IEEE Circuits and Systems Magazine - Q4 2021 - 14

made to overcome the limitation of time-scale separation
and propose a theory for epidemics on dynamic networks
which allow to model and study the coevolution of the two
dynamical processes. In the rest of this section, we present
and discuss some of these fascinating paradigms.
B. Temporal-Switching Networks
The use of temporal-switching networks to study epidemics
in time-varying systems has been initially proposed
in [70]. In its original incarnation, a switching network
is generated by repeating a deterministic sequence
of T static networks, characterized by the adjacency
matrices
AAT
1 f so that () =At A modtT
,, ,
. Therein, the
discrete-time deterministic SIS model has been studied,
extending the results found for static networks. Specifically
(in the homogeneous scenario), if one defines
T
PI At
t
:(() ),
1
=-nm+
=
% 1
If () ,
(21)
then, the behavior of the system is determined by the
spectral radius ().Pt
t P 11 then the disease-free
equilibrium is asymptotically stable and the disease is
quickly eradicated. Otherwise, the disease-free equilibrium
becomes unstable and the disease becomes
endemic. Such a framework has been extended to the
discrete-time stochastic SIS model in [71], showing the
same epidemic threshold by mapping the time-varying
system onto a multi-layer network structure. In [72], sufficient
conditions for stability have been established for
a more general scenario, where the network switches arbitrarily
among a set of topologies, possibly according
to stochastic mechanisms, such as Markov switching
rules where, given a Markov process ()tv with the state
space
" 1 ,,Tf ,, we set () =At A ()t
v
. This sufficient condition
is expressed in terms of the maximum possible
norm of products of matrices Pt in the set
P:( ).== -+ " PI A ,tt1 nm
For a review of the most recent developments of this
theory, including the computation of a unified formula
for the epidemic threshold, we refer to the following paper
by Zhang et al. [73].
The use of temporal-switching networks to study epidemics
on dynamic networks has been recently extended
to continuous-time processes. Specifically, in [74], the
theory of positive linear switched systems is leveraged
to derive conditions for global asymptotic stability of the
disease-free equilibrium. Such conditions are obtained
by combining the stability analysis of the linearized
switched system an appropriate notion of irreducibility
for the linearization. Specific results are obtained if the
topology evolves stocastically according to a Markov
14
IEEE CIRCUITS AND SYSTEMS MAGAZINE
switching rule. This approach is followed in [75] to design
Markov switching laws to enforce quick eradication
of the disease via geometric programming. In [76], a continuous-time
network SIS model is studied in a scenario
where the network topology switches deterministically,
at discrete-time steps, following a sequence of adjacency
matrices. Therein, the epidemic threshold is computed in
terms of the Lie commutator bracket of the adjacency matrices,
showing that adjacency matrices that are noncommuting
yield a lower epidemic threshold, favoring the epidemic
spreading. In [77], the scenario of continuous-time
switching networks is considered. The epidemic threshold
is explicitly computed in the case when the adjacency
matrix A(t) commutes with the aggregated adjacency matrix
up to that time
#t
A ,r
As ds. Under this condition, the
()
order of the switching matrix has no effect on the dynamics,
which is fully determined by the average adjacency
matrix
defined in Eq. (18). If // (),A11mn t
r
then the
disease-free equilibrium is asymptotically stable; otherwise,
it is unstable and the disease becomes endemic.
In [78] the analysis of the continuous-time deterministic
heterogeneous SIS model from [31] is extended to slowly
changing time-varying systems by leveraging Lyapunov
arguments. In that work, the effect of stochastic perturbations
of the model is also studied.
A major limitation of the theoretical analysis of temporal-switching
networks is that it is often assumed
that the network switches between different adjacency
matrices determined a-priori and that the switches typically
take place at deterministic (often equally-spaced)
time-instants. In the following, we will present two alternative
paradigms that do not rely on this assumption.
C. Activity-Driven Networks
In all the network models presented so far, interactions
were determined by some pre-determined connectivity
patterns, represented by an adjacency matrix, or a sequence
of adjacency matrices. Activity-driven networks
(ADNs), originally proposed by Perra et al. in [79] suggested
to perform a paradigm shift. In ADNs, interactions
are not seen as a consequence of a network structure
which identifies pre-determined dyadic relationships
between pair of nodes, but are rather generated by individuals'
properties, according to a stochastic process. In
their original incarnation, each individual i is characterized
by a unique parameter
a [, ]01i
!
, termed activity,
which represents an individual's propensity to generate
interactions. Then, starting from t = 0, the dynamic network
is generated according to the following algorithm:
i) the edge set is initialized as ()tE
= Q;
ii) each individual i activates with probability ai,
independent of the others. If an individual i activates,
then the individual generates m links with
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