IEEE Circuits and Systems Magazine - Q4 2021 - 17

Geometric programming, convex optimization, and model predictive control
are powerful tools for controlling epidemics, even in a distributed fashion.
minimize the expression in Eq. (9), which characterizes
the epidemic threshold. Such an approach can be interpreted
as a resource allocation problem, where limited
amounts of antidote can be distributed to the population.
Such an intuition is formalized in the following optimization
problem.
Problem 2. Given a network and a fixed budget B > 0
imize
min
n
subject to
where :f RRn
"
nn n
K -
i
!!
n
t AM
fB
#
[, ],i 6i V,
()
() ,
i
+ is a cost function associated with the cost
of increasing the recovery rate and n i ( in ) is the minimum
(maximum) admissible recovery rate for node i V!
Even though the objective function is a spectral ra.
dius,
which in general is nonconvex, under reasonable
assumptions on the cost function, tools from geometric
programming and convex optimization can be leveraged
to tackle the problem. Solutions have been proposed in a
centralized fashion [101]-[104], and through distributed
approaches [105]-[109]. Some of these works deal with a
more general problem, in which, besides increasing the
recovery rate, the controller can also reduce the infection
rates im modeling, for instance, the distribution of
personal protective equipment. A resource allocation
problem similar to Problem 2 is studied for an extension
of the SIS model, in which complication phenomena of
the illness are considered [110], [111].
When considering more complex dynamics than the
standard SIS epidemic model, ideas from optimal control
have already been applied a decade ago [112], using
a linear-quadratic regulator, and, more recently, in [113],
leveraging the Pontryagin's Maximum Principle.
More recently, researchers have identified impossibility
results which reveal the possible limitation of
feedback control. In [114], [115], the authors have proved
that, utilizing the recovery rate in as control input, a
large class of distributed controllers cannot guarantee
convergence to the disease-free equilibrium. In [116],
a similar result has been proved for more complex dynamics
that involve two concurrent epidemic processes.
The limitations may become even more profound when
examining the effect of optimal control in real-world disease
management [117].
B. Control of Stochastic Models on Static Networks
Because of the stochastic nature of the models, the related
control results are centered around evaluating the
FOURTH QUARTER 2021
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control performance in terms of bounding as tightly as
possible the epidemic thresholds on different classes
of networks. In [118], [119], the studied control problem
is how to distribute a fixed amount of antidote to nodes
in the given network that may have special topological
features, e.g., scale-free networks. In [118], two different
methods are compared, one based on contact tracing,
which augments the recovery rates of all neighbors of an
infected node, and the other based on degree-centrality,
which augments the recovery rates of all nodes, proportional
to their degrees. Surprisingly, it is found that
contact tracing may only succeed when the number of
infected individuals is small (e.g., in early stages of the
epidemic outbreak), since it requires a total amount of
antidote B i
= R !V in
that grows super-linearly in the
number of contacts; otherwise the degree-centrality
based approach outperforms contact tracing, as stated
in the following result.
Theorem 6. Consider the stochastic network SIS model
then the exin
Eq. (12) on a generic network. If nm$
pected eradication time verifies []E TK ln ,n for some
constant K .02
iik ,i
# l
However, from this result, we note that the needed
amount of antidote B scales linearly with the sum of the
weights of the links in the network, hindering its practical
implementation
even in sparse networks, where
such a sum grows linearly in the number of nodes.
In a similar setup, in [119], [120], another control
method is proposed, in which the antidote is dynamically
allocated to the nodes. The allocation method in
[119] requires that all the antidote B is concentrated at
a single node at each time; using martingale theory, the
following result is established.
Theorem 7. Consider the stochastic network SIS model
in Eq. (12) on a generic network with the control policy
proposed in [119]. Let us define the maximum degree and
the cutwidth of the network as
D== /
i V
!
iij
ij
and
011
respectively. IfBn and BW ,42 then the expected
eradication time verifies []E Tn/B26#
if BK /lnnn2
KK ,
$ 16D ln
. Moreover,
, then []E TK ln ,n for some constants
# l
l 2 0.
Briefly, the proposed control technique guarantees
fast eradication with a sub-linear amount of antidote,
depending on the network topology. In [120], a fundamental
limitation is further established for any dynamic
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IEEE Circuits and Systems Magazine - Q4 2021

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