IEEE Circuits and Systems Magazine - Q4 2021 - 16

Although epidemics control problems are typically straightforward
to formulate, they are often very difficult to be solved analytically.
disappearance of the links (geometric in the discretetime
model, exponential in the continuous-time model),
which is not consistent with the presence of burstiness
and temporal clustering discussed in the above [62],
[63]. In [95], this paradigm has been extended to account
for general inter-event time distributions, establishing a
sufficient condition for (almost sure) exponential stability
of the disease-free equilibrium.
IV. Control of Epidemics on Networks
For engineering researchers who are familiar with the
monitoring, intervention or control of complex systems,
it is of great interest to know what research works have
been carried out to study how to influence, mitigate,
and even stop the epidemic processes, especially those
based on the models we have presented in the previous
sections. Although much fewer control results have
been produced compared to the epidemic modeling activities,
it is beneficial to give an overview of the existing
results, which will serve to inspire researchers, with
or without control theory background, to work in this
critically important research area. In what follows, we
categorize the corresponding results into control of deterministic
and stochastic epidemics respectively, and
to underscore the importance, we devote a separate subsection
for the discussion of the distinct features when
the underlying networks are dynamic. Note that an earp
1
- p
jj
ii 1 - q
q
Figure 9. Transitions of the two-state Markov chain associated
with the existence of a generic link (i, j) of a discretetime
edge-Markovian dynamic graph.
lier survey [25] has summarized some main results on
control epidemics on networks by then, and thus we
give special attention, wherever appropriate, to those
that appeared in the last five years.
A. Control of Deterministic
Epidemics on Static Networks
The simplest idea of controlling epidemics processes
on networks comes from the intuition that removing infected
or high-risk individuals and the links associated
with them will slow down the transmission, which in
practice translates to quarantine and vaccination policies.
Following the discussion in the previous sections,
this intuition implies that one can lower the epidemic
threshold, e.g., by reducing the spectral radius ()At
of the adjacency matrix A. Another intuitive idea is to
optimize the distribution of antidote, which in practice
translates into modifying the entries of matrix M in
Eq. (7). These key control actions are illustrated in Fig. 10.
Naturally, there is always a cost associated with the control
actions and consequently optimal or sub-optimal
control objectives can be formulated. However, to get
a flavor on why such intuitive control problems under
cost constraints are difficult to solve, we look at the following
direct formulation of the control problem.
Problem 1. Given a network of E ,=
fixed number m ,1 . Find those m links R eem
1
r = ER=
links and a
= f {, ,}
of the network after removing which the adjacency matrix
of the resulting network
GV ( ,) has the minimum
spectral radius among all the reduced networks obtained
by removing m links from the original network.
Although this control problem is straightforward to
Node Removal
µi
Antidote
Link Removal
Figure 10. Schematic of the main control actions that can be
taken in the control of static networks.
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IEEE CIRCUITS AND SYSTEMS MAGAZINE
formulate, it is very difficult to be solved analytically. In
fact, it was shown in [96] that Problem 1 is NP-hard. A
similar spectral minimization control problem through
removing nodes is discussed in [97]. In fact, the difficulty
in solving such control problems is rooted in the fact that
the formulated optimal control problems are variations
of the constrained combinatorial optimization problems,
which are in general hard problems. For this reason,
various heuristics have been proposed to solve the
control problems, and a lot of them have smartly taken
advantage of the network structures, e.g., the degree
distribution of the nodes, centrality indices, and connectivity
patterns [98, 99], or solving a nonconvex quadratically
constrained quadratic program [100].
Another intuitive approach lies in tuning the values
of the parameters by increasing the recovery rates in to
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