IEEE Circuits and Systems Magazine - Q4 2021 - 11

n
m
where ()At
cency matrix A, then
E #
T
[]
if // ()A12mn t , then
E TK ,eKn
2
[] $ 1
where KK ,
(16)
12 02 are two nonnegative constants that
depend on the model parameters and on the network
structure.
The proofs of these results are quite technical and
based on the theory of Markov processes [33]. Briefly,
the key idea is that, in the fast extinction regime, the
probability that the number of infected individuals in
the population increases as determined by Eq. (12) is always
less than the probability that it decreases, yielding
a drift in the direction of the disease-free equilibrium.
Above the epidemic threshold, instead, the inequality is
reversed when the process is close to the disease-free
equilibrium, implying that the infections tend to rise
and large stochastic fluctuations are needed to reach
the disease-free equilibrium. Similar techniques have
been used to study other stochastic epidemic models
on networks, including the SIR model. For more details,
we refer to these review papers [25], [38].
The results summarized in Theorem 3 show the existence
of a sharp phase transition between a regime
where the epidemic is on average quickly eradicated, and
a regime where the disease lasts on average for an exponentially
long time. However, these results on the average
eradication time may fail in characterizing the actual
behavior of a single instance of the epidemic process. On
the one hand, quick eradication can be guaranteed by
directly applying the Markov inequality to Eq. (15), yielding
()
Tn#
ln
a with probability converging to 1 as n " 3,
for any a 2 1. On the other hand, an accurate analysis of
the eradication time has been performed in [39], where
it has been established that, if
n 2m
t ()A
a constant that depends on the network structure (more
specifically, on the isoperimetric constants of the network),
then the eradication time is exponentially large in
the number of nodes with probability converging to 1 as
n " 3. Note that, if
lnn
nmt()
A
;
(15)
1
1
t ()
A
,
(14)
is the spectral radius of the (weighted) adjakovianity
assumption, allowing different forms for the
process ()
Xt . In particular, an SIS model in which the
statistical distribution of the contagion time and/or of
the recovery time differ from the exponential distribution
associated with Markov processes was proposed
in [40]. Therein, a mean-field approach is used to determine
conditions for fast eradication of the disease. Further
extensions of this approach can be found in [41],
[42]. Without relying on any mean-field approximations,
in [43], a lower bound on the decay rate to the infectionfree
equilibrium is rigorously computed. Equivalences
and differences between Markovian and non-Markovian
epidemic models have been extensively discussed in
[44], [45]. All these works suggest that the non-Markovianity
of the mechanisms that govern the epidemic
process may have a significant impact on the spread
of a disease and outline an important avenue of future
research in the field of stochastic epidemic models toward
shedding lights on how the distributions of infections
and recovery times shape the spreading process.
D. Discrete-Time Epidemic Models
This survey focuses mostly on continuous-time epidemic
models. However, it is important to mention that the continuous-time
formulations of deterministic and stochastic
models presented in this survey (both the population
and the network models) naturally have discrete-time
counterparts, where differential equations and Markov
processes are replaced by difference equations and Markov
chains, respectively. Here, we report the equations
for the discrete-time deterministic network SIS model,
which will be used in some of the models of epidemics
on dynamic networks presented in Section III. For each
node i V! , the health state is updated as follows:
xt 11 xt xt-+ -- -^h (17)
ii ii i
11 1
() () () (()) () ,
+= nmmti()
Table I.
Notation for network epidemic models.
Symbol
Description
v , where v $ 1 is GV E= (, ,)A
t()A
n
v 2 1 , there may be a gap between
the two regimes. In some specific cases (e.g., for complete
graphs and Erdo˝s-Rényi random graphs), v may be
equal to 1, yielding a sharp phase transition not only in
the expected duration of the epidemic disease, but also
in its actual duration, with high probability.
FOURTH QUARTER 2021
Xti
m i
n i
x)
T
()
network (node set, edge set, adjacency
matrix)
spectral radius of the adjacency matrix A
number of nodes
health state of node i (stochastic models)
infection rate of node i
recovery rate of node i
endemic equilibrium
eradication time (stochastic models)
IEEE CIRCUITS AND SYSTEMS MAGAZINE
11
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