IEEE Circuits and Systems Magazine - Q4 2021 - 10

to approximate the solution of Eq. (7) about the epidemic
thresholds has been proposed in [32].
Similarly, the SIR model and more complex compartmental
models have been embedded and studied on network
structures. For more details on these implementation
and their analysis, we refer to the following review
papers [12], [24], [25].
C. Stochastic Network Models
In their stochastic implementation, network epidemic
models are defined as follows. Each node of the network
represents an individual and is characterized by the
health state ()
Xti
Xt = '
i
1
, which coincides with one of the compartments.
In the stochastic network SIS model, we have
()
if is susceptible at time ,
if is ected at time .
it
it
inf
Xt 01 n
!
(10)
The nodes' health states are gathered into an n-dimensional
vector () {, }
state of the population.
Most of the literature on stochastic epidemic models relies
on the assumption that the evolution of the epidemic
process ()
Xt can be represented by a Markov jump process,
where the state transitions are triggered by Poisson
clocks. Such an assumption, although simplistic, allows to
use the rich theory on Markov processes [33] to perform
rigorous analyses of the model, determining its asymptotic
and transient behavior, as detailed in the following.
We assume that the vector ()
, which represents the health
pends only on the recovery rate of the individual, that is,
it is a Poisson clock with the rate equal to (( ))
mni
=
i XtR
.
These two mechanisms yield the following transition
probabilities
P
P ii i
ii i
TT T
T
01
10
m
m
TT (12)
+
+
,
for all
[( )| () ]( ())( ),
[( )| () ]( ())( )
Xt tXtXtT oT
Xt tXtXtT oT
+= == C
+= == R
i V! which unequivocally determine the dynamics
of the Markov process, as illustrated in Fig. 6.
At this stage, one may observe the presence of strong
similarities between the expressions of the transition
probabilities of the Markov process and the differential
equations that govern the deterministic network SIS
model. Indeed, these two models have strong connections.
In fact, recently, a different formalization of the
deterministic network model has been proposed. In this
formalization, each node of the network is a single individual
of the population (similar to the stochastic framework)
and the corresponding health state ()
xti
xX tEii
=
denotes
the probability that node i is infected at time t, that is,
[( )]
[30], [34]. Under the so-called N-intertwined
mean-field approach, the following approximation
is made: [()( )] [( )] [( )]
EEXt XtE
Xt Xt .
Xt . This approach, even though not
ij ij , showing
that the deterministic system of ODEs in Eq. (7) approximates
the evolution of the expected value of the stochastic
process ()
Xt evolves according
to a continuous-time Markov process, governed by
the contagion and the recovery mechanisms, which act
on the health state of each node. The former regulates
a node's state transitions from the susceptible state to
the infected state and is modeled by a Poisson clock
with the rate proportional to the sum of the connection
strengths of the infected individuals that node i is in
contact with and to the contagion rate im , that is,
mm=
C (( ))
i Xt
iij
j
/ aX t
!V
j().
(11)
The recovery mechanism determines the state transitions
from the infected state to the susceptible and deλC
i
λR
i
(x)
Figure 6. Transitions of the Markov process X(t) for a stochastic
network SIS model are determined by the contagion
and recovery mechanisms. Nodes in green are susceptible
and nodes in red are infected.
10
IEEE CIRCUITS AND SYSTEMS MAGAZINE
i (x)
i
exact (since it is based on an approximation that typically
does not hold true), is often used to approximate
and study the evolution of more complex stochastic network
epidemic models.
From Eq. (12), we observe that the disease free equilibrium
x 0= is the unique absorbing state of the Markov
process and is globally reachable. Hence, different
from its deterministic counterpart, in the stochastic SIS
model, the disease is always eradicated with probability
1 in finite time [33]. However, in the following, we will
show that an epidemic threshold is still present in terms
of the duration of the transient evolution of the epidemic
outbreak, before reaching the disease-free equilibrium.
Formally, it has been observed that a phase transition
with respect to the eradication time can be established,
where the latter is defined by
Tt Xt0 :{ :()}.==$
min
(13)
Specifically, a set of results that characterize the expected
value of the eradication time []TE
depending on
the model parameters and on the network structure has
been established [35]-[37]. The following theorem gathers
some key results from the cited literature.
Theorem 3. Consider the homogeneous stochastic network
SIS model in Eq. (12) with imm= and inn= , for all
i V! , on a strongly connected network. If
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