IEEE Computational Intelligence Magazine - August 2022 - 58

the simulation model to s returns a set of
vectors with compartment values for each
time step ,, .t
1 f
n Therefore, s(S) refers to
the resulting vector of S compartment
values, s(E) to the vector of E compartment
values, etc. Three objective functions
(f1, f2, f3) assess the performance of the
solutions and their values should be minimized
by the optimization algorithms.
The first two are based on [7], and the
third is a new addition.
a) Health Objective
The first objective (f1) corresponds to
the peak of concurrent infections.
fs
1()=+ ++ (12)
ma (( )( )( )( ))
x sE sE sI sI
qq
It can be assumed that minimization
of this function is most important
for preventing a collapse of the
health care system. Using the total
number of infections over the simulated
period does not necessari ly
penalize high infection peaks, therefore
allowing overloads of the health
care systems.
b) Economy Objective
The second objective (f2) represents
the goal of minimizing economic
setbacks.
fs sGDP t
2 =- () () n
(13)
In the reference work [7] this objective
is modelled by measuring the peak
damage to GDP. Since the GDP is
defined to start at 0 in this model, the
objective value cannot become better
than that. Any solutions which keep the
GDP curve at or above 0 for the entire
duration would be equal and optimal
for the algorithm. The approach used in
this paper defines the objective as the
negative value of the GDP at the last
time step (tn), which enables the distinction
between solutions that successfully
prevent an economic setback compared
to the starting value.
c) Intervention Efforts Objective
The third objective (f3) measures the
overall strength of the employed containment
strategies. Let P be the set of
policy strength curves for all policies in
s, with height
$ 0 and # 1. Furthermore,
let pr for each pP! be the average
height of the curve. Then f3 is given
by the following.
fs
3 ()=
/ !
r
pP p
P
(14)
In this study, less regulatory influence is
considered to be desirable for a healthy
society. As this objective is in direct conflict
with the usage of strong policies, its
purpose is to force the optimization
Construction of a Policy Influence Curve
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
050
100
150
Time
FIGURE 2 A possible policy influence curve showing the underlying control points, through
which the curve is defined.
200
250
300
Control Points
Influence Curve
process to find efficient and effective
solutions simultaneously. This objective
is calculated by taking the average normalized
influence of all parameter
adjustment curves for an experiment.
IV. Experiments
Four different multi-objective optimization
algorithms are used in the presented
multi-objective approach to find
optimal control strategies: NSGA - II
[18], MOEA/D [19], as well as GLMO
[20] in two different configurations:
once with NSGA - II and once with
NSGA - III [21]. GLMO is a large-scale
multi-objective optimization algorithm,
which is useful for this study because it
deals with a large-scale problem-a
maximum of 130 decision variables in
the largest experiment.
The presented approach is compared
with the model and findings in [7], as
both studies use multi-objective optimization
to obtain optimal control policies
(details in Section IV-B). Table II
shows the parameter settings for the
control policies.
Each experiment employs a different
combination of policies: all possible containment
strategies consisting of just a
single policy, as well as several strategies
with multiple policies (see Table I). The
five experiments with multiple policies
are as follows: social distancing and lock
down measures (sd.+ld.), NPIs, PIs, the
combination of NPIs and PIs, and the
combination of NPIs, PIs, and measures
that lessen the negative impact of the
pandemic on the economy (abbreviated
by ECO). PIs refer to pharmaceutical
interventions such as treatment or vaccines,
which usually come with a large
time delay when compared to NPIs [1],
[2]. All experiments use identical initial
model parameters (see Table III). The
compartment values (S, Sq, E,...) are
inherently dynamic, as they are continuously
recomputed. The other parameters
are constant, unless the containment
strategy of a given experiment affects
them. In that case, the values are adjusted
based on the used policies. All predefined
parameters are based on rough
estimates of realistic values representing
the SARS-CoV-2 pandemic and follow
58 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2022
Policy Strength
IEEE Computational Intelligence Magazine - August 2022

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