IEEE Circuits and Systems Magazine - Q4 2021 - 18

allocation by showing how the network topology influences
the possibility of eradication the epidemics under
the given budget of the antidote to be allocated. A similar
control policy based on dynamical resource allocation
has been proposed in [121].
Very recently, the powerful tool of model predictive
control (MPC), already used for deterministic epidemic
models [122], has been employed to deal with the control
of stochastic epidemic processes without relying on
mean-field approximation [123]. The use of MPC has
also allowed to deal with more complex epidemic models.
For instance, in [124], the authors deal with a model
with a pre-symptomatic phase, by utilizing MPC with a
robust moment closure technique. Optimal control has
also been considered for stochastic SIS model with different
assumptions on the information available [125].
C. Differences in Control When the
Networks are Dynamic
When the networks are dynamic, the optimization
problem in control may have to face time-varying constraints.
Fortunately, some of the geometric program
techniques still apply, although the complexity in seeking
the solution increases [126]. Optimal control theory
can be applied to time-varying systems; however, it is
well known that the corresponding stability conditions
might be more conservative and more difficult to check
[127]. In [75], the authors deal with a resource allocation
problem similar to Problem 2, on temporal-switching
networks, under the assumption that the switching is
determined by a Markov process. Therein, the problem
is solved via geometric programming, finding a solution
with a computational complexity that grows super-linearly-but
polynomially-in the network size.
For some specific cases, control strategies for eradicating
the outbreak have been successfully proposed.
For instance, in [128], a distributed control scheme has
been designed for a deterministic SIS model on periodic
time-varying networks. In this scheme, which recalls the
antidote distribution proposed in [118] and summarized
in Theorem 6, each node dynamically sets its recovery
rate proportional to the sum of the weight of the links to
its neighbors at that time, that is,
nm=
ii ij
j V
()
ta ().t
!
/
(25)
Under this scheme, global asymptotic stability of the
disease-free equilibrium is guaranteed, under the assumption
that all the instances of the time-varying network
are strongly connected [128].
An interesting research line is to incorporate the human
behavioral mechanisms that lead to variations of the
dynamic network structures [65]. Understanding human
18
IEEE CIRCUITS AND SYSTEMS MAGAZINE
behavior will be critical to draft and implement vaccination
policies [129], which affects the dynamics of the
networks and at the same time is deeply affected by the
dynamics of the networks [129]. Further study may dive
in how isolation of infected individuals may adaptively
change and reshape the network dynamics and how this
can be leveraged to devise effective intervention policies
[130], [131]. In particular, in [130], the authors explicitly
compute the following epidemic threshold for a network
SIS model on activity-driven networks, depending on the
possibility of isolating infectious individuals by decreasing
their social activity to a factor
p [, ],01!
where p = 1
means that no interventions are enacted and p = 0 models
complete isolation of infected individuals:
v =
mp ap ap a11 4
2
(( )( ))++ -+
GH
22 2
GH GH
.
(26)
By comparing Eq. (26) with Eq. (22), one can assess
the effect of isolation policies on increasing the epidemic
threshold.
More recently, awareness-based control strategies,
which were developed a decade ago [132], have been
extended to study temporal networks such as activitydriven
networks [133]-[135]. The proposed formalism
allows to study scenarios in which aware individuals
reduce the risk during their physical interactions [134]
or they dynamically rewire themselves to avoid interactions
with infected individuals [133], possibly in combination
with other control policies, such as isolation of
infected individuals [133], [135]. In the model proposed
by Yang et al. in [134], awareness is modeled as a process
that co-evolves with the epidemic spreading on a twolayer
network. The epidemic process spreads on a physical
contact layer, while awareness spreads on a virtual
communication layer. Aware individuals reduce their infection
probability, as a consequence of the adoption of
self-protective practices. The epidemic threshold is then
computed as a function of the individuals' activities on
the two layers and the coupling between them. In the activity-driven
adaptive-SIS model proposed in [133], two
modifications are made to the ADN algorithm illustrated
in Section III-C. First, in step ii), the activity of an infected
individual i, is multiplied by a parameter
lar to [130]. Second, in step iii), a link (, )iih
p [, ],01i
!
ri [, ].01!
simito
a node ih
that is infected is added to the edge set with probability
The
epidemic threshold is computed, in
terms of the decay ratio to the disease-free equilibrium,
finding a rather complicated expression that depends
on the joint distributions of the activities and the parameters
pi and πi. Such an analytic expression is used
to devise an optimization problem, to optimally allocate
resources into isolation of infected individuals and
awareness, solved via geometric programming.
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