IEEE Circuits and Systems Magazine - Q4 2021 - 12

In most of the real-world epidemic outbreaks, the underlying network of social interactions
changes dynamically, co-evolving with and influenced by the spread of the disease.
where mti () is def ined in Eq. (5); here, m i
and in
have to be interpreted as the per-contact infection
probability and the per-time-unit recovery probability,
respectively.
Most of the results discussed in this section concerning
the existence of a phase transition between a
fast extinction regime and a regime where the epidemics
becomes endemic and its dependence on the model parameters
and on the network structure can be extended
with some minor adjustments to the discrete-time counterparts
of the models. For more details on the analysis
of discrete-time epidemic models and their main results
for deterministic models, we refer to [46]-[50], for which
a recent review paper by Parè et al. covers most of the
results [51]; for stochastic models, we refer to [52], [53].
Detailed discussions on the main differences between
continuous-time epidemic models and their discretetime
counterparts can be found in [54], [55].
E. Challenges for Network Epidemic Models
A few years ago, Pellis et al. in a perspective paper outlined
eight important challenges for network epidemic
models [56]. Besides other-more practical-directions,
calling for the integration of network computational
modeling and epidemiological relevant data, two
key challenges were identified, which are of great interest
for the engineering community. The first challenge
concerns the study of epidemic models on dynamic
network structures, leading to the analysis of nonlinear
time-varying dynamical systems. The second focuses
on understanding how the network structure (static or
dynamic) can be exploited to effectively design intervention
policies to stop or mitigate the disease spreading;
control-theoretic tools are key to address this second
challenge. In the rest of this survey, we focus on these
two research directions, presenting the state of the art
in terms of key progresses of the last few years and most
promising lines of current research.
III. Epidemics Models on Dynamic Networks
The extension of the classic compartmental models
to static networks and the corresponding rigorous
analysis have allowed the scientific community to
understand how the architecture of human social interactions
affects the spread of epidemic diseases in
interconnected populations. However, in most of the
real-world epidemic outbreaks, the underlying network
of social interactions is not static, but dynami12
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cally changes, co-evolving with and influenced by the
spread of the disease [57]-[60].
Several endogenous and exogenous reasons may be
adduced to explain and motivate the dynamic evolution
of the network structure. First, dynamical changes of
the individuals' patterns of interactions may be directly
or indirectly caused by seasonal factors, such as school
holidays and weather conditions, which may favor or hinder
gatherings and social events [58], [61]. Second, social
interactions are often characterized by an intermittent
behavior, whereby individuals' propensity to generate
connections is subject to burstiness, yielding clusters
of connections separated by latency periods [62], [63].
Third, the infection events themselves may affect the
network structure. In fact, not only infected individuals
may reduce their interactions as a consequence of the illness,
but also susceptible individuals-driven by awareness
and risk perception cognitive mechanisms-may
dynamically modify their behavior, reducing or rewiring
their interactions to reduce the risk of contagion [57],
[64], [65]. Finally, nonpharmaceutical intervention policies
often entail a dynamical modification of the pattern
of human interactions, which may be reduced through
the implementation of lockdown or social distancing policies,
or reshaped by travel bans and mobility limitation,
as observed during the ongoing COVID-19 pandemic [66].
All these evidences of the dynamic nature of social interactions
call for the extension of the epidemic models presented
in the previous sections to dynamic networks.
In the following, we review and discuss some successful
modeling paradigms that have been recently
developed to capture this dynamic nature of the network
of social interactions within analytically tractable
mathematical models of epidemic diseases. Through
the analysis of these models, we shed lights on how the
dynamical nature of human interactions plays a key role
in shaping the evolution of the epidemic outbreak. As
we shall illustrate in the next section, the insight gained
into the epidemic process through these analyses has
enabled researchers to propose and assess valuable
intervention strategies to control and mitigate the epidemic
spreading, taking into account and leveraging the
dynamical properties of social interactions.
A. First Approaches: Time-Scale Separation
The first class of approaches to deal with dynamic networks
has extensively relied on time-scale separation techniques.
These techniques are based on the assumption
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