# A survey on modelling of infectious disease spread and control on social
contact networks

Md Shahzamal  Macquarie University, Sydney, Australia<EMAIL_ADDRESS>Saeed Khan  University of Queensland, Brisbane, Australia<EMAIL_ADDRESS>
###### Abstract

Infectious diseases are a significant threat to human society which was over
sighted before the incidence of COVID-19, although according to the report of
the World Health Organisation (WHO) about 4.2 million people die annually due
to infectious disease. Due to recent COVID-19 pandemic, more than 2 million
people died during 2020 and 96.2 million people got affected by this
devastating disease. Recent research shows that applying individual
interactions and movements data could help managing the pandemic though
modelling the spread of infectious diseases on social contact networks.
Infectious disease spreading can be explained with the theories and methods of
diffusion processes where a dynamic phenomena evolves on networked systems. In
the modelling of diffusion process, it is assumed that contagious 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 a
population through individual social interactions. The evolution behaviours of
the diffusion process are strongly influenced by the characteristics of the
underlying system and the mechanism of the diffusion process itself. Thus,
spreading of infectious disease can be explained how people interact with each
other and by the characteristics of the disease itself. This paper presenters
the relevant theories and methodologies of diffusion process that can be used
to model the spread of infectious diseases.

###### keywords:

diffusion; disease spread; contact networks, human movements, and covid-19

##  1 Introduction

Infectious diseases are a significant threat to human society which was over
sighted before the incidence of COVID-19, although according to the report of
the World Health Organisation (WHO) about 4.2 million people die annually due
to infectious disease. Due to COVID-19, more than 2 million people died during
2020 and 96.2 million people affected by 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 is very important to
understand the theories and methods for modelling spread of infectious disease
and its control. The popular approach to model infectious disease spread is
using theories and techniques of diffusion process on contact networks. The
availability of high computational power allows researchers to simulate and
analyse dynamics of infectious disease spread using the models of diffusion.
Therefore, building simulation tools based on the diffusion modelling theories
can help authorities to make policies and strategies to 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 the networked system through inter-node interactions. In the
diffusion process, contagious items (infectious particles, a piece of
information, innovation and a specific behaviour etc.) initially grow on one
or more nodes of the networks and then spread through neighbouring nodes over
the network. The interactions among nodes are responsible for transmitting
contagious items from one node to the other nodes. The interactions can happen
through physical contact such as being in a common location, touch the same
object within a time frame and transferring of objects from one node to other
nodes through medium such as air. In infectious diseases scenarios, infected
individuals interact with susceptible individuals and disease is transmitted
to susceptible individuals. Thus, the nodes of a networked system underlying
of a diffusion process are represented by individuals and the inter-node
interactions causing transmission of infectious particles are called contacts,
edges, or links in the study of disease spreading on social contact networks
[4, 5, 6].

Modelling of diffusion dynamics on networked system or social contact networks
is required to understand what factors shape the diffusion behaviours. In
reality, the factors governing the behaviours of diffusion process vary
depending where it happens. However, the spreading of contagious items, e.g.
infectious particles, over a contact network is often treated as the coupling
of results of three factors, namely individual interactions, characteristics
of contagious items, and environments. These three factors make strong
contributions to diffusion phenomena on a contact network. The roles and
influence of these factors depend on the context of diffusion processes.

Researches on diffusion modelling identified interactions between individuals
are the main drivers for spreading contagious items on contact networks. The
individual interaction patterns provide pathways to spread contagious items on
contact networks. For example, how many susceptible individuals an infected
individual meets during their infectious period determine the spreading speed
of the infectious disease. If an infected individual is connected to many
other individuals, there is a high probability to transmit disease to others
by him. On the other hand, if an infected individual has no contact with other
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 on the contact patterns distribution in the population. There
are several interaction properties such as how contact happens, contact
frequency and contact duration etc. at the individual levels and contact
degree distribution, clustering coefficient etc. at the network level that are
studied to understand and model diffusion processes on social contact
networks.

The contagious items can be infectious particles for spreading infectious
diseases. The contagious items themselves play strong roles in their spread in
a population and is identified as the critical factors to spread disease in
the literature. The internal spreading potentiality of contagious items varies
based on its characteristics. For example, the infectiousness of contagious
particles is defined by the disease types and highly infectious diseases
usually spread faster in a population. The spreading potential is also
affected by the process of infectious particles generation. The impacts of
contagious items also vary depending on the recipient individual behaviours.
For example, the impacts of infectious particles varies according to the
susceptibility of the individuals. Therefore, the characteristics of
contagious items are often considered in the modelling of diffusion processes.
In case of modelling infectious disease spreading, the characteristics of
infectious items varies widely due to the various nature of disease.

The spreading of contagious items is often influenced by the environment. The
environments represent the characteristics of the space where diffusion
processes occur. For example, the impacts of infectious particles are
determined by weather conditions such as temperature and humidity etc. The
infectious particles generally lose their infectiousness over time and may
also depend on the weather conditions. The underlying medium where contagious
items spread can also be heterogeneous. For example, a disease can spread
through multiple platforms such as proximity contact networks, transportation
contact networks, and air-travel contact networks. Thus, diffusion modelling
is also required to consider the heterogeneity of diffusion medium.

To model the diffusion on contact networks, these factors should be included
to capture realistic diffusion behaviours. The impacts of contagious
properties and environment have been studied for a long time in the literature
[7]. However, analysing contact structure to understand the spreading of
contagious items is comparatively new. The recent exploration of data on
individual interactions have fuelled research on unravelling contact patterns
affecting contagious spreading [8, 9, 10, 11].

A spark has been seen on modelling of COVID-19 spread and control using social
contact networks properties [12, 13, 14, 15]. The authors of [12] shows that
there is significant variation in the number of connections and social
contacts each individual make in a social interaction networks. Individuals
with more social contacts are more likely to attract and spread infection.
These individuals are likely the drivers of the epidemic, so-called
superspreaders. When many superspreaders are immune, it becomes more difficult
for the disease to spread, as the connectedness of the social network
dramatically decreases. Authors of [16] also show that properly taking some
relevant network features into account, linear growth can be naturally
explained. Further, the effect of nonpharmaceutical interventions (NPIs), like
national lockdowns, can be modeled with a remarkable degree of precision
without fitting or fine-tuning of parameters.

This paper analyse the approaches to integrate individual contact patterns
with diffusion modelling and discuss how these approaches are used in
modelling of infectious diseases.

##  2 Diffusion modelling

Diffusion modelling is an intensively researched area due to its wide
applications. As the area 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 their purposes: i) explanatory models,
and ii) predictive models [17, 18, 19, 20]. The explanatory models are usually
developed to understand the factors affecting diffusion dynamics on a contact
network. This often allows one to answer the questions such as which nodes are
influential, what is the underlying reason for the way diffusion occurs, and
what is appropriate diffusion controlling strategy? On the other hand,
predictive models usually predict the spreading intensity and the final number
of individuals received contagious items based on certain factors. It is often
the case that the explanatory models find the key influential factors and
apply them for developing predictive models. This paper focuses on
understanding the diffusing approaches for explanatory models. There has been
a range of approaches for explanatory models of diffusion. This section
discusses some of them that are widely used in different fields of diffusion
ranging from disease spreading on individual contact networks to information
spreading on online social networks (OSN) [19, 20].

###  2.1 Compartmental models

Compartment epidemic models are frequently applied to study diffusion in many
applications such as information diffusion, innovation diffusion and computer
virus spreading [21, 22, 23]. The fundamental concept of these approaches is
to divide the system into several compartments or partitions. Each compartment
represents a set of individuals having a specific status. Then, the dynamics
of the diffusion are determined by the flows between these compartments.
Widely used compartments are Susceptible having individuals who are exposed to
the contagious items, Infected having individuals who have adopted the
contagious items and started forwarding items (infecting) others, and
Recovered having individuals who were infected but now recovered. This
compartment model is called SIR propagation model and the dynamics of the
compartments are given by

$\frac{dS}{dt}=-\beta IS,\quad\frac{dI}{dt}=\beta IS-\tau
I,\quad\frac{dR}{dt}=-\tau I$ (1)

where $S$, $I$ and $R$ are the fractions of the population in the Susceptible,
Infected and Recovered compartments respectively, $\beta$ is the transmission
rate from Susceptible to Infected compartment and $\tau$ 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 diffusion modelling, $\beta$ is given
as $\beta=\gamma\beta_{0}$, where $\gamma$ is the number of potential contacts
on average individuals has with others through which contagious items transmit
with a rate $\beta_{0}$. By changing the value of $\gamma$, $\tau$ and
$\beta_{0}$ one, therefore, can study the diffusion dynamics of diffusion
processes.

The compartment model can be analysed easily mathematically in the simple
case. It requires no more details than needed to reproduce and explain
observed behaviours. It reduces data collection cost and computational cost.
Clearly, it can be applied in many situations where high precision is not
necessary. However, the assumption of having homogeneous interaction $\gamma$
between individuals is not realistic. Many researchers have pointed out that
the interaction between individuals is clearly heterogeneous as individuals do
not have the same level of contact with all its neighbours [24, 25, 26, 25].
Moreover, the constant rate of transmission probability $\beta_{0}$ and
constant recovery rate $\tau$ are not realistic in many diffusion processes.
This is because individuals have different contact intensity with infected
individuals and heterogeneous susceptibilities to the contagious items [27,
28, 29].

Heterogeneity is often integrated into the compartment model by dividing the
main compartments (such as S, I, R) into sub-compartments. These sub-divisions
can be constructed based on the age, risk behaviours, or spatial diversities
of individuals. Then, the transmission probability can be divided into $k$
sub-classes and the model can be parameterised by means of a $k\times k$
transmission rate matrix instead of a constant transmission rate $\beta$ [30,
31]. For example, some disease spread models split individuals spatially
(divide population with different regions) and assign heterogeneity for
infection risks [32, 33, 34]. These approaches are called meta-population
models. Similarly, different infectious periods can be implemented in the
model by dividing population into $k$ sub-classes which resolve the
limitations of the constant rate of recovery [27].

The integration of heterogeneity with sub-compartmentalisation relax some of
the most unrealistic assumptions of basic compartmental models. However, many
limitations of 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 in these models. Hence, individuals
in the divided sub-populations behave homogeneously. In case of meta-
population models, dividing a population into the various spatial groups can
create asynchronous between these groups, but for time $t\rightarrow\infty$
they become homogeneous [36]. Therefore, the real steady state heterogeneity
cannot be captured by meta-population models. Thus, high partitioning in
compartmental models lose the simplicity. It also requires more data to fit
the model and increases the data collection cost.

###  2.2 Network models

To overcome the limitations of the compartment models and capture realistic
contact patterns, network science is adopted for modelling diffusion processes
on the networked systems [37, 38]. The network based diffusion modelling is
also empowered by the graph theory where contact networks are often generated
by graph models. In addition, graph theory is applied to study the
characteristics of contact networks. The core entities in the network-based
modelling are nodes, representing individuals, and links connecting one node
to other nodes in the network, which represent interactions between
individuals. A contact network can be represented by a graph where vertices
correspond to nodes and edges to links. The network models provide a range of
flexibility for assigning nodes various attributes and defining links with a
range of properties. A wide range of efficient network based models has been
developed in the literature for studying diffusion processes [37, 38, 39]. For
generating contact networks, a fundamental aspect is to build network
structure called the network topology based upon which nodes interact with
each other. There are four types of network structure namely regular lattice
networks, random networks, small-world networks and scale-free networks that
are frequently used to study diffusion processes [40, 41, 42, 43]. In
addition, data-driven network models are also derived from real-world data and
they may assume some properties of the theoretical models.

Regular lattices are the simplest representation of contact network structures
where nodes are only connected to their nearest neighbour nodes in a lattice
with a regular fashion [42, 43]. The regular lattice networks assume large
path lengths, i.e. the average distance 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 regular lattice model
where nodes contact with each other in a random fashion and each pair of nodes
has an equal probability to be connected. Furthermore, the average path
lengths in the random networks match with many real-world networks with
appropriate contact 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 six degrees of separation phenomena which
states that if you choose any two individuals anywhere 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 the
neighbours of a given node are likely to be neighbours of each other. The
Watts-Strogatz model [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 clustering co-efficient and short path length [48].
The latest approach to generate contact structure is scale-free networks
developed by the Barabasi-Albert model [49] where node’s degree follows a
power-law distribution. In the Barabasi-Albert model, the scale-free networks
are self-organised with growing and preferential attachment processes. The
research found that many real-world systems have a power-law degree
distribution [50, 51].

The above structured contact networks can be analysed mathematically and
numerically. The authors of [52] has presented surveys on the methods of
designing contact networks applying real contact data. These networks are,
however, static in nature where node attributes and link properties are not
changed during the observation period. These contact networks are often
represented with adjacency matrices of binary values. This allows the use of
algebra to calculate various network properties and corelate it with diffusion
dynamics unfolded on it. While having some strong benefits over compartmental
models, these network models still have several shortcomings, namely that in
these models, the quality of contacts is overlooked. For example, the duration
of contacts affects the transmission probability, and frequency of contact
etc. There have, however, been some 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 found to have an impact on
spreading dynamics [56]. The other crucial temporal factor is that the contact
sequences among individual are completely missing in these network models.

For studying diffusion processes with realistic contacts, there have been
several approaches to make the contact network dynamic as well [39, 57, 58].
The dynamic networks assume the links are transient in status, i.e., links
appear and disappear. However, the relationship between two linked nodes is
often permanent. In the dynamic contact network models, the above static
network models can be implemented as the underlying structure (capturing
permanent social relationship among individuals) and an additional mechanism
is added on top of that to maintain the link dynamic. The dynamic contact
network models are often difficult to analyse with exact mathematical
solutions. Thus, an approximation is often used to characterise the system
dynamics [59]. There are no analytical solutions for many dynamic contact
network models and such models are only used for simulations to explore
diffusion dynamics for wide scenarios of developed models. These classes of
contact networks are often efficient tools to validate the simulations results
of data-driven individual-level diffusion models.

###  2.3 Individual-based models

The other trend of diffusion modelling on contact networks is to apply
individual-based models [60, 61, 62, 63]. In these models, all operations are
executed at the individual-level and thus the integration of many realistic
contact properties becomes easier. The other fundamental concept is that
individual-based models are implemented upon a community of targeted
individuals and that are situated in an environment. In these models, every
individual plays its role and interacts with its respective environment. Thus,
infectious items 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 is not so easy to define the boundaries of the
model class based on individual compared to compartmental models or network
models as the assumptions in the individual based models varies largely.

The system dynamics in the individual-based model are generated with all
individual actions happened simultaneously within the respective simulation
environment of the individual [64]. The respective environment depends on the
modelling approaches and it may include parts or sometimes all of the other
individuals. Thus, all individuals are affected 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 stochastic events. The
reactions process to a simulation environment is often implemented with a set
of rules (e.g. IF-THEN operations). For disease simulation, such a rule can be
IF 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 the inclusion of natural mechanisms for every desired aspect of the model
to be as realistic as possible. They can offer characterisation even at the
link level and environmental conditioning including complex biological
mechanisms [1, 69]. The current exploration of data on social interactions
leverages the benefit of this class of model as they easily allow modelling of
individual interactions over time. The complexity of implementing higher-level
architecture such as clustering and community structure are reduced as the
network formation mechanism is implemented at a lower level. The contact
networks created by dynamic contact network models can also be simulated with
individual-level models. However, the individual based models become difficult
with detailed information and require more effort to analyse sensitivity. To
achieve stable insights, repeated simulations are conducted with a high number
of parameter combinations. Therefore, the individual based model requires more
computing resources and computation time. These models often cannot be
analysed mathematically due to their stochastic nature and large number of
parameters.

##  3 Contact graphs for diffusion study

Contact networks are widely used tools to study diffusion on social contact
networks. This allows to characterise the diffusion behaviours and simulate
the spreading dynamics. These contact networks can mimic the social
interaction and reveal the influence of individual interactions. Contact
networks are often generated by graph models. It is discussed in the previous
section, dynamic contact networks are realistic to model diffusion processes.
However, the development of dynamic graph models for generating dynamic
contact networks is still at an early stage compared to the static models.
There have been limited number of approaches to develop dynamic graph model.
This section presents a brief details of current dynamic graph modelling
approaches which consist of representing contact networks and links
generation. Here, dynamic contact graphs/networks represent temporal or time-
varying graphs/networks where edges between a pair of nodes are dynamic as
their availability for transmission are not permanent.

###  3.1 Dynamic graphs representation

The evolution of a dynamic contact graph can be captured in many ways. The
evolution in the graph can occur due to changes in the status of nodes and
status of links. The links in the static graphs represent a relationship
between a pair of nodes and is created there 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 different meaning)
[39, 72, 58]. The contacts indicate interactions between a linked pair of
nodes occurring at certain times during an observation period. Dynamic graph
modelling is required to incorporate timing information of these contacts with
link dynamics. The development 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
entries containing identities of interacted nodes and the time when the
interaction happened, even with other some meta information such as gender and
locations. The interaction time 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. This representation is a
straightforward and practical format computationally. However, analysing
diffusion processes on the graphs with this format would be difficult as they
do not count some properties such as contact duration. It is also difficult to
visualise the contact graphs and hence representing it to audience.

Multi-layer graphs: The dynamic graph can be visualised well if it is
represented with as a sequence of static graphs. In this method, the
observation time is divided into 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 static graph as a single layer.
This allows one to understand and analyse the dynamic graph using static graph
theories and then combine the results for the sequence of times to obtain
overall results. This method is applicable where the time resolution is high
(or continuous) compared to dynamic process on the studied graph. For studying
infectious disease, this method has a limitation as the disease cannot be
transmitted over a path during a sequence of the graph and thus analysing
multi-layer graphs cannot capture real dynamics. The time-lines representation
of contacts is one of the extended approaches where nodes are placed in one
axis and times in another axis. The advantage with this representation 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 the time dimension. The structure of time-respecting paths can be
represented as a binary matrix as it is in a static graph with an adjacency
matrix. Thus, the dynamic graph can be expressed 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 to process such
graph [59, 76].

Dynamic links graph: In this representation, temporal variations are captured
with only one dynamic graph where nodes and links change their status over
time. The underlying graph is a static graph with the fixed links among nodes.
In fact, the underlying graph captures the fixed topology of the dynamic graph
and can be treated as the foot-print of nodes [77, 78]. Then, the static graph
structure evolves over time where contacts can appear and disappear. This
means a time dimension is added with the static network. This approach is
considered for the class of graphs where the targeted research question is to
understand how the structure has evolved and how it affects the diffusion
process unfolded on the graph. The dynamic graphs are typically data-oriented
where the focus of study is on a data set, its structure, and how something
behaves on it e.g. how disease spreading would behave on the graph. In
addition, these observations may 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 contact networks.

Time-node graphs: The recent trend of dynamic graph modelling is to extend the
concept of node into temporal node i.e at each time step the same node is
considered as a different node. Then, the graph is built among the temporal
nodes. This approach is called the static expansion of a temporal graph [79].
This type of graph can be practical since 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. This requires high
computational power which is available in the current technology. However, the
applicability is limited by the size of the networks.

###  3.2 Dynamic graph modelling

A general representation of the dynamic contact graph can be described as
follows. Consider 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 property such as links weights,
set of node attributes; that is, $L\subseteq Z\times Z\times Y$. The relations
between nodes are assumed to take place over a time span
$\Gamma\subseteq\mathbb{T}$ denoting the lifetime of the system. The temporal
domain $\mathbb{T}$ is generally assumed to be $N+$ for discrete-time systems
or $R+$ for continuous-time systems. The dynamics of the system can be
subsequently described by a dynamic contact graph,
$G_{T}=(Z,L,\Gamma,\phi,\psi)$, where

$\phi:L\times\Gamma\rightarrow\\{0,1\\}$

$\phi$ 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 to create links at a certain time. Thus,
the model can be extended by adding a node status function

$\psi:Z\times\Gamma\rightarrow\\{0,1\\}$

where the activation function $\psi$ of nodes depend on a given time. Given a
$G_{T}=(Z,L,\Gamma,\phi,\psi)$, 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 and indicates only the pairs of
nodes that have relations at some time in $\Gamma$. The connectivity of
$G=(Z,L)$ does not imply that $G_{T}$ is connected at a given time instant
with the connectivity of $G$. Within this general graph definition, various
contact networks can be generated by varying details in links and nodes
definition. Thus, graph models are often accompanied with a network generation
method. This network generation method handles incorporating details to links
and nodes while graph defines the relationships at the abstract level. Some
network generation methods that have been designed under the frameworks of
various graph models are presented here.

Static graphs with link dynamics: The simplest method of generating a dynamic
graph is first to generate a static graph with links using a static graph
model and then define a sequence of contacts for each link generated. To avoid
complexity, contact generation process is often kept independent of the
network position of the links in this approach. The authors of [77] have
applied the following procedures to generate a dynamic graph.

> i) a static graph is constructed from a multigraph that is generated using
> the configuration model [80] and deleting the duplicate links and self-links
>
> ii) 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
> generated using a truncated power-law. For a link, the active interval
> starts at a uniformly random starting time within a sampling time frame
> (observation period)
>
> iii) a sequence of contact times is generated following an inter-event time
> distribution over the observation period. This also generates burstiness in
> the contact patterns.
>
> iv) finally the contact time sequence is wrapped with the corresponding
> active interval of each link. The contact sequence times (step 2) which are
> within the active time interval are taken and other are deleted for a link.
> The wrapping is done for all links in the graph and a dynamic contact graph
> is obtained

The authors of [78] used a similar method to generate contact graph. These
methods easily 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
topological elements and sub-graphs such as triangles and stars. The ERGM
model generates a class of graphs and the model parameter inferred from an
empirical data capture the corresponding graphs. A similar modelling framework
is used by the authors [82, 83] to generate temporal exponential random graphs
(TERGM). Thus, the temporal dynamics of links are influenced by the network
topology in TERGM. In TERGM, a set of states of the nodes observed over a time
window for an ongoing dynamical process is applied to estimate the model
parameters. Thus, the probability of making a contact between a pair of nodes
is bounded with the time window of applied data set for modelling. The
temporal exponential random graph models are not node-oriented model (nodes
connectivity is not build at the node level, but connectivity is defined based
on the network topology) that makes possible to change the network with its
basic building block such as links. Thus, it is 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 node and
link level operations such as which neighbour nodes a host node contacts
frequently, how recent changes in contact sequences affect the future
contacts, whether the contact creation and deletion follow any specific social
mechanism. There have been several works to incorporate these characteristics
of dynamic contact graphs [85, 86, 87, 71]. The work of [71] has developed a
dynamic graph model for face-to-face interactions. This is a spatio-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
assigned to it. Therefore, the more attracted a walker is to its neighbour
nodes, the slower its walk becomes. A similar approach is implemented in the
dynamic graph model developed by [88]. This model for online setting and their
assumption is that some individuals are much 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 a positive feedback
mechanism. The authors have applied stochastic point processes to model
dynamic contact graphs. In this model, a node creates and breaks links
according to a Bernoulli process with memory. The probability of an event
between two nodes increases with the number of events recently occured between
them.

Activity-driven graph modelling: A comparatively simple dynamic graph
modelling approaches is proposed by [57]. This model is called activity driven
network modelling (ADN). They adapt the graph sequence framework of dynamic
graph modelling and generate a simple graph $G_{T}$ at (it is a discrete time
system) time $t$. The graph generation procedures are as follows:

> i) node $i$ is assigned an activity potential $a_{i}$. This is usually done
> with a power-law distribution. The activity potential is assigned to all
> nodes in the graph
>
> ii) 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
> contact memory from previous time step
>
> iii) activate node $i$ with a probability $a_{i}\Delta t$. If node $i$ is
> activated, it is connected with other $m$ randomly chosen distinct nodes.
> Repeat the step 2 and step 3

The distinguishing characteristics of this model are that the activity of the
nodes governs the link creation. In contrast, the previous models are
connectivity driven where the network’s topology is at the core of the model
formation. The ADN overcomes the timescale separation assumption and
explicitly accounting for the concurrent evolution of 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 characterised
using a heterogeneous mean-field approach. Interestingly, it is found that
some system properties are directly related to the activity potential of
nodes. The diffusion dynamics are quite different from that of aggregated
static networks [91]. However, the basic ADN model has several limitations
such as nodes contact with a fixed number of links during each activation, the
contacts are not repetitive, and social structure among individual is not
maintained.

###  3.3 Realistic activity driven graph models

When nodes are active in the basic activity driven network, they randomly
create connections with other nodes. In other words, at any time, a node may
connect with any other nodes in the graphs. That means this method does not
apply any prior knowledge of social or geographic relationships that could
alter the selection of one link over another. However, this assumption
severely challenges the feasibility of the ADN modelling when it is
implemented for real graphs. There have been great efforts to incorporate
realistic features with the basic ADN. An individual can interact with other
individuals arbitrarily or by choice. Thus, some links tend to be persistent
in time and such links are generated in the household, at the office and with
close friends. This property is integrated introducing memory effects in the
link formation. Therefore, social graphs can have two types of links. The
first class describes strong ties that identify time repeated and frequent
interactions among specific couples of nodes. The second class characterises
weak ties among agents that are activated only occasionally. It is natural to
assume that strong ties are the first to appear in the system, while weak ties
are incrementally added to the contact set of each node. This approach is
studied in the work of [92] where it is assumed that a node will connect to a
new node with a probability $P(n+1)=\frac{\eta}{\eta+n}$, where $n$ is the
current contact set sizes of the node and $\eta$ is the tendency to broaden
their contact set sizes. Therefore, the probability of contacting with a node
from previously contacted nodes is $1-P(n)$. This method of repeating with the
old contacted nodes and extending contact set size is called the reinforcement
process. In the above process, every node has the same tendency to extend the
contact graph. In reality, however, individual have heterogeneous tendency to
extend the contact set size. This issue is addressed by the work of [93] where
they proposed to assign a heterogeneous value of $\eta$. For a node $i$, the
probability of contacting a new node is given by

$p(n_{i})=\left(1+\frac{n_{i}}{\eta_{i}}\right)^{-\alpha_{i}}$

where $\alpha_{i}$ is the reinforcement of node $i$ and $\eta_{i}$ is the
characteristic number indicating the size of contact set size before
reinforcement start. The value of $\eta_{i}$ is often assigned with power-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 and make a community. Therefore, the underlying social structure of a
dynamic contact graph also should maintain community structure. The community
structure emerges in a graph by creating triadic closure when making a new
connections [94]. One typical mechanism to make triadic closure is to use
common neighbours (CN) indices, where two nodes $i$ 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 the
number of common neighbours. In other words, triadic closure can be created if
the host node chose a new neighbour from its neighbour’s neighbour [94, 95].
The random new neighbour selection mechanism of basic activity driven graph
can not emerge the community saturate. The work of [72] has upgraded the basic
ADN integrating a triadic closure creation mechanism. If a host node $i$ has
not contacted any node yet, it randomly picks another node from the entire
graph $j$ and creates a link. Otherwise, the host node tries to make a new
link with the triadic closure mechanism. As the first step, it selects
randomly one neighbour node $j$ from his contact set with a probability. If
node $j$ is not selected or has no other neighbours node except node $i$, node
$i$ looks for another random new node and creates a link. If node $j$ is
elected and has neighbour nodes, then it selects a 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 on social 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]. This bursty 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
defined as $a_{i}=\frac{1}{<t_{i}>}$. The inter-event time usually spans over
several orders of magnitude. The authors of [97] shows a mechanism to capture
this bursty nature of human dynamics using a power-law distribution. The basic
ADN can generate heterogeneous contact degree distribution based on the value
of $a_{i}$. However, the growing of contact set size is completely dependent
on $\eta$. In addition, the model can capture link heterogeneity of directed
graphs where the in-link propensity can be different to out-link density. This
issue is addressed by the work of [98] where each node is assigned an
attractiveness. When a node selects a new neighbour node, a node will be
chosen based on its attractiveness. However, any current modification of ADN
can generate contact networks with links for indirect interactions. The study
of [58] shows how to add repetitive contacts and indirect contacts in
generating contact networks.

##  4 Infectious disease diffusion

The theories of diffusion process can be applied to model infectious disease
spreading. It is required to understand how the infectious disease spreading
is modelled. This allows to integrate the above diffusion model with the
disease spread. In the infectious disease, two key aspects are disease
transmission and calculation of transmission force.

###  4.1 Disease transmission

An infectious disease is transmitted from an infected individual to
susceptible individuals via transferring organisms/microbes capable of causing
infection [99, 100]. These organisms/microbes are called pathogens. In this
paper, contagious items or infectious particles refer to these pathogens. The
infectious items enter the body of susceptible individuals 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 to persist
within a population, relevant contagious items are required to be transmitted
continuously to new bodies. The contagious items are transmitted through two
mechanisms: 1) direct transmission and 2) indirect transmission [8, 101, 6,
58, 102]. Direct transmission occurs through individual-to-individual
interactions transferring contagious items without any intermediate
transmission medium between these two individuals. The example includes
physical touches (such as shaking hands, kissing etc.) and contact of blood
and body fluids. Direct transmission is found in infectious disease such as
common colds, sexually transmitted diseases etc. For many infectious diseases
[103, 104, 99, 100], infected individuals generate particles containing
infectious microbes by their respiratory activities like talking, laughing,
coughing or sneezing. These particles are scattered into the environment of
the proximity of the infected individuals. The infectious particles then
deposit onto objects or surfaces and survive long enough time to transfer to
other susceptible individuals who subsequently touch the objects. This creates
the indirect transmission of diseases where intermediate medium or objects are
required to transmit infectious 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
classified into different modes which are based on the roles of the
intermediate medium and the properties of the infectious particles when
transmitting through the intermediate medium. The respiratory activities of
infected individuals generate droplets containing infectious particles and the
sizes of the droplets often define the mode of transmission. Droplet whose
size is comparatively large, often assumes to be greater than 5$\mu$m, are
transmitted through the air to nearby susceptible individuals. This mode of
indirect transmission is called droplet transmission. However, the droplet
whose size is small, often assumes to be less than 5$\mu$m, evaporates quickly
and becomes droplet nuclei. These droplet nuclei are suspended in the air for
a long time and can travel large distances. Thus, they can transmit to
susceptible individuals with a long time delay after their generation, even to
susceptible individuals who are far away up to 100m from the source infected
individual. This mode of indirect transmission is called airborne
transmission. Indirect transmission of infectious particles can also happen
through vectors (mosquitoes, flies and mites etc.) that carry infectious
particle from an infected individual to susceptible individuals with delay and
at substantial distances. In this situation, the contagious items present in
the blood or skin of an infected individual are ingested by vectors. Then, it
is developed in the vectors itself. Susceptible individuals are usually
infected through the bite of an infectious vector, though other ways of entry
are possible. Examples of vector-borne disease transmissions are yellow fever,
malaria, plague and dengue etc. [103, 104, 99, 105]. Another indirect mode of
transmission is to spread disease through contaminated objects spatially.
Examples of such diseases includes water-borne diseases, food-borne diseases
[106, 107].

It has been observed that the spreading of some infectious diseases is
dominated by only direct transmission and can be modelled by creating direct
transmission links for co-presence interaction between infected and
susceptible individuals. However, the infectious diseases that spread based on
indirect transmission or have additional indirect transmission along with
direct transmission cannot be modelled by creating only direct transmission
links for co-presence interactions. This thesis considers airborne infectious
disease spreading as a case study for understanding and modelling the impacts
of indirect transmission links. For airborne diseases, infected individuals
generate droplets containing infectious particles through various respiratory
activities. The authors of [108] have found that an infected individual
generates on average 75,000 particles/cough but it can be up to 500,000
particles/cough. They have also found that 60% of these particles can reach
the alveolar region of lungs if the particles are inhaled by another
individual. It is found that the cough frequency of an infected individual is
on average 18/hr [109]. Thus, an infected individual deposits about
$1.36\times 10^{6}$ particles during a one hour stay at a location. Up to 50%
of these particles evaporate and become droplet nuclei (airborne particles)
which are suspended in the air for a longer time [110]. Airborne particles are
also added to the environment by breathing, talking and laughing. There have
been a wide range of studies to understand the viral load of airborne
particles. The studies show that most of the influenza virus is contained in
the droplets whose sizes are $<5\mu$m. The works of [111, 112, 111] have found
that up to 75% virus is contained within droplets with sizes $<5\mu$m. The
exhaled breath of an influenza patient can generate on average 0.5 plaque-
forming units (PFU) for influenza viruses [113]. The study of [112] found that
a cough can generate up to 77 PFU virus. Inhalation of 0.7 - 3.5 PFU of
influenza is sufficient to cause infection in 50% of susceptible individuals
[114]. Therefore, it can be concluded that the generated airborne particles
have sufficient viral load to cause infection if they are inhaled.

The impact of airborne transmission is different to the other model of
indirect transmission as airborne particles can travel spatially while large
droplets settle nearby. The literature indicates the various range of travel
distances for airborne particles. The travel distances depend on the weather
conditions and air-flows. The authors of [115] show that airborne particles
can travel up to 100m in the direction of air-flow. The travel distance can
also be interpreted from the analysis of SARS outbreak occurred in the Amoy
Gardens Hong Kong in the year 2003. The study of [116] has revealed that the
infection had reached the Block-E which was at 60m distance from the Block-B
where infection had started, although there was no indication of physical
interaction among the residences of these buildings. Thus, it was concluded
that the infection particles travelled to the Block-E through airborne
transmission. A number of studies have also shown that it is also possible to
disperse airborne particles between flats in a building [117, 118] and between
wards in a hospital [119]. The experiment of [118] shows that airborne
particles can also travel from one building to the nearby buildings. The
travel distance of airborne particles is extended in the open area. The
authors of [120] have studied the presence of influenza A virus around pigs
farms by collecting air samples at different distances from the farms. They
have noticed a significant amount of RNA copies of the virus at the distance
of 1.5Km from the pig farms that had influenza A infected pigs. Therefore, the
airborne indirect transmission mode of infectious disease has strong potential
to spread diseases. The airborne infectious diseases spreading is an important
application of diffusion process with indirect transmissions.

###  4.2 Infection risk

An interaction (e.g. being in the same location) between infected and
susceptible individuals poses an infection risk for the susceptible
individual. Infection risk assessment can be divided into two steps:
determining the intake dose of infectious particles and finding the
corresponding infection probability [121]. The infectious particles that reach
the target infection site are called the intake dose. The intake dose is
estimated based on the exposure level to the infectious particles, the
pulmonary ventilation rate of susceptible individuals, the exposure time
interval, and the respiratory deposition of the infectious particles. Then,
the infection probability is calculated by a mathematical formula. Two
approaches are applied to determine if an infection occurs: deterministic and
stochastic. The first approach assumes that each individual has an inherent
resistance up to a dose of infectious particles. Thus, a susceptible
individual contracts the disease when a target infection site is exposed to a
dose equivalent to or exceeding the threshold dose. In the stochastic
approach, any amount of intake dose causes disease with a certain probability.
The infectious particles are usually randomly distributed in the suspension
medium. Thus, the estimated exposure level and intake dose of airborne
particles are always expected values rather than exact values. Therefore, the
stochastic models are appropriate for studying airborne disease spread. Models
that are frequently used for assessing infection risk for airborne diseases
are now discussed.

A wide range of models has been developed for the spread of airborne disease.
These range from simple models that are easy to apply to complex models that
require greater detail of the disease spreading process. Unfortunately, these
details is not always available for many diseases. In the literature, the
Wells-Riley model or its modification are widely used to estimate infection
risks [121, 122]. The Wells-Riley equation is given as

$P_{I}=1-exp\left(-\frac{Igpt}{Q}\right)$ (2)

where $P_{I}$ is the probability of causing infection to a susceptible
individual for the intake dose $E=Igpt/Q$, $I$ is the number of infected
individuals at the interaction room, $p$ is the breathing 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 susceptible individuals. This model is based
on the concept of quanta which is the number of droplet nuclei 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 to determine disease spreading dynamics for the large
population. The model parameter quanta generation rate $\vartheta$ is required
to be estimated from the real outbreak cases. This is very difficult for many
diseases as it requires data from real outbreak scenarios. The model is also
limited due to its assumption that particles are homogeneously distributed in
the air, and that every particle reaches to the target infection site. It does
not consider the 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 inhaled infectious particles by multiplying a fraction term
with the intake dose as $E=\frac{Igpt\theta}{Q}$, where $\theta$ is the
fraction of infectious particles reached to a target infection site. Air
disinfection and particle filtration are used in the many buildings that
reduce the effective infectious particles to cause infection. These factors
are included for the Wells-Riley equation in the work of [123]. However,
collecting such data is difficult and expensive for large scale simulation.
The assumption about the homogeneity of particle distribution in the
interaction area is addressed by [124]. They considered the time-weighted
average pathogen concentration in the room air to incorporate the non-steady-
state conditions in the Wells-Reily equation. This model is given by

$P_{I}=1-exp\left(-\frac{pIg(Qt+e^{-\varphi t}-1)}{VQ^{2}}\right)$ (3)

where $Q$ is the air change rate or disinfection rate, $\varphi$ is the
particle accumulation rate and $V$ is the volume of interaction area. In spite
of these improvements, the Wells-Reily model still requires the total exposure
during an outbreak to find the quanta generation rate and that is not possible
for many diseases.

Rudnick and Milton [125] developed a model where the exhaled air volume
fraction is used to estimate the number of quanta that the susceptible
individuals are exposed to:

$P_{I}=1-exp\left(-\frac{g\bar{\omega}It}{N}\right)$ (4)

where $\bar{\omega}$ is the average volume fraction of room air that is
exhaled breath and $N$ is the total number of people in the premises. To find
the quanta generation rate based on the $\bar{\omega}$, one requires a
knowledge of carbon dioxide concentration in the room. These models still
follow the well-mixed assumption of particle concentration. Some works [118,
126] address this problem by experimenting the dispersion of tracer gas and
integrating impacts with model.

In the models discussed above, the quanta generation rates are not well
understood for many diseases. However, the infectious particles generation
rates, their formation, pathogen loads and their survivable time etc. are now
becoming available. The authors of [127] first introduce a dose response model
based on the infectious particles concentration instead of quanta. The model
is

$P_{I}=1-exp\left(-\frac{Ig\theta pt}{Q}\right)$ (5)

where $g$ is the number of infectious particles released per infected per unit
time and $\theta$ is the fraction of infectious particles reaches the target
site. In this equation, the quanta generation rate $g$ is replaced by $\theta
g$. The authors defined the source strength $g$ with cough frequency, pathogen
concentration in the respiratory fluids and the volume of expiatory droplets
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
infectious particle concentration considering non-steady-state conditions as

$P_{I}=1-exp\left(-\frac{I(g-r)\theta
pt}{Q}\left[1-\frac{V}{QT}\left(1-e^{-\frac{Q\tau}{V}}\right)\right]\right)$
(6)

where g is the particles generation rate, $r$ is the mortality rate of the
generated particles, $\theta$ is the deposition fraction of the inhaled
particles, $\tau$ 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 captured using a non-steady-state model. However, this model assumes that
all infected individuals arrive at the same time and this may not happen in
reality. The variations in the arrival time of infected individuals also
introduce the fluctuations in the particles concentration. The current models
also do not capture the exposure that susceptible individuals receive after
infected individuals leave the interaction locations. Thus, it would be more
appropriate to find exposure level due to contact with each infected
individual and sum them up to find total exposure. Therefore, the arrival and
departure of each infected individual can be tracked independently and hence
the exposure during indirect interactions. This also allows one to assign a
random value of $Q$ to each contact to capture heterogeneous particle
concentrations at different locations.

The disease transmission probability can also be calculated for both the
direct and indirect contacts as follows [8]. This model resolves the problems
mentioned in the previous section. This model define a contact called same
place different time (SPDT) that has both direct and indirect infection
transmission probability. This accounts when an infected individual and
susceptible individuals have been to a place. If a node in the susceptible
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 indirect transmission links according to the following equation

$E_{l}=\frac{gp}{Vr^{2}}\left[r\left(t_{i}-t_{s}^{\prime}\right)+e^{rt_{l}}\left(e^{-rt_{i}}-e^{-rt_{l}^{\prime}}\right)\right]+\frac{gp}{Vr^{2}}\left(e^{-rt_{l}^{\prime}}-e^{-rt_{s}^{\prime}}\right)e^{rt_{s}}$
(7)

where g is the particle generation rate of infected individual, p is the
pulmonary rate of susceptible individual, V is the volume of the interaction
area, r is the particles removal rates from the interaction area, $t_{s}$ is
the arrival time of the infected individual, $t_{l}$ is the leaving time of
the infected individual, $t_{s}^{\prime}$ is the arrival time of susceptible
individuals and $t_{l}^{\prime}$ is the leaving time of susceptible
individuals from the interaction location and $t_{i}$ is given as follows:
$t_{i}=t_{l}^{\prime}$ when the SPDT link has only a direct component,
$t_{i}=t_{l}$ if the SPDT link has both direct and indirect components, and
$t_{i}=t_{s}^{\prime}$ otherwise. If $t_{s}<t_{s}^{\prime}$, $t_{s}$ is set to
$t_{s}^{\prime}$ 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

$E=\sum_{k=0}^{m}E_{l}^{k}$ (8)

where $E_{l}^{k}$ is the received exposure for kth link. The probability of
infection for causing disease can be determined by the dose-response
relationship defined as

$P_{I}=1-e^{-\sigma E}$ (9)

where $\sigma$ is the infectiousness of the virus that causes infection [99].

##  5 Diffusion control

###  5.1 Diffusion control strategies

Controlling diffusion dynamics on individual contact networks has a wide range
of applications 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 the spreading of items to the largest
proportion of populations [129, 130] while diffusion controlling for
infectious disease focuses on minimising the number of infections reducing the
number of individuals received spreading items [131]. However, the key task in
all controlling methods is to find a set of individuals and change their
behaviours to alter spreading rates. These individuals often have high
spreading potential and are called super-spreaders. The size of the set should
be minimal to reduce vaccination cost as well as achieve the control goals.

Most of the efforts of developing a control strategy are put on finding the
optimal set of individuals. Accordingly, researchers search for the contact
properties of individuals and their behaviours relevant to the spreading of
contagious items [132, 133, 134]. Individual’s preference and exposure
intensity to contagious items, personal status and their surrounding
environment often define the spreading potential. Thus, the influential
individuals are often searched based on the individual’s behaviours [135,
136]. The network properties such as the number of connection of individuals
to others are also key factors to determine one’s spreading potential.
Understanding personal behaviours and modelling is a complex process. In
addition, research focuses on understanding the impacts of contact properties
on diffusion dynamics.

Various measures of network properties are applied to find influential
individuals for controlling diffusion of contagious items. The widely used
measures to find the important individuals are degree centrality, betweenness
centrality, k-core score and PageRank centrality etc [132]. Individuals
contact degrees defined the number of connections to other individuals is
frequently used as topological measures of influence. In the social contact
networks with broad degree distribution, it is observed that individuals with
high degree determine the diffusion dynamics [137]. However, degree based
methods sometimes underestimate the low degree individuals that can be
influential through connecting high degree individuals. The page ranking
algorithm developed to rank the content in the World Wide Web is also adapted
to find the pivotal individuals in social contact networks. The ranking
mechanism of PageRank is simple and straight forward [138] where the
importance of page is measured by counting the number and quality of links to
that page. The PageRank algorithm is only applicable to directed networks.
Betweenness centrality is also a good candidate in many applications as it is
a measure of the number of shortest paths passing through one individual
[139]. Thus, it is most likely that individuals having high betweenness
centrality will play key roles in shaping the diffusion 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
positions of individuals in the social networks with the k-index obtained by
iteratively removing k-degree nodes [140]. These networks measures are based
on static networks and may not provide optimal performance in dynamic contact
networks.

The one way of finding influential nodes in dynamic contact networks is to use
temporal versions of traditional centrality metrics [133, 134]. In this work,
time respecting paths, paths are created based on the time order of links
availability, are used to calculate the betweenness centrality and closeness
centrality. Eigenvector centrality is also modified for dynamic networks
[141]. These methods require complete information regarding contact networks.
The random walk is applied in [142] for measuring temporal centrality. This
does not require global information. However, all these algorithms require
huge computational resources. The temporal centrality measurement is in the
early stage and is 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 social contact networks as the contact information often
is available [132, 143].

###  5.2 Disease spread control

It is difficult to apply the above approaches for controlling disease
spreading as it is quite difficult to collect contact information of a
population. The vaccination strategies are required 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 the local
contact information. The simplest way of selecting a set of individuals is to
choose randomly from the population and is called random vaccination [144,
145, 146]. This approach does not consider the disease spreading behaviours of
the chosen individuals. Therefore, the information collection cost is minimal.
However, it requires a large set of individuals to be vaccinated for achieving
hard immunity to disease. Thus, the research is directed to select the
individuals who have strong disease spreading potentials [147]. These methods
are called targeted vaccination. In the targeted vaccination, the number of
individuals to be vaccinated is often small and the effectiveness of
strategies is substantially high if an appropriate set of individuals are
chosen. Therefore, the infection cost can be substantially lower in a targeted
vaccination strategy with the reasonable cost of information collection and
vaccination cost.

There has been a wide range of vaccination strategies using obtainable contact
information [148, 149, 150, 151]. All these methods do not develop vaccination
strategies based on local contact information. Sometimes global information is
also used and methods are developed to find the global metrics with locally
obtainable contact data. There have been several other methods that depend on
the movement behaviours of individuals instead of collecting information on
interactions between individuals. Vaccination strategies are also varied based
on the implementation scenarios. There are two specific vaccination scenarios:
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 vaccination strategies in the real-world scenarios are
expensive and difficult, empirical contact networks or synthetic contact
networks are frequently applied to test and validate developed vaccination
strategies. The discussion includes the network model based vaccination
strategies.

The authors of [145, 149] present an elegant way of implementing vaccination
using local contact information called acquaintance vaccination (AV).
According to this strategy, 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 network are required. In fact, it selects
the node that has a large number of connection to other nodes. Its efficiency
greatly exceeds that of random vaccination. The acquaintance method 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 frequently. 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 vaccinated then the performances are improved
significantly.

The acquaintance based strategies still require a large number of individuals
to be vaccinated to achieve control goal. Thus, there have been a fair amount
of works to search for other contact properties that can be obtained locally.
The analysis of social contact networks shows that individuals are connected
to various communities [155, 156]. These properties of the contact network are
exploited by several works where the concept of the bridge nodes is introduced
as these nodes provide the pathways for a disease to propagate from one
community to another community. Therefore, vaccinating such nodes will be a
more effective strategy than of selecting random acquaintance. However, it
requires the searching methods that use only the local contact information.
The searching algorithm developed by the work of [157] can find the bridge
nodes using stochastic searching methods that need only local structural
information. They found that the developed 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 between two communities, are
chosen for vaccination.

The above vaccination strategies have been developed based on static network
properties. However, the real-world social contact networks are dynamic which
has a strong impact 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 appropriate nodes to be vaccinated. For
example, the infection risk for being in a contact with an infected individual
is relevant to the contact duration. Moreover, there is a higher risk if one
susceptible interacts frequently with the infected individual. The works of
[159, 160, 161] consider this information in neighbour selecting instead of
selecting random neighbours. The authors of [159] use the most recent contact
for vaccination and they also apply weight to capture the contact rates with
the neighbouring nodes. Theses vaccination strategies outperform acquaintance
vaccination.

The collection of contact information is often difficult. Thus, vaccination
with detailed local contact information may be infeasible in real-world
scenarios and lose the benefit of using the local contact information. There
have been several other vaccination models based on individual movement
behaviours where individuals contact properties are not consider explicitly
[150]. Beyond the contact properties, the work of [162] consider the
individuals 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
the contact information about neighbouring nodes. However, in the diffusion
processes with indirect interactions, it is difficult to identify the
neighbours contacted through indirect interactions. Thus, there is a need to
understand the efficiency of strategies with indirect interaction and find the
best strategy. The authors of [102] addressed this issues to build vaccination
strategies with the indirect links.

##  6 Conclusion

For modelling disease spread within a society, a proper infection risk
assessment model and a proper contact network are required. This paper
analysed a wide range of networks models and infection risk models. There are
simple infection risk assessment models that capture practical situation
through simple model parameters. The complex model can capture the temporal
dynamics of individual interactions and estimate more accurate infection risk.
But, it require granular level contact information which may not available all
time. In addition, it is observed that many factors affect the spreading
dynamics of contagious items on contact networks. However, it is clear that
interaction pattern of individuals 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 the impacts of interaction
patterns with diffusion modelling [25, 24, 26, 164]. There are, however, still
some critical factors to be addressed in constructing proper diffusion models
that capture realistic contact patterns. In addition, current opportunities
for gathering individual-level contact data have attracted the researchers to
deep dive further in this field by looking at contact patterns at the granular
level [165, 62, 5, 166, 101]. The models presented in this paper would provide
a guideline to model infectious disease like COVID-19.

##  7 Introduction

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