high-rise residential buildings: A review.
Building and Environment , 94:516–531,2015.[118] NP Gao, JL Niu, Marco Perino, and Per Heiselberg. The airborne transmissionof infection between flats in high-rise residential buildings: tracer gas simulation.
Building and Environment , 43(11):1805–1817, 2008.[119] CB Beggs. The airborne transmission of infection in hospital buildings: fact orfiction?
Indoor and Built Environment , 12(1-2):9–18, 2003.[120] Cesar A Corzo, Marie Culhane, Scott Dee, Robert B Morrison, and MontserratTorremorell. Airborne detection and quantification of swine influenza a virus in airsamples collected inside, outside and downwind from swine barns.
PLoS One , 8(8),2013.[121] GN Sze To and CYH Chao. Review and comparison between the wells–riley anddose-response approaches to risk assessment of infectious respiratory diseases.
In-door Air , 20(1):2–16, 2010.[122] Kevin P Fennelly and Edward A Nardell. The relative efficacy of respirators androom ventilation in preventing occupational tuberculosis.
Infection Control & Hos-pital Epidemiology , 19(10):754–759, 1998.[123] William W Nazaroff, Mark Nicas, and Shelly L Miller. Framework for evaluatingmeasures to control nosocomial tuberculosis transmission.
Indoor Air , 8(4):205–218,1998.[124] Laura Gammaitoni and Maria Clara Nucci. Using a mathematical model to evaluatethe efficacy of tb control measures.
Emerging infectious diseases , 3(3):335, 1997.[125] SN Rudnick and DK Milton. Risk of indoor airborne infection transmission esti-mated from carbon dioxide concentration.
Indoor air , 13(3):237–245, 2003.[126] Yun-Chun Tung and Shih-Cheng Hu. Infection risk of indoor airborne transmissionof diseases in multiple spaces.
Architectural Science Review , 51(1):14–20, 2008.[127] Mark Nicas. An analytical framework for relating dose, risk, and incidence: anapplication to occupational tuberculosis infection.
Risk Analysis , 16(4):527–538,1996.[128] Chacha M Issarow, Nicola Mulder, and Robin Wood. Modelling the risk of airborneinfectious disease using exhaled air.
Journal of theoretical biology , 372:100–106,2015. 36129] Christophe Van den Bulte and Yogesh V Joshi. New product diffusion with influ-entials and imitators.
Marketing science , 26(3):400–421, 2007.[130] Renana Peres, Eitan Muller, and Vijay Mahajan. Innovation diffusion and newproduct growth models: A critical review and research directions.
Internationaljournal of research in marketing , 27(2):91–106, 2010.[131] Lu-Xing Yang, Moez Draief, and Xiaofan Yang. The optimal dynamic immunizationunder a controlled heterogeneous node-based sirs model.
Physica A: StatisticalMechanics and its Applications , 450:403–415, 2016.[132] Mohammed Ali Al-Garadi, Kasturi Dewi Varathan, Sri Devi Ravana, Ejaz Ahmed,Ghulam Mujtaba, Muhammad Usman Shahid Khan, and Samee U Khan. Analysisof online social network connections for identification of influential users: Surveyand open research issues.
ACM Computing Surveys (CSUR) , 51(1):16, 2018.[133] I. Scholtes, Nicolas W., and A. G. Higher-order aggregate networks in the analysisof temporal networks: path structures and centralities.
The European PhysicalJournal B , 89(3):61, 2016.[134] Miray Kas, Matthew Wachs, Kathleen M Carley, and L Richard Carley. Incrementalalgorithm for updating betweenness centrality in dynamically growing networks. In
Proceedings of the 2013 IEEE/ACM international conference on advances in socialnetworks analysis and mining , pages 33–40. ACM, 2013.[135] Jing Ma, Dandan Li, and Zihao Tian. Rumor spreading in online social networksby considering the bipolar social reinforcement.
Physica A: Statistical Mechanicsand its Applications , 447:108–115, 2016.[136] Jing Ma and He Zhu. Rumor diffusion in heterogeneous networks by considering theindividuals’ subjective judgment and diverse characteristics.
Physica A: StatisticalMechanics and its Applications , 499:276–287, 2018.[137] R´eka Albert, Hawoong Jeong, and Albert-L´aszl´o Barab´asi. Error and attack toler-ance of complex networks. nature , 406(6794):378, 2000.[138] Sergey Brin and Lawrence Page. The anatomy of a large-scale hypertextual websearch engine.
Computer networks and ISDN systems , 30(1-7):107–117, 1998.[139] Linton C Freeman. Centrality in social networks conceptual clarification.
Socialnetworks , 1(3):215–239, 1978.[140] Stefan Wuchty and Eivind Almaas. Evolutionary cores of domain co-occurrencenetworks.
BMC evolutionary biology , 5(1):24, 2005.37141] Dane Taylor, Sean A Myers, Aaron Clauset, Mason A Porter, and Peter J Mucha.Eigenvector-based centrality measures for temporal networks.
Multiscale Modeling& Simulation , 15(1):537–574, 2017.[142] Luis EC Rocha and Naoki Masuda. Random walk centrality for temporal networks.
New Journal of Physics , 16(6), 2014.[143] Mark EJ Newman. A measure of betweenness centrality based on random walks.
Social networks , 27(1):39–54, 2005.[144] Nilly Madar, Tomer Kalisky, Reuven Cohen, Daniel Ben-avraham, and ShlomoHavlin. Immunization and epidemic dynamics in complex networks.
The EuropeanPhysical Journal B , 38(2):269–276, 2004.[145] Reuven Cohen, Shlomo Havlin, and Daniel Ben-Avraham. Efficient immuniza-tion strategies for computer networks and populations.
Physical review letters ,91(24):247901, 2003.[146] Marc Lelarge. Efficient control of epidemics over random networks.
ACM SIG-METRICS Performance Evaluation Review , 37(1):1–12, 2009.[147] Romualdo Pastor-Satorras and Alessandro Vespignani. Immunization of complexnetworks.
Physical review E , 65(3):036104, 2002.[148] Maria Deijfen. Epidemics and vaccination on weighted graphs.
Mathematical bio-sciences , 232(1):57–65, 2011.[149] Tom Britton, Svante Janson, and Anders Martin-L¨of. Graphs with specified de-gree distributions, simple epidemics, and local vaccination strategies.
Advances inApplied Probability , 39(4):922–948, 2007.[150] Liang Mao and Ling Bian. Efficient vaccination strategies in a social network withindividual mobility.
UCGIS 2009 Summer Assembly , 2009.[151] Adam J Kucharski, Timothy W Russell, Charlie Diamond, Yang Liu, John Ed-munds, Sebastian Funk, Rosalind M Eggo, Fiona Sun, Mark Jit, James D Munday,et al. Early dynamics of transmission and control of covid-19: a mathematicalmodelling study.
The lancet infectious diseases , 20(5):553–558, 2020.[152] Petter Holme. Efficient local strategies for vaccination and network attack.
EPL(Europhysics Letters) , 68(6):908, 2004.[153] Li Chen and Dongyi Wang. An improved acquaintance immunization strategy forcomplex network.
Journal of theoretical biology , 385:58–65, 2015.38154] Lazaros K Gallos, Fredrik Liljeros, Panos Argyrakis, Armin Bunde, and ShlomoHavlin. Improving immunization strategies.
Physical Review E , 75(4):045104, 2007.[155] Michelle Girvan and Mark EJ Newman. Community structure in social and biolog-ical networks.
Proceedings of the national academy of sciences , 99(12):7821–7826,2002.[156] Roger Guimera, Leon Danon, Albert Diaz-Guilera, Francesc Giralt, and Alex Are-nas. Self-similar community structure in a network of human interactions.
Physicalreview E , 68(6):065103, 2003.[157] Kai Gong, Ming Tang, Pak Ming Hui, Hai Feng Zhang, Do Younghae, and Ying-Cheng Lai. An efficient immunization strategy for community networks.
PloS one ,8(12):e83489, 2013.[158] Marcel Salath´e and James H Jones. Dynamics and control of diseases in networkswith community structure.
PLoS computational biology , 6(4):e1000736, 2010.[159] Sungmin Lee, Luis EC Rocha, Fredrik Liljeros, and Petter Holme. Exploiting tem-poral network structures of human interaction to effectively immunize populations.
PloS one , 7(5):e36439, 2012.[160] Michele Starnini, Anna Machens, Ciro Cattuto, Alain Barrat, and RomualdoPastor-Satorras. Immunization strategies for epidemic processes in time-varyingcontact networks.
Journal of theoretical biology , 337:89–100, 2013.[161] Naoki Masuda and Petter Holme. Predicting and controlling infectious diseaseepidemics using temporal networks.
F1000prime reports , 5, 2013.[162] Liang Mao and Ling Bian. A dynamic network with individual mobility for design-ing vaccination strategies.
Transactions in GIS , 14(4):533–545, 2010.[163] Joel C Miller and James M Hyman. Effective vaccination strategies for realisticsocial networks.
Physica A: Statistical Mechanics and its Applications , 386(2):780–785, 2007.[164] Jos´e Luis Iribarren and Esteban Moro. Impact of human activity patterns on thedynamics of information diffusion.
Physical review letters , 103(3):038702, 2009.[165] Michele Starnini.
Time-varying networks approach to social dynamics: from in-dividual to collective behavior . PhD thesis, Universitat Polit`ecnica de Catalunya,2014. 39166] Andrew J Tatem, Zhuojie Huang, Clothilde Narib, Udayan Kumar, Deepika Kan-dula, Deepa K Pindolia, David L Smith, Justin M Cohen, Bonita Graupe, PetrinaUusiku, et al. Integrating rapid risk mapping and mobile phone call record datafor strategic malaria elimination planning.