IEEE Circuits and Systems Magazine - Q4 2021 - 8

Network theory has emerged as a powerful tool to capture the complexity of real-world
contagion patterns and formalize it in mathematically tractable models.
powerful tool to capture such a complexity and formalize
it in mathematically tractable models [12], [23]-[25].
In the rest of this survey, we will focus on network
epidemic models, mostly discussing the SIS model,
which is among the simplest and most studied models.
In Section II, we will review some classical results for its
deterministic and stochastic implementations on static
networks. In Section III, we will move to the most recent
developments and the state of the art for the research
on epidemics on dynamic networks. Section IV is devoted
to the discussion of control of deterministic and stochastic
network epidemic models, with specific attention
on the control of dynamic networks. In Section V,
we present a focused discussion on the current challenge
of the ongoing COVID-19 pandemic and on how
the mathematical modeling of epidemics can provide
powerful tools toward fighting against and mitigating its
spreading. Finally, in Section VI, we briefly summarize
the content of this survey and outline some promising
challenges for the future research.
II. Classic Models of Epidemics on Networks
Here, we present the network SIS model and report the
main results for its deterministic or stochastic implementations.
Then, we briefly mention some important
extensions of these results to scenarios that incorporate
further features of real-world systems. Towards
this end, we first gather some definitions and basic notions
on network and graph theories, which will be used
throughout the survey.
A. Basic Notions on Networks and Graph Theory
A static network is represented by means of a time-invariant
graph with n nodes, denoted by a set of positive
integer indices
through a set of directed links E VV ,#3
.
2
4
1
3
1
2
2
2
5
3
3
4
A =
2
3
1
2
4
2
4
3
3
2
1
2
3
2
Figure 4. Example of a weighted undirected static graph with
n = 5 nodes and its weighted adjacency matrix A. Note that
the graph is connected.
8
IEEE CIRCUITS AND SYSTEMS MAGAZINE
The intensity of such connections is measured by a
nonnegative quantity. Specifically, for any pair of nodes
,
ij V!
, we define the connection strength a 0ij
$
ai .
j
E
=
!
that
quantifies how strongly node i is connected to node j,
and so,
ij 2 + (, ) ! The connection strengths
#
$0
.
are collected in a (weighted) adjacency matrix A Rnn
The triple GV E (, ,)A defines the graph. An example
is illustrated in Fig. 4.
A network is undirected if the corresponding adjacency
matrix is symmetric, that is,
AA ;=
<
otherwise
it is said to be directed. A network is connected (strongly
connected, for directed networks) if its adjacency matrix
A is irreducible, that is, if for any pair of nodes i and
j, there exists a sequence of nodes
such that (, ),, 1 E!+
vi ,, ,vv jk12
vv
, ,f for =- A network is
11k
,,
==f
.
unweighted if the adjacency matrix of the corresponding
graph A has binary entries, that is, all nonzero entries
(corresponding to links) are equal to 1. Given a node
,
i
j
a
Note that if the network is unweighted, then the degree
ki
nected to, which are called neighbors of i.
A dynamic network is represented by means of a
time-varying graph () (, (),( )),tt AtGVE
=
and connected through a time-varying set of links ().tE
The matrix ()
At Rnn
!
#
$0
i V! we denote by k = R !V ij its (weighted) degree.
is equal to the number of nodes that node i is conwhere
the
time t can be a discrete or a continuous index. The n
nodes in the node set
V 1 f= {, ,}n are time-invariant
is the time-varying (weighted)
adjacency matrix and measures the strengths of the
connections between nodes at time t.
V 1 f= {, ,}n . Nodes are connected
such that
(, )ij E! if and only if node i is connected to node j.
B. Deterministic Network Models
From the first implementation of a deterministic SIS
model on a (static) network, proposed by Ana Lajmanovich
and James Alan Yorke in 1976 to study the spread
of gonorrhea in a population [26], network epidemic
models have become an established and successful
paradigm to study the spread of epidemic diseases in
complex populations [12], [23]-[25].
In the network SIS model, each node represents a
group of individuals and is characterized by its health
state, that is, (),( ).
^h The health state of node i V!
st xt
ii
represents the fraction of individuals per health states
in the ith group [26]. Hence, () [, ]
st 01 , () [, ],
i
!
xt 01
i
!
and, similar to the population SIS model described in
the previous section, it holds true that () ()
xt st 1ii
xti
+= .
Hence, the health state of each node can be fully determined
by the unique variable ()
and the entire state
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