Doing more with Models: Illustration of a SD Approach
for exploring deeply uncertain issues, analyzing models, and designing adaptive robust policies
Erik Pruyt? Jan H. Kwakkel & Caner Hamarat
Delf University of Technology
July 2013
Abstract
Many grand challenges are both dynamically complex and deeply uncertain. Combining
System Dynamics with Exploratory Modeling and Analysis allows one to generate, explore,
identify and analyze all sorts of plausible scenarios related to such issues, and design and
test adaptive policies over many scenarios, This paper explains and illustrates different uses
of the resulting computational System Dynamics approach by means of an applied case, the
outbreak of a new flu strand like the 2009 A(HIN1)n flu. First, we illustrate the use of this
approach for generating and exploring different types of plausible pandemic shocks. Second,
we illustrate the use of machine learning techniques to analyze contributions and effects of
uncertainties, and d
ver and select scenarios. Finally, we illustrate the use of this approach
for supporting the design of robust adaptive policies in order to be prepared for any new flu
outbreak, especially those that really require action.
Introduction
In terms of applications, our research team addre:
are characterized by high degre
illustrate how developments in sc involved in model-based decision support can be combined
with System Dynamics (SD) modeling and simulation (Forrester 1961; Sterman 2000). Combining
useful for addressing the combined challenge of dynamic complexity and deep uncertainty
through generation, exploration and analysis of many plausible scenarios! and through robust
optimization of adaptive policies.
The remainder of this paper is structured as follows. First we define deep uncertainty and
introduce Exploratory Modeling and Analysis for dealing with deep uncertainty, as well as its
combination with SD modelin,
Then we use
ss grand challenges and important i
of dynamic complexity and deep uncertainty. In this paper we
on
for dealing with deeply uncertain dynamically complex issues.
a single case to illustrate multiple uses of this approach, more specifically (i) open
exploration, (ii) advanced analysis using machine learning algorithms, (iii) open s
and selection, (iv) directed scenario discovery and selection, (v) adaptive policy design, robust
optimization and regret analysis, and (vi) model testing (verification and validation). The case
used to illustrate this computational SD approach is the ‘A(HIN1)n’ case -the 2009-2010 flu
pandemic. This case was chosen mainly for explanatory reasons: the model is relatively simple (a
small SD101 simulation model2), the s easily understandable (everyone is familiar with the
2009-2010 pandemic)
enario discovery
and is used in the tutorial on our web site? which explains how to use our
“Corresponding author: Erik Pruyt, Delft University of Technology, Faculty of Technology, Policy and Manage-
ment, Policy Analysis Section; P.O. Box 5015, 2600 GA Delft, The Netherlands ~ E-mail: e.pruyt@tudelft.nl
‘In this paper we use the word ‘scenario’ for the time evolutionary behavior of a simulation run or computational
experiment which is a combination of specific instantiations of uncertaintie
?'This case is available in the ‘small System Dynamics Models for BIG Issues’ case book (Pruyt 2013) available
for free at http: //simulation. tbm. tudelft -n1.
3The tutorial is available at http: //simulation.tbm. tudelft .nl/ema-workbench/tutorial .html
and an older_~—version of ~—s the = EMA — workbench is available for_— free — at
http: //simulation.tbm. tudelft .nl1/ema-workbench/download.html.
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Mod
EMA Workbench software. Finally we draw some lessons and conclusions for this computational
SD approach and, more generally, the SD field.
Dealing with Deeply Uncertain Dynamics
complexity nor to SD.
The audience of this paper does not require an introduction to dynami
However, an introduction to uncertainty, deep uncertainty in particular, and Exploratory Modeling
and Analysis may be useful.
Deep Uncertainty
In general, uncertainty could be defined as limited knowledge about future, past, or current
events. A variety of conceptual schemes, definitions, and typologies of uncertainty have been put
forward in different scientific fields (Morgan and Henrion 1990; Hoffman and Hammonds 1994;
van Asselt 2000; Walker et al. 2003; Kwakkel ct al. 2010b). Three such taxonomies were used by
Pruyt (2007) to assess how SD deals with different types of uncertainties. Interestingly, System
Dynamicists have assumed for decades that uncertainty is omnipresent and matters to such an ex-
tent that models are referred to as ‘plausible’ models, and SD model results are mostly interpreted
in terms of general modes of behavior, not specific point or trajectory predictions or probabilistic
outcomes. This stance fits well with Level 4 or deep uncertainty as defined in Table 1 adapted
from (Kwakkel et al. 2010b; Kwakkel and Pruyt 2013).
Approaches for dealing with the level
Level of Description
Uncertainty
Level 1: Recognizing that one is not absolutely certain Performing sensitivity analyses on model
parameters by changing default values with
some small f
Being able to enumerate multiple possible
marginal but that uncertainty is a marginal issue.
uncertainty
Level 2: Recognizing that uncertain is more than marginal,
futures or generate alternative model outcomes,
shallow and being able to enumerate multiple alternative
uncertainty and provide probabilities (subjective or objective) and to specify their probability of occurrence
Level 3: Being able ta enumerate rvulilplaspossielliiegiand Being able to enumerate multiple possible
uncertainty
rank p ities in terms of perceived likelihood.
However, haw wauch wore likely or unlikely one alter-
native is compared to another cannot be specified
futures or alternative model structures, and
Belg able Cy judge en la tarmas‘o! parcalvel
likelihood, not in terms of probabilities
Being able to enumerate multiple possible
Level & Being able to enumerate/generate multiple
deep possibilities without being able to r: futures or specify multiple alternative
uncertainty |] order the possibilities in terms of how model structures and generate alternative
likely or plausible the with ifying likeli!
Level 5 Being unable to enumerate multiple possibilities, Fully accepting the possibility of being wrong
recognized
e
because one does not or cannot know the generative
at play nor the possibilities that may
mechanisms
or being surprised because existing mental
and formal models are known to be inadequate
ignoray
be generated
Table 1: Five levels of uncertainty adapted from (Kwakkel and Pruyt 2012b)
Contrasting deep uncertainty to other levels of uncertainty, it could thus be defined as pertain-
ing to those situations in which one could generate or enumerate multiple ~several to even millions
of- possibilities without being able or willing to rank order the possibilities in terms of how likely
or plausible they are judged to be (Kwakkel et al. 2010b). Deep uncertainty could also be defined
as pertaining to those situations in which it is not unambiguously clear which of many plausible
ing mechanisms will generate the real-world dynamics, for which it is uncertain which
probabilities may be attached to plausible real-world outcomes, and for which different experts
and/or policymakers may disagree about the acceptability of the outcomes (Lempert ct al. 2003).
In other words, models could be used to generate many plausible scenarios if a plausible model
-preferably different! plausible models could be specified. Deep uncertainty is of particular inter-
underh
in system boundaries, conceptual models, model stru
4Models are considered different if there are differenc
tures and functions, internal and external parameter values and other input data, and model implementations (
different simulation methods). The word ‘model’ thus refers to a particular set of assumptions: two different se
of assumptions are therefore two different model
3S
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models. 3
rs dealing with grand challenges and other
A ; mplex uncertain i
nce most of them are indeed characterized by deep uncertainty and/or
recognized ignorance.
Exploratory Modeling and Analysis
The definition and explanation of deep uncertainty already suggest that SD models could also
be used differently for dealing with deeply uncertain issues — still respecting the SD stance that
uncertainty is omnipresent and truly matters, and exact predictions cannot be made: plausible SD
models could be used to generate many plausible scenarios which corresponds to a development
in model-based decision support, namely the emergence of a different way of developing and using
models (Bankes 2009), which we refer to as Exploratory Modeling and Analy
EMA can be useful when relevant information exists that can be exploited by building models,
but where this information is insufficient for specifying a single model, i.c. a single set of assump-
tions. In many such circumstances, multiple models could be constructed that would be consistent
with the available information. A single model run drawn from a model or a set of plausible models
is then merely a computational experiment that reveals how the system would behave if the vari-
s this particular model makes about the various unresolvable uncertainties were correct.
i computational experiments allows one to explore the implications of
s combinations of assumptions.
EMA thus refers to the explicit representation of a set of plausible models, the process of
exploiting the information contained in such a set through a large number of computational ex-
periments or very specific direct searches, the analysis of the results of these experiments, and the
use of the set of robust policy design (Bankes 1993; Agusdinata 2008).
Important steps in EMA are to (i) conceptualize the decision problem and the ass
(ii) develop an ensemble of fast and easily manageable models of the system of intere
(iii) specify the uncertainties that are to be explored. Next, depending on the purpose for whi
EMA is applied, various subsequent steps are possible. Depending on the particular applic
or use of EMA, different subsequent steps are possible. In case of an open exploration, aimed
at identifying the diversity of dynamics implied by the models and the associated uncertainties,
the next steps are (iv) to generate a series of computational experiments, (v) execute these ex-
periments, and through various visualization techniques (vi) develop insight into the types of
possible dynamics. In case of a more advanced analysis, the steps of open exploration would be
followed up by (vii) defining types of dynamics or of outcomes that are for some reason
of interest, and (vil) reveal the for the occurrence
aus
’ of dynamierecenario
discovery (Kwakkel et al. 3013) the typical 1 subsequent steps are to (iv) analyze the behavioral
landscape resulting from (iii) through time series clustering; (v) identify the combinations of un-
ainties from which regions of interest in the behavioral landscape originate; (vi) assess these
combinations of uncertainties using various model quality metrics and related machine learning
techniques for assessing model quality (Bryant and Lempert 2009); (vii) qualitatively or quantita-
tively communicate the typical scenarios in these regions of interest, ic. exemplary scenarios, and
the combinations of uncertainties from which the regions of interest in the behavioral landscape
originate to the actors involved in the decision making problem.
Quite a different series of subsequent steps is used in case of directed search, aimed at answering
targeted questions such as what is the worst that could happen? In this case, the next steps are
(iv) define an objective function that lates the targeted question; (v) perform non-linear
optimization; (vi) translate the results from the optimization into an answer to the targeted
question.
Both open exploration and directed se: can be combined, for example for the devel-
opment of robust adaptive plans or policies. Here, an iterative pr is often used based on
first identifying the causes for undesirable behavior, translating the resulting insight into possible
solutions, and testing the solutions for their efficacy. Directed search techniques can be used to
fine-tune actions, assess which combinations of actions are the most efficacious, or help in speci-
fying weak signals that can be monitored for triggering actions only when and if they are needed
ic
rch uses
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models. 4
as in (Hamarat et al. 2013).
The practice of EMA is still being developed. However, it has been applied to a variety
of decision making problems. It has been applied to climate change problems in an effort to
identify policy options that are on the one hand acceptable to a wide variety of countries, de-
pending on their state of development and their belief about climate change, and on the other
hand robust across a wide variety of different plausible future climate change developments
(Lempert et al. 2003). Other EMA applications are found in the field of energy generation:
Agusdinata (2008) studied how CO, emissions could be reduced in the Dutch houschold sector and
(Kwakkel and Yucel 2012) in the electricity sector. A third area in which EMA has been applied is
transport planning. Van Der Pas et al. (2010) report a case study related to intelligent speed lim-
iters. EMA has also been applied to the field of airport strategic planning (Kwakkel ct al. 2012).
Sea and Lempert (2007) report on the use of EMA for addressing water resources management
uues in California. Other case studies of the application of EMA in other fields are reported
on in (Bankes and Margoliash 1993; Bankes 1994; Lempert ct al. 1996; Park and Lempert 1998;
Brooks ct al. 1999; Bryant and Lempert 2009; Kwakkel et al. 2013). An overview of the field of
EMA can be found in (Lempert 2002; Bankes et al. 2002; Bankes 2009) and most recently in
(Bankes et al. 2013).
Exploratory System Dynamics Modeling and Analysis
Since EMA is appropriate for systematically exploring and analyzing deep uncertainty and testing
the robustness of policies but requires models, SD is appropriate for generating plausible dynamics
but requires techniques to handle deep uncertainty, and EMA and SD are philosophically similar,
it follows that their combination —wh all Exploratory System Dynamics Modeling and
Analysis (ESDMA) i ematically generating, exploring, and analyzing
many different plav and for testing the robustness of policies over all sorts of
plausible dynamics. Note that ESDMA, in spite of the different label, is just another SD strand.
SD models used for ESDMA are easily-manageable models and consequently rather small
(Ghaffarzadegan et al. 2011; Pruyt 2010), are slightly more exogenous than traditional SD models
(but still largely endogenous), and contain additional SD structures for injecting different types of
uncertainties like the ones discussed by Pruyt et al. (2011). Uncertainties we are currently able
to deal with include: uncertainties related to initial values and parameters, functions, lookups,
generative structures, model formulations, model boundaries, different models, different modeling
methods and paradigms, different preferences and perspectives related to different world-views,
and different policies with uncertain imp
Many real world ESDMA studies were recently performed, such as (Pruyt and Coumou 2012)
and (Logtens et al. 2012; Auping ct al. 2012) in health and societal aging, (Auping et al. 2012) in
resource scarcity, (Kwakkel and Slinger 2012; Kwakkel and Timmermans 2012) in water security,
et cetera.
We currently build these exploratory SD models in Vensim DSS (Ventana Systems Inc. 2010)
and use a shell written in Python (Van Rossum 1995) to generate computational experiments that
cover the space spanned by the specified uncertainties. Through our Python shell we force Vensim
DSS to execute experiments (i.c. combinations of uncertainties and models) to generate transient
simulation runs ios). This shell is also responsible for storing the data when generated so
that the ensemble and individual runs can be explored, searched, compared, used for debugging
and other purposes. We then use a library of machine learning algorithms coded in Python, C,
and C++ integrated in our so-called EMA Workbench to analyze the ensemble of scenarios, and
visualize the most interesting findings®. We also use various techniques and algorithms, some
5Reasons for this particular choice of tooling, are (i) the ease with which different types of uncertainties can
be handled and ensembles can be generated, (ii) the ease with which existing algorithms (in Python libraries) can
be adapted and used, (iii) the ease with which new algorithms can be developed, tested, used, and compared to
existing algorithms, (iv) the ease with which large data sets can be handled and explored, and (v) the possibility of
sampling and using algorithms across multiple models, even multiple modeling methods, and policies. Note however
that this computational SD approach could also be performed without our EMA Workbench by using commercial
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models.
of which have been used already in SD. Many other are new to the field. Some of thes
will be introduced and illustrated below while illustrating ESDMA using one
the same is done in (Pruyt and Kwakkel 2012a;
and not just SD models.
and technique
but for different purposes.
Kwakkel and Pruyt 2012), but then with different
¢
Illustration of multiple uses of ESDMA
The 2009-2010 A(H1N1)v Pandemic
In the
More or
ible dynami
and conseque
as Swine flu or Mexi
tools
days, weeks, and months after the first reports about the outbreak of a new flu variant
in Mexico and the USA, much remained unknown about the po’ ;
of this possible epidemic/pandemic of the new flu variant, first known
an
flu and known today as new influenza A(H1N1)v. Table 2 shows that, more information became
available over time, but still many uncertainti
it was poss
remained. However, even with these uncertainties
ble to model this flu variant, since it was flu, and flu outbreaks can be modeled.
Date 24 April 30 April 08 May 20 May 12 June 20 July 21 August
Tnfectivity unknown unknown unknown unknown unknown unknown unknown
Ro unknown unknown prob. 1-2; pr [R up to 2]
1.4-1.9 14-16
Immunity unknown unknown indications idem idem idem idem
(elderly)
Virulence unknown unknown unknown unknown unknown mild and idem
self-limiting
Incubation unknown unknown long tail? median 3-4d idem idem
(up to 8d) e L-Td
17%? 4%? 2%? 0.4-1.8%?
unknown unknown 0.1%? 0.1%? 2%? 0.4%?
unknown unknown unknown unknown unknown 0.3%(1%)? 0.1-0.2%?
(0.35% - ex.)
unknown unknown —_ older people skewed tow idem idem
less affected? younger
unknown possible indications
unknown unknown unknown unknown unknown indications 33-50% (ass.)
unknown unknown, unknown unknown unknown unknown unknown
Source: (ECDC 20098) __(ECDG 2009F) _ (BCDC 2009) _(BCDE 20096) __(ECDG 2009a) __(ECDE 20090) __ (ECDC 20096)
Table 2: Information and unknowns provided by the ECDC from 24 April until August 21. CFR
stands for Case Fatality Ratio
Many nations were at first particularly concerned about the potential loss of human life.
moderately low~ about the potentially
at large in case large frac-
simultaneously by the flu. Hei
later —after it became clear that the
tions of the (active) pc
fatality ratio w
disruptive effects both on health care systems and socicties/economies
bili
would be i
and
, we will,
in what follows, mainly focus on the deceased population and the highest peak of the fraction of
the population that is infected at a given point in time.
Use 1: Open Generation and Exploration
Open generation and exploration can be us
deep uncertainty. It relies
Monte Carlo sampling, Latin Hypercube sampling, or factorial method
be used to answer questions suc
cumstances would this polic
1?” An open explor:
el
behaviors possible and to create insight into plausible dynamic
to imagine many p and think the unthinkable.
on provides
ssible futures
on the careful di
pos
SD software together with advanced mathematical programs.
as ‘What kinds of dynamic
ibly do well? Under what c
insight into the full rid
mble of models. Hence, ESDMA is used in open exploration to generate the full ric
that could occur. As such it helps
'd to systematically
ign of experiment:
an thi
explore plausible models under
and can use techniques s
An open exploration can
tem exhibit? Under \ what
ich as
ors of a model or an
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models. 6
~ acnon to raceme deseo
scchation deeiion aeion delay = _rarcnaned sen a
total population
— "region
ae
veasent pon cae a 4
aidan rin pen ee vs tend ant § 4
einen pope / \ \
nee on i [ot comaet \ \
cto / sae) // yore \
/ - ‘ \
- i WY wpiduad sxc | |
sosept to me / option on outort
sitcer |, “i, aes e)/ |
— i; ‘moruul coutset / |
~
a sal —. EF =a a=
pemelaney ay
‘
the opidaion ofreajon
~~. 9
Til value temine
a Populaionrepon 2
popubtion
~~ ital vabe suscepti
popubtionregion 2
inpact nfcied popshtion
a contact rte reso 2
¥
poral nme <
popula region?
Figure 1: Stock-flow diagram of the ESD two-region flu model with the core structure for region
1 in bold (basic variables are displayed in black connected by dark blue arrows, initial values
and parameters in orange, switches to turn structures of/off in light blue and yellow, and policy
structures in green)
In the case of A(HIN1)v, or H1N1/09 as it is referred to today, a SD simulation model was
developed shortly after the first signs of a potential outbreak were reported in order to foster
understanding about the plausible dynamics of the flu outbreak (Pruyt and Hamarat 2010). The
model developed at the time, displayed in Figure 1, was small, simple, high-level, data-poor (no
complex/special structures nor detailed data beyond crude guestimates), and history-poor given
the information in Table 2. The model was used in an ex-ante exploratory way: developments were
not waited for and uncertainties were amplified and explored instead of reduced or ignored. In the
model, the world is divided into two regions: the Western World, and the densely populated Devel-
oping World. For a more elaborate description of the model, see (Pruyt and Hamarat 2010). Table
3 lists the uncertainties used in combination with this flu model. These uncertainties were loosely
based on the various unknowns and guestimates as reported by the European Center for Disease
Control over the period of early April 2009 up to late August 2009 (Pruyt and Hamarat 2010).
The ranges were set somewhat wider, given our explorative purpose and
trophic
ecial interest in catas-
ble: the core of the
sonal flu variant—
. At first, the model was purposefully kept as simple as pos
model is a SIR model —not a SEIRS model as would be appropriate for a s
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Mode
because of the focus on the first (pandemic) wave, and the very short incubation time (1-2 days).
Later, a more refined model was developed to test whether the conclusions obtained with the
and dirty’ model would hold when analyzed with a more refined model®,
‘quic
(Parameter Lower Limit_| Upper Limit
additional seasonal immune population fraction region T 0.0 05
additional seasonal immune population fraction region 2 0.0 0.5
fatality ratio region 1 0.0001 O41
fatality ratio region 2 0.0001 O41
initial immune fraction of the population of region 1 0.0 05
initial immune fraction of the population of region 2 0.0 0.5
normal interregional contact rate 0.0 0.9
permanent immune population fraction region 1 0.0 05
0.0 0.5
0.2 08
0.2 0.8
1.0 10.0
1.0 10.0
infection ratio region 1 0.0 0.15
infection ratio region 2 0.0 0.15
normal contact rate region 1 10 100
normal contact rate region 2 10 200
Table 3: Parameter ranges for the LHS with parameterised fatality ratios (0.01% — 10%) and
ranges for the infection ratios (0% ~ 10%)
reduced —more ‘credible*
The combination of this model, these uncertainties and the Latin Hypercube sampling plan
used, generates an ensemble of thousands of flu scenarios of which the envelope and the 37 worst
scenarios are displayed in Figure 2a. These 37 scenarios result from selecting the 20 worst
in terms of deceased population in region 1 (the Western world) and the 20 worst scenarios both
in terms of infected fraction in region 1 out of the ensemble of 20000 flu scenarios with 3
being among the 20 worst in terms of deceased population and peak infected fraction in region 1.
These graphs should resonate with System Dynamicists since they show the behavior over time.
However, many policymakers are not familiar with behavior over time graphs and may prefer
different visualizations. The 20000 runs could for example also be represented in a 3D tter
plot as in Figure 2(b). Since different: types of visualizations are useful for different. purposes and
convey different insights, we use many different types of visualization, such as lines, envelopes,
multiplots, heat maps, 3D graphs, interactive graphs, etc
At the time, we learned from this open exploration that this flu variant could turn into any-
thing from a small flu episode to a catastrophic pandemic, but also that the most catastrophic fu
outbreaks would either take place overnight or within approximately one year. Hence, adequate
adaptive social distancing measures were needed for dealing with pandemics that would happen
before vaccines could be rolled out and vaccination development had to be started up without
delay for high priority groups for dealing with pandemics that would happen within the year, but
not necessarily for the whole population since the information available at the first signs of the flu
variant were too uncertain to be used to justify 100% coverage: Delaying the vaccine stock order
decision for the rest of the population in order to gain information for better decision making
would have been a good idea at the time if collectively agreed upon by a large coalition of nations.
enarios
‘enarios
Use 2: Advanced Analyses Using Machine Learning Algorithms
To get an idea of the relative contribution of separate uncertainties with regard to diversity of
we use Random Forest (Breiman 2001) and Feature Selection (Kohavi and John 1997)
algorithms. Since uncertainties do not need to be continuous parameters, it is also possible to use
these techniques to explore the relative contribution of structures, loops, policies and models. Table
4 shows the ranking of un s from highest contribution to the number of s for 20000
flu scenarios according to the random forest attribute selection and the feature selection algorithms.
individually, not for combinations thereof.
rtainti ualti
These relative contributions are for these uncertainties
®Both the simple core model and a more extended model can be accessed via (Pruyt 2013)
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models.
deceased population region |
67
‘deceased population region 1
‘Time (months)
infected fraction RI
2.2
Infocte fraction RI
ET
0
Time (months)
(a) Envelopes for the deceased population and the infected fraction in region 1 of 20000 scenarios with lines
for the 37 worst scenarios in terms of deceased population and/or infected fraction
Y-axis
Cumulative number
of fatalities
590,0000.000
02
Laxis
Infected fraction
X-axis
Moment of peak wave (time in months)
(b) 3D scatter plot with projections of the LHS 20000. X-axis: 0-48 months; Y-axis: 0-50% infected
fraction; Z-axis: 0-50.000.000 fatal cases
Figure 2: Envelopes and lines versus scatter plot of the flu scenarios
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Mod
This information may for example be used to remove uncertainty ranges (all those with zero and
negative values for the random forest attribute selection or extremely low feature selection scores)
from generation with computationally expensive techniques (factorial methods) or further analysis.
Random forest attribute selection RF scores | Feature selection FS scores
infection rate region 1 normal contact rate region T 0.0463
aormal contact te seksi fatality ratio region 1 0.0157
recovery time region 1 recovery time region 2 0.0156
fatality ratio region 1 root contact ratio region 2 0.0138
permanent immune pop. fraction R1 recovery time region 1 0.0130
root contact rate region normal contact rate region 2 0.0069
add. seasonal immune pop. fraction R1 fatality rate region 0.0031
recovery time region ini. immune fraction of the pop. of R2 0.0031
ini. immune fraction of the pop. of R2 ate region 0.0029
infection rate region aasonal’ immune pop fraction R1 0.0024
add. seasonal immune pop. fraction R2 ction rate r 0.0017
normal contect rate region J permanent immune population fraction R1 0.0015,
policy normal interregional contact rate 0.0010
model 0 | add. seasonal immune pop. fraction R2 0.0005
permanent immune population fraction R2 -0.001 | infection rate 12 0.0004
normal interregional contact rate “0.002 | ini. immune fraction of the pop. of R1 0.0001
fatality rate regi -0.002 | permanent immune population fraction R2 00002
ini. immune fraction of the pop. of R1 -0.009 | policy 0
root contact ratio region -0.020 | model 0
Table 4: Random forest attribute selection and feature selection on 20000 flu scenario:
and model have zero scores because alternatives were not included in the analys
If the goal is to create insight into the combinations of assumptions that produce particular
kinds of dynamics or outcomes, then methods and algorithms could be used like the Patient Rule
Induction Method (PRIM) (Friedman and Fisher 1999; Lempert et al. 2006; Groves and Lempert 2007).
PRIM is useful if one
of output variables are considerably different from the average value or a classifier threshold over
the entire input domain. In the context of this paper, the input space is the uncertainty spa
PRIM then generates box-like subspaces (with the fraction of positive matches and the mass of
the box relative to the total scenario space) that perform below/above a particular threshold or
are characterized by particular features (e.g. acute crisis behavior). PRIM could thus be used to
find subspaces in the global uncertainty space that result in highly desirable or undesirable out-
comes or dynamics which makes PRIM particularly useful for discovering uncertainty subs
with catastrophic consequences or behaviors, and identifying the corresponding root causes which
allows one to develop adaptive policies consisting of specific adaptive actions for dealing with
different scts of plausible futures (see below and (Hamarat et al. 2012).
ks a set of subspaces of the input variable space within which the values
box T Test box
PRIM box bouding uncertainties min: max:
Tiormal contact rate region T 59.490 99.986
infection ratio regio I 0.060 0.150
time regio 0.282 0.750
Salditlenal aeasonal misile pop AaHeH RA 0.023 0.470
fatality ratle regia 2 0.010 0.100
nf ate region 2 0.014 0.150
yoot contact fate regio 0.012 4.693
permanent immune population fraction Rt 0.000 0.478
ceptible to immune pop delay time region 1 0.561 1.999
Table 5: Prim box ranges for more than 1.5 million flu fatalities
Figure 3 shows the uncertainty space box obtained with PRIM for cases that result in more
than 1.5 million deaths. This box consists of the combination of ranges of these particular unce
tainties displayed in the second column of Table 5 relative to the full ranges in the last column.
This PRIM box, that is box 1, covers more than 37% of all cases with more than 1.5 million
deaths, and more than 97% of the runs within this box lead to more than 1.5 million deaths. The
most determinant uncertainties in this box are related to uncertainties that determine the infec-
tivity and its speed, not ~surprisingly— the fatality ratio which may intuitively look like a more
important determinant for the number of deaths on which basis the runs were classified. A more
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models. 10
uncertatory bandwidth
i i: i : i :é
3 H . Z E
4 zg 2
Figure 3: Prim box plot for more than 1.5 million flu deaths in region 1
sophisticated analysis, based on preprocessing the data using Principal Components Analysis as
in (Kwakkel et al. 2013), can improve the coverage to 90%.
If the aim is to provide insight into the types of dynamics that could possibly occur, then results
from the series of computational experiments were clustered based on the type of dynamics. This
requires a form of time series clustering. The goal of clustering in general is to organize an
unlabeled data set into homogenous groups where the similarity within the group is minimized
and the dissimilarity between groups is maximized (Theodoridis 2003; Liao 2005). Time series
clustering approaches try to modify existing clustering approaches for static data so that they
can cope with time series data. Either the algorithm is modified to deal with the raw time series
data, or the time series are processed in such a way that static clustering methods can be used
directly (Keogh and Kasetty 2003). A relatively recent review of the state of the art in time-
series clustering can be found in (Liao 2005).
» currently use an agglomerative hierarchical
tering approach. That is, we start by positioning each time series in its own cluster, and
then hierarchically merge e ster into larger and larger clusters (Liao 2005). Similarity of
dynamica is determined based on an extension of the behavior pattern features discussed by
Yucel and Barlas (2011) and extended further by Yucel (2012). An example of the use of this
clusterer is provided in the context of open scenario discovery and selection in the next subsection.
clu
cl
Use 3: Open Scenario Discovery and Selection
Open Scenario Discovery and Selection refers to exploring and analyzing an ensemble generated
by means of open generation to identify and select scenarios of particular interest or exemplars.
Scenarios
‘ould for example be of particular interest because of their own dynamics, outcomes,
and/or origin, or because of their representativeness for dynamics, outcomes, and/or origin of
a subset of the ensemble. Various techniques could be used for open scenario discovery and
ection. Multi-dimensional classification could be used to discover and select (representative)
scenarios on multiple outcome indicators. PRIM could be used to discover and s
that are representative of scenarios that share distinctive features (e.g. very undesirable outcomes)
cenarios
and which are highly concentrated in terms of origin in the multidimensional uncertainty space
(Bryant and Lempert 2009; Kwakkel et al. 2013). And time seri
used to identify and select
astering techniques could be
cenarios for their (representative) behavior.
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models. 11
EMAScenario Scenario Discover Single-hazard Integration of CAs
af Automated
Generation and Set Selection Capability Analyses yer all represent.
All-Hazard CA,
ESD Modeling
scenarios of all risks
x uncertainties tONRAscores —settoCAuDU CA policies
Flu
Radicalization
Design & testing of potential
all-hazard capability sets
Optimization of the robustness
of the capability set over all risks
Figure 4: Integrated All-hazard Risk-Capability under deep uncertainty
Open scenario discovery and selection based on the dynamics of multidimensional effe
example been used on an adapted version of the flu model (with rather catastrophic setting:
context of an Integrated Risk Capability Analysis (IRCA) for the Netherlands (Pruyt et al. 2012).
SD simulation models are used in the model-based IRCA described there and displayed in figure
0 ds of plausible scenarios for each of many different risks. Next,
a subset of 100 scenarios that is representative in terms of dynamics, multi-dimensional effect
and origin in the multi-dimensional uncertainty space is identified and selected for subsequent use
pability analysis (CA) model to test the effect of different capability policies under deep
uncertainty for all sorts of risks. The aforementioned clusterer was applied to the total National
Risk Assessment (NRA) scores (labeled ‘total score’ in Figure 5a) of 10000 flu scenarios in view of
open scenario discovery and selection. Figure 5a shows a lines plot of the evolutions of the infected
fraction and Figure 5b the total National Risk Analysis impact for 10000 plausible outbreaks of a
s in the Netherlands. Total NRA impact scores above 0.33 are considered catastrophic,
scores between 0.11 and 0.33 are considered very serious. Note that almost all new
atastrophic happen very fast, that most flu in this ensemble are
ious and happen slower or build up over time, and that a si :
classified as less than very serious. Using the clusterer on the total NRA score, 16 different types
of behaviors were found (see Figure 5(d)). Two exemplars from cach of these 16 different time-
s were selected and supplemented with 68 hand-picked exemplars, especially from the
largest clusters (proportional to the size of the clusters), resulting in the subset of 100 scenarios
displayed in Figure 5(c) selected from the larger ensemble of 1000 runs. Figure 5(e) displays a
‘tisk envelope diagram’, which could be used to plot deeply uncertain risks. This risk envelope
diagram shows that the small ensemble (blue line) is indeed representative in terms of the total
NRA impac mble (red line) and could therefore be used to represent the
larger ensemble in the ensuing capability analysis under deep uncertainty.
4 to generate
ina
new flu viru
enari
aller subset of flu scenarios
series clus
scores of the entire e1
Use 4: Directed Scenario Discovery and Selection
Particular questions can be answered through directed searches. Directed search, in contrast to
less refined open exploration, is a search strategy for finding particular cases that are of interest.
Directed search can be used to answer questions such as: What is the worst that could happen?
What is the best that could happen? How big is the difference in performance between rival policies?
7A risk envelo| iagram is a risk diagram in which the cumulative relative number of runs in each of the total
impact classes ing with the highest impact class are plotted. In other words, 20% of the 1000 risk scenarios
have a catastrophic NRA impact, about 83% of these 1000 runs have at least a very serious impact, and 98% have
at least a limited impact.
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models.
infected fraction rl
10 20 30 40
(a) Infected fraction of the Dutch population for 1000 plausible flu scenarios
total score
(b) Total NRA scores of the 1000 plausible flu scenarios in the framework of the Dutch National
Risk Assessment (NRA)
infected fraction rl
30 a
(c) Small representative ensemble of 100 flu scenarios discovered and selected with time series clus-
tering
nig tee
(a) Time series clusterer dendrogram (flu) (ec) Risk envelopes diagram (flu)
ion for the Dutch IRCA
enario discovery and sele
Figure 5: Open flu s
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models. 13
Figure 6: Two almost identical extreme flu cases obtained with two directed searches with different
objective functions
A directed search provides detailed insights into the dynamics of specific regions of the full uncer-
not the entire uncertainty space. Directed search relies on the use of optimization
techniques, such as genetic algorithms and conjugant gradient methods. Active non-linear testing
is an example of a directed search strategy (Miller 1998). A suitable optimization algorithm for
directed search in the context of SD should be able to cope with the non-linearity of the model, a
' in the search space, a seai ace that is rife with
local optima, and noise (Miller 1998). In the context of ESDMA, two additional complications are
added, namely a potentially very large search space, and a discontinuous search space arising out
of the ir tions in e.g. structural equations. On top of this, a suitable optimization
algorithm should be economical. That is, it should be able to find the optimum relatively rapidly,
without requiring a very large number of computational runs. As argued by Miller (1998), Ge-
netic Algorithms (GA) mect the outlined requirements. Open exploration and directed search can
complement each other. For example, if the open exploration reveals that there are distinct types
of dynamics, then directed search can be employed to identify more precisely where the boundary
is located between these distinct regions.
Relevant questions in the case of A(HIN1)v
loss of life and worst social disruption and what should be done to addr
Kwakkel and Pruyt (2013) addressed these questions with directed searches and found two worst
case is the maximum
|
are: what are the worst cases in terms of total
those worst cases?
case scenarios, displayed in Figure 6, that are almost identical. The first wor
number of casualties (‘deceased population’). The second worst case is the highest social disruption
(ie. the peak infected fraction). The socially most disruptive case is thus almost identical to the
with the highest number of casualties. In both cases, the flu spreads very quickly. Thus leaving
$ to react, let alone leaving time for the development of vaccines.
ion makers are social distancing related measures that
¢
very little time for policymak
The only type of actions avai
reduce the speed with which the pandemic spreads
Note that the two runs obtained here are worse than any of the runs in the ensemble generated
with open exploration: the directed see enarios were arrived at through optimization, whereas
the open exploration ensemble was generated randomly and ‘missed’ the exact combination leading
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models. 14
to these worst case scenarios. This stresses an important point:
thous with ‘brute computational force’ to answer specific question
or intelligently performed open explorations are often better for answering spec
obtained by directed searches may also be too narrow: in the of an imminent flu
pandemic, focussing only on the worst possible pandemic may not be a good idea since that requires
a completely different response than in almost any other flu scenario. Note also that directed
searches are often but not necessarily faster than open explorations: performing a directed
with multi-objective optimization to find the Pareto front of non-dominated worst case scenarios
on multiple dimensions, instead of optimizing a single criterion to find a single wort
on a single criterion as illustrated in this subsection, is computationally expensive, i.e. tak
of computing time.
ands of runs
answers
arch
nario
a lot
Use 5: From Simple Adaptive to Robust Adaptive Policy Design
One of the most important uses is the design of effective robust policies in the presence of deep
uncertainty, i.e. policies that —given the re » at hand- lead to acceptable outcomes
no matter what happens. Pruyt and Hamarat (2010) intuitively designed various adaptive policies
for the flu case and tested them on the full ensemble. Figure 7 shows the effects of some of these
s starting from the no policy ensemble in Figure 7(a). Figure 7(b) shows that an adaptive
ccination policy in function of the population fatality ratio® is only effective towards the end of,
and following, the vaccination campaign. This policy was therefore combined with two adaptive
social distance policies that could help out before the end of the vaccination campaign. The
first adapti ial distancing policy with which it is combined is based on monitoring of the
infected fraction. Figure 7(c) shows that this combined adaptive policy ally helps to reduce
impacts in terms of this infected fraction, but not so much the cumulative number of fatalities.
The second adaptive social distancing policy with which it is combined is based on monitoring
of both the fatality ratio and the infected fraction. Figure 7(d) shows that the latter combined
adaptive policy —although still simplistic and solely based on intuition and trial and error testi
significantly reduces the infected fraction as well as the cumulative number of fatalities. However,
many una scenarios remain present.
Hence we reflected on smarter and more refined ways to design adaptive robust polic’
that are conditional upon the context and robust, i.e. effective in the presence of a wide va-
riety of uncertainties ally when really needed. For doing so, one needs to know which
es of scenarios require (additional) policies, what policies are most effective given the
root causes of these subensembles of scenarios, and when these policies would need to be acti-
vated. To that purpose, we developed an approach, rooted in the emerging literature on plan-
ning under deep uncertainty (Albrechts 2004; Kwakkel et al. 2010a; Walker et al. 2001), called
‘Adaptive Robust Design’ (Hamarat et al. 2012). A common characteristic of these approaches
is the combination of time urgent actions to be taken immediately with pre-specified actions
taken in response to how the future unfolds. In order to achieve a robust and adaptive pol-
y design, it is important to correctly specify when to respond with these pre-specified ac-
tions. To this end, signposts to track specific information can be defined for monitoring the
stem. Specific values of these signposts are called triggers and they are triggered when pre-
specified conditions occur in the system (Kwakkel et al. 2010a). We perform robust optimiza-
tion? (Ben-Tal and Nemirovski 1998; Ben-Tal and Nemirovski 2000; Bertsimas and Sim 2004) us-
ing Genetic Algorithms (Fraser and Burnell 1970; Holland 1975) to determine these trigger values.
Operationally speaking, our iterative approach combines (i) open exploration to generate all
sorts of plausible scenarios, (ii) identification of undesirable scenarios and their root causes (c.g.
using PRIM), (iii) design of policy actions that address these root causes and triggers to activate
ing
8The vaccination coverage is at least 20% —assumed to be the minimum to start up vaccine development and
production. It is assumed that the coverage finally planned for will be based on information regarding the observed
population fatality ratio according to pre-specified values. The low-coverage variant shown here assumes a coverage
of 60% in case of a total population fatality ratio >2.5%, else the coverage is 20%.
°Robust optimization methods aim at finding optimal outcomes in the presence of input parameter uncertainty.
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models. 15
(a) No Policy (b) Adaptive vaccination based on the fatality ratio
(c) Adaptive vaccination + social distancing based on (d) Adaptive vaccination x social distancing based on
monitoring of infected fraction monitoring of fatality ratio
Figure 7: Effect of simple adaptive polici
ensemble. Source: (Pruyt and Hamarat 2010)
starting from and compared with the No Policy
these policy actions, (iv) robust or stochastic optimization of the strength of the policy actions
and the trigger values, (v) testing of the robustly optimized policy on the level of the ensemble,
(vi) identification of remaining undesirable futures and their causes, etc. Regret analysis as in
(Lempert et al. 2003) could be used to choose one adaptive robust policy if multiple adaptive
robust policies would have been developed.
Figure 8 contains outputs generated by Hamarat et al. (2012) applying this approach to the
flu case. The basic policy referred to in that figure consists of those actions that are non-regret and
time-urgent and are taken from the start. In the adaptive policy two adaptive actions! are added
to the basic policy. In the optimized adaptive policy, the triggers of these adaptive actions are
determined using robust optimization. Comparing these policies and the outcomes with Figure 7c
—in which case social distancing depends on the infection rate too~ shows that smarter adaptive
policies are better, cheaper, and less disruptive than the simple adaptive policies in Figure 8.
A major lesson learned from combining SD with stochastic optimization is that this type of
optimization under uncertainty leads to much better results if the policy is adaptive (both in
terms of strength and in terms of being triggered) and flexible (different actions are triggered for
different circumstance
10(j) For an observed case fatality ratio (cfr) of 0.1%, the vaccination level is increased to 60%. If the observed
CER is 1%, then vaccination level is 80% and for CFR of 10% then the vaccination level is 100%. (ii) If the rate of
increase for the infected fraction is positive for three consecutive weeks, then the alert is activated and an additional
50% emergency contact rate reduction is applied.
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models. 16
(a) No Policy (b) Basic Policy
Tow ne Tae th
(c) Envelopes of the basic, adaptive, and optimized adaptive policy ensembles
Ba z
# 3 2
& 25 £
ai i
Bo. Es 2
0.00 0.05 0.10 OS 0.21 0.00 0.05 0.10 O18 Or
deceased population region { deceased population region |
(f) Basic Policy (g) Adaptive Policy (h) Optimized Adaptive Policy
Figure 8: Effect of smarter adaptive policy making — comparing the effects of a basic policy,
adaptive policy, and robustly optimized adaptive policy: 3D-scatter plots, envelopes and heat
maps
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models. 17
Use 6: Model Testing
The EMA workbench is also referred to by some as the ‘EMA torture rack’ si is very useful
for verification and validation of SD models too. It can be used to perform stress test and try to
break models. Each run that requires debugging can be identified in ‘interactive mode’, singled
out of the ensemble, saved!!, and instantiated directly in a Vensim model that could be used to
debug that particular run, and hence, the model. Identifying subsets of results with impossible
outcomes, followed by the identification of the joint causes of the impossible behavior can be used
to remove subensembles or to set constraints on ranges.
The workbench allows performing comprehensive!” sitivity anal
entire parameter space, automated sensitivity analysis over parameterized functions, sensitivity
testing with regard to delay times and orders of delays, sensitivity testing with regard to lookup
functions with Hearne’s method!* (Hearne 2010; Eker et al. 2011) and other methods, as well
as automated extreme condition testing. Other validation tests (like family member tests) can
be performed with open exploration and directed searches. Using multiple models allows for
multi-model triangulation and multi-model behavioral comparison too. And last but not least
could open explorations and directed searches be used to test policy robustness. See for exam-
ple (Kovari and Pruyt 2012) for policy robustness testing of what started out as a more tradi-
tional quick and dirty SD study. Semi-automated loop knock-out analysis was also implemented
(Keijser et al. 2012) and other formal model analysis approaches and algorithms are next on the
agenda.
automated se over the
Conclusions
In other scientific fields interested in model-based decision support, developments have taken place
that can be summarized in terms of using models for systematic exploratory use. Exploratory
Modeling and Analysis (EMA) aims at performing suc ematic model-based explorations and
directed searches. We have tried to explain and illustrate in this paper how SD and EMA can be
combined in order to address grand societal challenges that are characterized by both dynamic
complexity and deep uncertainty. Addressing such problems requires the systematic exploration
of different hypotheses related to model structure, model parametrization, and input uncertainties
on the kinds of behavioral dynamics that can occur, as well as directed searches. The resulting
ESDMA approach is in fact a computational SD approach (Pruyt and Kwakkel 2012a). However,
ESDMA makes SD more generally applicable, i.e. it extends the usefulness of SD from dynam
complex to deeply uncertain dynamically complex iss
While developing this computational SD approach, we became gradually convinced that some
s of old, may, given the current and near future state of science and
computing, become les ntial for the practice of SD, whereas other characteristics and ideas
may require much more emphasis and development into what they were intended for in the first
place.
We believe in that respect that SD models should still be largely endogenous, but in ESDMA,
it actually makes sense to ‘pollute’ SD models with ‘open’ elements (time
to bring in exogenous uncertainty, as well as with clements to include uncertainty in the internal
functioning of models
The original idea to resort to qualitative modes of behavior as a way of dealing with the
unavoidability of uncertainty may require rethinking. Interpreting model outcomes in terms of
modes of behavior may ; i s for dealing
with real-world issues. And with today’s cc ing power.and tectiniques, wecfinally have the
means to go beyond modes of behavior for dealing with ubiquitous uncertainties.
e
serve many goals but may not be s:
Mot just the run, but also all parameter values, functions, and model that generated that particular run
12Univariate + 2 Multivariate + 3 Multivariate + ..
13 The basic idea of Hearne’s method is multiplication of model functions by ‘distortion functions’ and varying the
parameters of these distortion functions in order to obtain various shapes and values of the model functions. Such
parameter-based generation of alternative function forms enables automated and extensive uncertainty analysis.
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models. 18
The same is true for the use of a reference base or reference
ases do not exist under deep
and with today’s computing power and advanced
techniques, there is no need for a single reference case nor for a very
uncertainty, there are at most base ensembles,
visualization and analy:
limited number of scenarios.
Deep uncertainty also puts another question on the table: whether developing and using one
plausible SD model is enough. If a model is just plausible, then maybe other plausible models need
to be made as well, before robustness could be tested properly. That brings us to the ontological-
epistemological stance of traditional SD and the m attempts to try to integrate very
different perspectives into one and the same model (Pruyt 2006; Pruyt and Kwalkel 2012a). From
our point of view, it makes more sense to model different perspectives separate
that are acceptable for all perspectives, 2
The illustration provided here was purposefully kept as
single-model. Single-model ESDMA is exceptional though: in ESDMA, it is much more nat-
ural to simultaneously use multiple simulation models. Examples of multi-model ESDMA include
(Pruyt and Kwakkel 2011; Pruyt and Kwakkel 2012b; Auping et al. 2012).
ESDMA is as much about analysis as it is about modeling, which is why much of our current
effort is into developing analytic tools to analyze outcomes and models (via the model outcomes).
The current analytic tools already generate a wealth of analytical and policy relevant insights,
partially filling another gap in traditional SD, namely the lack of advanced analysis of model
outcomes (and models).
We hope in that sense that this paper is also an answer to ac
few respectable System Dynamicists, namely that th
that first needs to be refined before being of intere:
approach is not a brute-force method in spite of the fact that more computational power needs
to be relied upon than in traditional SD. Although sufficient computational power is needed,
this approach should be performed intelligently and with advanced analytical tools, sophisticated
techniques, refined analyses and directed searches.
ESDMA also operationalizes the concept of policy robustness by allowing testing robustness
ticism we recently received from a
th is too much a brute-force method
of policies and comparing policies over the entire multi-dimensional uncertainty space. Small,
manageable and partly open SD models are most appropriate for ESDMA. Since ESDMA allows
exploring policy robustness over the entire uncertainty/scenario space and design of adaptiv
policies addressing the entire scenario space, it actually makes the job of policymakers much
lighter. On the one hand, that is also true for the analyst: more analytic tools are available
to perform policy relevant analyses. On the other hand does it make the job of a modeler and
analyst more difficult, since a larger tool set needs to be mastered, multiple plausible models may
need to be developed, analyses need to be performed systematically, complex outputs need to be
interpreted, and policies need to be designed iteratively and compared with all other pol:
Another complicating factor is the fact that easy software for doing all of the above is not
commercially available yet. Our software, which requires python coding skills, makes it accessible
and useable to a select few only. But even if user-friendly software would become available to all,
then System Dynamic ine learning, data mining, time
ies clustering, formal modeling techniques.
However, it is our experience that performing excellent ESDMA may not be enough. Policy-
makers need to become part of the ESDMA and need to experience uncertainty first hand before
being aligning heart and mind (Pruyt 2011). Gaming sessions as in (Pruyt 2011), new types
of modeling workshops as in (Logtens et al. 2012), and embedding ESDMA in recurrent. pol
process with real world pilots to further the understanding of the real world system and reduce
uncertainty may be necessary complements to ¢3 5
ESDMA modelers need to have more than just modeling and basi
, facilitator skills, modeling skills, programming skills, sampling skills, :
ills are all needed — in other words, super(wo)men or well-functioning
s covering the whole skill set may be required. Hence, more team work and cooperation is
needed. Everyone wanting to join our team effort to further the computational science of SD is
therefore more than welcome.
need advanced education in macl
and the like.
advanced analy
Pruyt, Kwakkel, Hamarat, 2012. Doing more with Models. 19
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