Lightening the Performance Burden of Individual-Based Models through
Dimensional Analysis and Scale Modeling
Nathaniel Osgood
Department of Computer Science
University of Saskatchewan
Email: osgood@ cs.usask.ca
1 Introduction
Systems models of larger populations (such as those used for health policy, ecological studies, marketing, urban
planning, etc.) have traditionally tended to be aggregate, in the sense that a particular state variable summarizes
information on many individuals. Such aggregate models offer many virtues, including greater transparency, ease
of analysis and calibration. However, modeling considerations sometimes motivate the use of finer-grained
models. Such factors include the need to evaluate interventions targeted at an individual level particular
individuals (e.g. such as are increasingly used in infectious disease epidemiology), to understand transfer effects
or dynamic impacts of heterogeneity on intervention effectiveness , the dynamics of complex and adaptive
network structures, and memoryful transition processes. Such concerns have fostered the popularity of
individual-based models, which explicitly describe and simulate the attributes and behavior of each member of a
population. Behavior within such individual-based models can be formulated in many different ways, including
using classic state-equation methods (using ordinary differential equations) or with discrete objects and rules (as
is typical in classic agent-based modeling).
While individual-based approaches offer considerable value for addressing certain types of questions and
problems, such models impose higher (and frequently dramatically higher) performance costs. The heavy
performance burden associated with large-scale individual-based models reflects several factors. The first is the
simple cost of updating a state vector of size proportional to the count of individuals in the population (which we
term n). Even a model where each individual evolves completely independent of others will experience linear
performance scaling; for large populations and significant amounts of individual state this can impose a very
heavy burden (not least because of memory hierarchy effects). Models with highly connected populations will
often exhibit performance costs that rise superlinearly with population size. For example, a model in which each
individual is connected to a small fraction « of other individuals in the population will impose performance costs
that rise at least proportional to 1 (i.e. Q(n)). Thirdly, because individual behavior and dynamics (e.g. infection
spread from one individual to another, or the decision of an individual to cease smoking) often involves some
stochastic factors, individual-based models often require consideration of stochastic processes. Gaining an
appreciation for the behavior of a given fully parameterized model often requires evaluation of a Monte Carlo
ensemble of realizations - a process that can require a day or more of simulation time.
In addition to inhibiting exploratory learning and user involvement, prolonged model simulation times can also
impose significant opportunity costs that can reduce the value delivered by the model. For example, long
simulation times can rule out interactive model exploration and considerable reduce the amount of focused model
exploration that is possible - thereby impeding a modeler’ s understanding of a model’s dynamics. The longer
cycle time associated with understanding each version of the model performance burden can also reduce the
opportunities for model refinement.
In light of the above performance concerns, creators of individual-based models involving large populations are
often tempted to draw insights from simulations involving subsamples of the full population [1]. Unfortunately,
while working with subpopulations can greatly lessen performance burdens, it can also be deceiving: Naive
intuition may mislead one into believing that reduction of a population size by some factor should be
1
accompanied by a similar reduction in all parameters - a transformation that may qualitatively alter the behavior
of the model and make results of the reduced-scale model completely unrepresentative of the full-scale model.
Even given an adequate reduced-scale model, it is often unclear how to scale up results from a reduced-scale
model to correspond the results that would obtain for a full population. Indeed, as we will see, this scaling can be
quite complex, involving power laws and ratios of other parameters.
Drawing inspiration from the widespread and critical role that dimensional analysis and scaling have played in
engineering and the physical sciences, this paper proposes the use of these tools to address the performance
challenges associated with individual-based models. Specifically, we propose a precise, rigorous, systematic and
general-purpose technique to formulating reduced-scale individual-based models. Measurements of output
parameters of such reduced-scale models can then be precisely transformed (in accordance with model scaling
laws) to yield comparable results for a full-scale model - without the need to run the full-scale model. It is
notable that these techniques these techniques are notable in relying only upon dimensional homogeneity of the
full-scale model, and on not the specifics of model behavior or use of a particular mathematical framework.
The remainder of the paper is organized as follows. The next section of this paper explains some basic concepts
of dimensional analysis. A key component of this section is the recognition that any real-world system - or any
dimensionally homogeneous model [2] - must be amenable to description in a fashion that is independent of unit
systems - a model, therefore, which involves only dimensionless variables. The following section introduces the
notion of similitude, which provides the basis for scale models. The next section simultaneously describes and
illustrates (with an example) a systematic means for deriving scale models from full-sized models with precisely
specified parameter values. The final section of the paper provides a brief summary of the paper and lays out
prospects for additional work.
2 Dimensional Analysis
Dimensional analysis rests upon the concept of dimensional quantities. Physical quantities can be associated with
semantic categories (the ‘Dimension’ of the quantity, such as “Person”, “Time”, “Length?” “Liquid Volume”), as
well as the standard of reference for measuring/sizing the quantity (the ‘Units’ of the quantity, such as “Day”,
“cubic foot”, “gallon”) [3, 4].
Dimensional quantities are associated with a formal dimensional algebra that differs in important ways from the
algebras on non-dimensional quantities, such as real numbers [5]. Within such an algebra, we associate each
dimensional quantity with a product of powers of different units (or dimensions), i.e. d;'d,°d,°...d," , where
there exist exactly n dimensions of concern to us, where each exponent ij represents the power to which the
corresponding dimension d is included, and where the ordering of the dimensions d is arbitrary. For example,
Mortality might be associated with units Person’/Day i.e. Person’*Day” (and of dimension Person'*Time''). A
notable class of quantities is those associated with unit dimension (commonly called dimensionless’, although this
is somewhat of a misnomer) - i.e. where i;=i)=...i,=0.
The dimensions associated with a given quantity (in this case Person’*Time’) can be used in a straightforward
fashion to indicate how the value of a given quantity would change given a change in unit systems. For example,
if we were to change our chosen measurement for people from individual persons to thousands of people (a factor
of 1000 increase) it follows that the numerical value of a quantity whose dimension of Person’*Time'' would be
divided by value 1000'. Similarly, if we were to change the measurement for time from days to weeks (a factor
of 7 increase), the numeric value of the resulting quantity would be divided by 77 (i.e. multiplied by 7). Given a
' Strictly speaking, the term “dimensionless” is a misnomer. While “Dimensionless” quantities have no dependence on the
units by which we measure any dimension, such quantities are no more dimensionless than quantities of 0 length are
“lengthless”. “Dimensionless” quantities have a dimension, just a very special dimension - one in which the exponent of
every dimension is identically 0. A better (or at least less abusive) name is “unit” dimension. Nonetheless, we cede to
convention in using the term “dimensionless” in this document.
2
dimensioned quantity whose dimensions include the term d’, increasing the magnitude of the unit of measurement
of a dimension d by a factor of x will lead the numeric value of that quantity to be divided by a factor of x’.
“Dimensionless” quantities are particularly notable in that their numeric values are independent of the units
system employed - that is, they retain their values regardless of our measurement conventions. As will be
discussed below, such quantities play a critical role in real-world systems.
Mathematical operators can be extended to dimensional quantities in consistent manners [5]. For example,
multiplication of quantities drawn from two different unit (dimension) systems yields a product with units
(dimensions)that are composed of a product of powers of each unit (dimension), where the exponent on a given
unit is just the sum of the exponents on that unit (dimensions) (possibly zero) for each of the quantities being
multiplied. In general, the presence of dimensional quantities aids the modeling process by imposing additional
structure that significantly constrains the set of legitimate formulations that can be considered in a given problem.
For example, addition of dimensional quantities are not defined unless both quantities possess identical units” and
dimensions. It follows from these results that a dimensioned quantity in an (for example) exponent or
trigonometric function must be of unit dimension. Matrix algebra, state equations, and other mathematical
constructs have been systematically extended to dimensioned quantities [5].
By far the most common use of dimensional analysis - and one widespread within the Systems Dynamics
community - is as a simple check the correctness of formulas. While dimensional analysis delivers considerable
value in this capacity, the ideas are vastly deeper than is commonly realized, and the full value of the approach
extends far beyond dimensional bookkeeping on exponents [6].
3 Similitude
3.1 The Buckingham Pi Theorem
3.1.1 Background
This section focuses on the notion of similitude, which provides the basis for scale models. We first introduce the
Buckingham’ Pi Theorem [7, 8], which states that any dimensionally consistent model of a physical system can be
formulated in a fashion that involves only dimensionless parameters. Because dimensionless parameters are
invariant to unit change, such a representation is independent of measurement system. It follows that that two
instances of a systems model having different population sizes and having identical values for all dimensionless
parameters must exhibit “similar” behavior". We then discuss a systematic approach for identifying the
dimensionless variables associated with a model and determining an exact relationship relating the scaling of a
given dependent variable in terms of other model governing parameters. Finally, we describe a procedure that,
given a complete specification for the parameters values of a full-scale model, permits us to derive the parameters
of a reduced-scale model offering desirable values for particular parameter values (such as the population size).
We begin this section by adopting the widely held assumption that the processes of the real world operate
regardless of which measurement systems we use to describe those processes. On the basis of this assumption, we
can say with confidence that whatever equations govern (and describe) those real-world processes, they must
operate independent of unit systems.
? Addition of quantities drawn from separate unit system but the same dimensions cannot be performed unless the quantities
are scaled by the appropriate unit conversions. It is simply nonsensical to add quantities drawn from different dimensions.
3 Note that although the Theorem carries Buckingham’ s name, there is good reason to think that the critical ideas originated
decades earlier.
“ We ignore here discretization effects, which play an important role for smaller models.
3
Atthe most basic level, the Buckingham Pi Theorem [7] builds upon two key observations: This invariance of
real-world systems to unit systems, and the fact that only dimensionless quantities are independent of unit systems.
Informally, the Pi Theorem states that one can respecify any legitimate dimensionally homogeneous equation
using an equation involving only dimensionless quantities.
More specifically, consider a dimensionally consistent (homogeneous) equation relating n dimensional quantities
(governing parameters) x;...x, and associated with r independent’ dimensional products. Without loss of
generality, assume that x; is the dependent variable:
X= £(X,, Xp Xyy0 K,)
This equation can always be reformulated as an equation involving only (n-r) dimensionless parameters Ij.
m= WY (Ay My Ryser Ty)
Each of these dimensionless parameters 7; is defined as a product of powers of the original governing parameters
X}...Xn such that the exponents of each dimension balance to 0 and their product is dimensionless. 1 is the unique
dependent dimensionless parameter, and is the sole dimensionless parameter involving the dependent variable x.
Given that 7 is defined as a product of powers of the x;, the relationship between x; and 7 is such that:
4
(Js
Where 0<k<n-d is the number of other governing parameters x; needed to multiply (divide) x; in order to make the
product dimensionless.
Moreover, a deeper result of the Pi Theorem indicates that any dimensionally consistent (homogeneous) model
can be reformulated in this fashion. It bears stressing that while the above equations appear algebraic, the
Buckingham Pi Theorem holds for equations in general - including Ordinary, Stochastic and Partial Differential
Equations.
3.1.2 Advantages of the II Theorem
While it may initially appear a mere curiosity, The Buckingham Pi Theorem has a number of significant
implications. While the Buckingham Pi Theorem describes a deep fact about relations between dimensional
quantities in the real world, we can take advantage of our knowledge of this theorem when we specify models of
the external world. Specifically, secure in the knowledge that relations between quantities in the world can be
described using only dimensionless quantities, we can confine our attention to building models of the real world
that use only dimensionless parameters (7).
Explicitly working with models involving only dimensionless parameters offers a number of significant
advantages. Firstly, consider the amount of effort needed to statistically estimate a mathematical model
mathematical model relating of n governing parameters (in d independent dimensions), as compared to the
corresponding mathematical model involving only (n-d) dimensionless quantities. Because of the curse of
dimensionality, the dimensionless mathematical model will often require orders of magnitude fewer parameters to
estimate than would the naive statistical model [9]. Secondly, formulating from the start a model involving only
dimensionless variables allows us to transparently switch between different unit systems, without any change to
the model.
5 We refer to independence in a linear algebraic sense, where the vector space has axes for each dimension of concern, and
each vector uniquely represents a unique product of dimensions. The value of the vector along a particular dimension is
given by the exponent associated with that product of dimensions.
4
Thirdly, the identification of the dimensionless parameters and knowledge that they are linked through some
(typically unknown) function ’ can provide insight into how a system must function. The dimensionless
parameters being used by such a model (e.g. the Reynolds Numbers, Fourier Number, Raleigh Number, Mach
Number) frequently have deep and recurring physical significance. Conceptualizing the world in terms of such
dimensionless quantities can provide insight into causal regularities, and particularly into scaling properties and
invariants. The level of insight gained through the Buckingham Pi Theorem does vary between problems; the
insights gained from models employing few dimensionless parameters can be particularly striking. Insights can be
sharpened by a variety of techniques, including refinement of the dimensional system via dimension splitting [8],
fusion of tightly coupled governing parameters [8], and reasoning based on intermediate asymptotics [9, 12].
3.2 Similar Systems
Consider two real-world systems A and B associated with the same set of governing parameters x; pertinent to the
phenomenon of interest, where those parameters are governed by the same fundamental processes. Although A
and B share identical sets of governing parameters, the two systems may differ with respect to the particular value
of these governing parameters. For example, A could be a full-size aircraft, and B a reduced-size aircraft for
wind-tunnel experiments. Alternatively, A may be a large antenna, and B electrical engineer's small-scale model
thereof, or perhaps A is a bridge, and B a smaller replica of that bridge for structural testing. For our purposes,
A could be a full-scale simulation model and B a reduced-scale model. Call the particular values associated with
those governing parameters in systems A and B X, ;; and X;, xs . (Note that while we adopt subscriptions
suggestive of “full-scale” and “reduced-scale” for the sake of clarity in later discussion involving reduced-scale
models, the notion of dimensional similarity by no means requires that either A or B is a systematically reduced
version of the other. Indeed, in comparison to A’s governing parameters, B’s governing parameters may be larger,
smaller, or some combination thereof.)
Following the previous section, such systems have a (frequently unknown) relation giving dependent variable x*,
(or x®,) as a function of the other governing parameters. Thus
Kips =f (X25, Xr 1Xapg rey Xnps )
and
Xirs = f (Xs 1X pg Xy pg sey Xa ps )
It bears emphasizing that because we assume that A and B are governed by the same processes, f is the same
function in both cases.
We know from the Buckingham Pi Theorem that these equations can be rewritten in the form
Ts =F (A285 1 73,pg 1 Ty pg r++) Ty 1,FS )
and
Tyas = Y (7245 1 T3R5 1 Taps re Tos)
WetermA andB similar if Vi,1<i< 1, (a5 = Tips ) . That is, two systems that may be very different in
terms of the particular values of their parameter sets X,-, and X, xs are considered similar if all dimensionless
variables share the same values in each of these two systems. The notion of dimensionally similar is important in
that the Buckingham Pi Theorem reveals that two similar systems are really “the same system,” the difference
between them being one merely of scale. Thus we can think of a model reformulated so as to use only
dimensionless parameters as defining equivalence classes of systems. Each member of these equivalence classes
5
shares with all others in the same equivalence class identical values for all dimensionless parameters in the
(reformulated) system.
We note that because 7; ,. is purely a function of the other 7; ., (and similarly for 7, ,, ) and because the relation
is identical for both A and B, A and B are similar if
Vi,2si<n(A zs =7ps)
From which it must follow that the dependent variables must hold identical values, that is, that 7, .5 = 7ps -
For many decades, scientists and engineers have exploited the concepts of similarity to build dimensionally
similar scale models that behave identical to a full-scale system with respect to clearly-defined class of measures.
We now tum to apply this notion to the particular case of individual-based simulations. The next section briefly
explains how such scale models are constructed.
3.3 Building Scale Models
Scale modeling is traditionally used to facilitate convenient experimentation with systems in the external world,
whether of natural or artificial origin. Reduced-scale models of bridges, buildings, ocean waves, and vehicles are
far cheaper and more tractable to work with than are their full-sized counterparts. In most such cases, the relation
between the relevant governing parameters is not fully known, but this poses no barrier: Such scale modeling
does not any knowledge of the underlying equations, and exploits only the invariance of the behavior of the
external world with respect to unit system choice.
Within this paper, we exploit the same basic idea, but with a slightly different twist. While perhaps all previous
scale models have focused on real-world artifacts, our modeling target is itself another model- an individual-
based simulation model. While traditional scale modeling relies upon the dimensional consistency (homogeneity)
of relations between real-world processes, we rely here upon the dimensional consistency of the individual-based
simulation model. The dimensional consistency of this model assures us that the Buckingham Pi Theorem applies
for any relation relating a dependent governing parameter x, to the other the governing parameters of the model
X)...Xn. While we need not explicitly derive the corresponding relation involving only dimensionless quantities I,
the Pi Theorem guarantees that this relation does exist and provides the basis for subsequent scale modeling.
As an alternative to simulating the full-scale system to measure on a large-scale individual-based model x;, we
can thus make use instead of a reduced-scale model (one with, for example, a much smaller population) by
following the steps below. In order to illustrate these steps, we introduce an ongoing example.
3.3.1 Step 1: Identify Governing Parameters and their Dimensions
Consider an individual-based model of the spread of a zoonotic infectious disease [10, 11]. For such diseases
(which include important tropical diseases such as Schistosomiasis, Malaria, and Dengue fever), all secondary
infections of a person (Host) occur from an animal (Vector; Snails for Schistosomiasis and Mosquitoes for
Malaria and Dengue fever). The etiology of such diseases can be complex, and perhaps our individual-based
simulation model includes detailed characterization of immune system response (as affects egg shedding in
Schistosomiasis), heterogeneity in disease response (notable for Schistosomiasis), and behavioral factors (as
might affect likelihood of exposure to a Vector or the successful transmission of the disease to vectors).
Suppose that we would ideally like to simulate a full-scale individual-based model’ having governing parameter
values’ as follows:
® While we would typically approach the scale-model identification with a specific model in hand, we purposefully omit
specification of the equations of such a model for the sake of brevity and communicating the generality of this approach -
particularly, the fact that it does not rely on any detail of model specification (except for its dimensional homogeneity).
6
Description Parameter | Dimension
Death rates for infected V ectors by 1/Time
Death rates for infected Hosts i 1/Time
Per-Contact likelihood of By 1 (“Dimensionless”)
Transmission from hosts to vector
Per-Contact likelihood of Bn 1
Transmission from vectors to host
Net migration rate for vectors M, Vector / Time
Net migration rate for hosts Mh Host / Time
Recovery rate for infected hosts a 1/Time
Initial infected hosts Tho Hosts
Initial infected vectors Ivo Vectors
Contact rate C Host/(V ector*Time)
Population size for vectors Py Vector
Population size for hosts Py Host
Current time t Time
Suppose further that the full-scale model using these parameters imposes an infeasibly high performance burden
due to high population sizes of vectors and hosts. We therefore wish to create a reduced-size model (specifically
including smaller populations for vectors and hosts) whose outputs we can extrapolate to the full-scale model.
While this goal seems obvious enough, it is not immediately clear how reducing the size of the populations should
impact the various parameter values. For example, if we were to reduce the population of vectors and hosts by
some uniform factor <, should we reduce the contact rate by the same factor? What of the transmission
probabilities? Would the reduced-scale model that is yielded have a timescale equivalent to that of the full-scale
model, such that we could apply times read off from the reduced-scale model (e.g. time of maximum prevalence)
directly in the context of the full scale model?
3.3.2 Step 2: Identify Dependent Variable
Suppose that we wish to calculate a dependent parameter 4 of dimension 1/Time - such as would be associated
with a frequency (e.g. frequency of oscillation), a fractional rate of growth, an eigenvalue of the linearized system,
etc. Because all of these have identical dimension, they can all be calculated using an identical reduced-scale
model and scaling procedure. (As we will see, dependent variables of different dimension can typically be
calculated using an identical reduced-scale model, but will require different scaling formulas to extrapolate from
the results of the reduced-scale model to those of the full-scale model.)
7 Note that some of these parameters are sizes of collections, as in the size of the host and vector population. We adopt much
of our notation from a compartment model of the same phenomenon included in the cited references.
7
3.3.3 Step 3: Identify Expressions for Dimensionless Variables
3.3.3.1 Basics
Recalling the notion of similarity from Section 3.3, we know that any scale model must preserve the value of all
independent dimensionless parameters 7;, Two models with identical values for corresponding dimensionless
parameters must yield an identical value for the dependent variable no matter how complex ‘Y may be. We thus
proceed to determine a set of governing parameters for the reduced-scale model (for example, ags and Mn,rs) that
are both computationally feasible and leave the dimensionless variables for that reduced-scale model (m,xs) equal
in value to the corresponding mrs.
The first step in this direction is to identify expressions for a set of dimensionless variables I in terms of the
dependent variable and plausible governing parameters. This set need not be unique and is straightforward (and
nearly mechanical) to derive, being identified through reference to a dimensional matrix populated through
reflection on the dimensionality of governing parameters and a bit of linear algebra.
For our case dimensional matrix can be defined as follows; the A ppendix describes a systematic means by which
this and similar matrices can be derived.
% | Mn | M Jun |p fo | tho | lo | Bn | By |e | Pa) PB ft
Host 1 0 0 0 0 1 0 0 0 1 0 0
Vector | 0 0 1 0 0 0 0 1 0 0 -1 7/0 1 0
Time -1 |-1 -1 -1 |-1 |-1 |0 0 0 0 -1 7/0 0 1
TT! 1 0 0 0 0 0 0 0 0 0 0 -1 )0 1
™ 0 1 0 0 0 0 0 0 0 0 0 -1 7/0 1
73 0 0 1 0 0 0 0 0 0 0 0 0 -1 1
1% 0 0 0 1 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 1 0 0 0 0 0 0 0 0 1
16 0 0 0 0 0 1 0 0 0 0 0 0 0 1
a 0 0 0 0 0 0 1 0 0 0 0 -1 7/0 0
1 0 0 0 0 0 0 0 1 0 0 0 0 -1 7/0
To 0 0 0 0 0 0 0 0 1 0 0 0 0 0
™0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
TM 0 0 0 0 0 0 0 0 0 0 1 -l1 ji 1
Table 1 Example Dimensional Matrix. Shaded portions are specific to the dimensionality of the dependent variable
chosen (1/Time).
The first column and 4" row of Table 1 (shaded for reference) are specific to dimensionality (Host/Time) of the
dependent variable (2) we have chosen; the remainder of the table is invariant of the choice of dependent variable.
We can read off the definition of each dimensionless parameter IJ; from the corresponding lower 11 rows of Table
1. Each entry in such rows describes the exponent of the associated governing parameter (as specified by the
column) within the dimensionless parameter (as specified by the row).
For our case, the dimensionless parameters are defined as follows:
at
n=
P,
M,t
m=
B,
M,t
1; =—
P,
1, = yt
Ts = byt
TM, =at
I
Eo
h
Is
eo
P,
7 = By
To = B,
cP,t
My = Pp
h
3.3.4 Step 4: Identify Values of Dimensionless Parameters in Full-Scale Model
The next step in deriving a reduced-scale model is to derive the value that each non-dependent dimensionless
parameter 7;,i>1 must hold in the reduced-scale model. We do so based on the known values of the non-
dependent governing parameters x; rs, i>1 within the full-scale model. To maintain similar results, we must
design the reduced-scale model to maintain identical values all of the dimensionless parameters.
For example, for x. we compute as follows:
ies My estes
5s
Pes
Similar calculations are performed for the other dimensionless parameters, except for the dependent
dimensionless parameter ;. The latter is to be determined through calculations on the results of empirical
measurement.
3.3.5 Step 5: Derive Governing Parameters for the Reduced-Scale Model
In the final step needed to complete the specification of the reduced-scale model, we derive the governing
parameters Xrs,i>1 for a reduced-scale system such that all of the non-dependent dimensionless parameters 7,i>1
in the reduced-scale model share the same values as their corresponding parameters in the full-scale model.
The derivation of these governing parameters will vary with respect to their position in the dimensional matrix
above (see Appendix). The governing parameters in the last several columns of the matrix depicted in Table 1
(those in submatrix C, as defined in the Appendix) are free parameters that can be assigned convenient values by
the modeler. By contrast, the other governing parameters for the reduced-scale model system are bound
parameters to be derived from the free parameters and the corresponding values of the governing parameters in
the full-scale model.
The next two subsections describe these parameters for our example.
3.3.5.1 Free Governing Parameters
The free (independent) governing parameters (the quantities associated with the last r columns the dimensional
matrix (where r is the count of independent dimensions) are for us to choose. In this model, we assume that the
computational expense arises predominantly from the need to simulate the transmission of infection in the
(individually represented) populations of vectors and hosts. Suppose that to construct the reduced-scale model we
wish to downsample the population by a coefficient of e<<1; thus we wish to have
Pus = &Phes
Pons = EP. es
For the final free governing parameter - time t - suppose that we wish to preserve identical timing for the
reduced-scale model as that which obtains in the full-size model, thus
s = tes
3.3.5.2 Bound G overning Parameters
Having determined the free governing parameters of the reduced scale model, we must determine the values of the
bound governing parameters in the reduced scale model. While we can perform this process in a fully automated
fashion using matrix manipulations (Gaussian Elimination) of the logarithms of coefficients of governing
parameters, this section illustrates the process through manual calculations. The A ppendix illustrates how similar
results can be obtained through inspection.
From the equivalence of dimensionless parameters in the full and reduced-size models, we know that the
governing parameters that happen to be dimensionless must hold identical values in the two models, thus:
Tons = 7,5 => Bsns = Pres
Tors = Tors = Burs = Burs
For other dimensionless variable, the situation is more complex, because they involve multiple governing
parameters. For example, consider m2:
h
Because we know that 77) g; = 7,ps it must be the case that
10
Muastas Myrstes
Pyrs Pies
Recall that
Puas = €Ph es
tas = ts
We can now solve for the size of governing parameter M, in the reduced-scale model, by simple algebra on the
equality above:
M nastes _ My rstes
(Pes ) - Pus
By following a similar procedure, we recognize that
M,t
13 = => My ns = EMy ps
,
, implying that M,, gs = €My ps -
Ty = yt Ls = Maes
Ts = Ut = Lyng = My rs
Mg = Ob Ags = Ags
= Tors = €lnors
7
P,
I
= v0 =
3 = phos = élyops
v
cP.t
™ = => Cys = Cog
h
A rapid and methodical approach to deriving such formulas is given in Appendix 2.
3.3.5.3 Parameters of the Reduced-Scale Model: Summary
We have now derived the values of the governing parameters of the reduced-scale model. A few of these (the free
governing parameters) were chosen so as to be computationally tractable and convenient, while (bound)
governing parameters of the reduced-scale model were specified in terms of a coefficients times the
corresponding parameter in the full-scale model - with the value of that coefficient chosen that the values of each
dimensionless parameters is equal between the two models.
% |Mb |M) Jun joy fo Jey jpn | Bo |B |e | t | Ph | PB
Host [ce Je Je |1 J1 J1 J1 J1 Jl fi fi | fe Je
Table 2: Scaling Coefficients for Coupled Infection-Health Services Model. An entry in this table describes the
coefficient by which a governing parameter in the reduced-scale model has been multiplied (in comparison to the
corresponding governing parameter in the full-scale model).
1
The results of this investigation are shown the table above. Note that the shaded scaling coefficients were
explicitly chosen, while the values of the others were derived by exploiting the equivalence of the dimensionless
parameters in the two models.
3.3.6 Step 6: Measure Dependent Governing Parameter from the Reduced Scale
Model
The next step in our analysis would be to parameterize the reduced scale model (using the scaling relationships
specified in Table 2). The reduced-scale model would then be run and a value for Ags measured.
3.3.7 Step 7: Calculate Dependent Variable for the Full-Scale Model
The dimensionless parameter 7 is distinguished from other dimensionless parameters in that 7 is to be calculated
from the measured output of the reduced-scale model. From a measured value for the dependent governing
parameter of the reduced-scale model (here Ars), we seek to extrapolate the value of the dependent variable for
the full-size model (here Ags).
Much as Step 5 (Section 3.3.5) calculated the values for bound governing parameters in the reduced-scale model
in terms of known scaling coefficients and governing parameter values for the full-scale model, here we use the
same process in reverse: We use the this measured value Ags from the reduced-scale model to calculated Ars for
the full-scale model.
For our example, by the process described in Step 5, we note that the definition z, = 2 and the fact that
h
mrs=71,rs implies that
es 1 Ld
yg lee 6°
and thus that
tpg = 2
€
We have now derived the desired quantity - the dependent variable for the full-scale model - in terms of known
quantities - the measured dependent variable of the reduced-scale model and a known (free governing parameter)
scaling coefficient. We have been able to deduce the value to be output by the full-scale model without bearing
the high cost of simulating the model with the full population.
As noted above, these results would change only slightly based on the dimensionality of the dependent variable.
For example, if the dependent variable were of dimension 1/Time (rather than Host/Time), the dimensionless
variables would be identical as to the previous case, with the exception of ;, which would then be given by the
expression
=A
71, alternative
and we would have Aj, = Ags
In general, we can change the dimension of the dependent variable without any need to change the
parameterization of the reduced-scale simulation model. Varying the dimensionality of the dependent variable
would only affect the scaling required to go from a measured value of Ags to a corresponding value for Ars. This
means that we can use a single reduced-scale simulation model to deduce relations involving dependent variables
of many dimensions, adjusting our only scaling strategy.
12
3.3.8 Empirical Results
To illustrate some of the issues that arise when building scale models in the manner described here, Figure 1
shows the fractional prevalence resulting from a full-scale individual-based model of a vector bone disease, and
similar output from a series of successively smaller reduced-scale models constructed in accordance with the
scaling relationships described above. The experimental model was constructed in an agent-based fashion
AnyLogic 6 (Advanced Educational Version) and is governed by the parameters shown in Table 1. This model
includes interacting populations of individually-represented V ectors (Mosquitoes) and Hosts (People).
Transitions (e.g. contracting or recovering from illness) and interactions (e.g. Vector contact with a particular
host) are Poisson distributed discrete events, according to transition rates given by the governing parameters.
Despite the stochastic nature of the simulation, the trajectory of prevalence shows a high degree of regularity for
human population sizes above 1000. Prevalence curves for the full-scale model and reduced-scale models match
closely down to and including a factor of 50x reduction in the human population (population 1000). With
populations below this point, however, the results increasingly diverge. We believe that these effects result from
the smaller sample sizes coupled with the discrete character of the population interactions.
The empirical results shown here demonstrate the potential for application of the scaling procedure to greatly
reduce computational burden of a non-trivial individual-based model. The results do highlight the need to
approach interpretation of scale model results with care. Confidence in the results of scale models can be
enhanced by performing sensitivity using different scaling levels, in a manner similar to that by which System
Dynamics modelers use sensitivity analysis using timesteps to build confidence in the reliability of numerical
integration results.
13
0.25
Full- and Reduced Scale- Model Output:
Host Prevalence
——Host Population 5C000(FullScale Madelt
Host Population 1€000(Epslon=.2)
—Hbost Population 1600 (zpsilan=
Prevalence
——Host Population 100 (Epsilor=.092)
——Host Population SC (Epsilon=.005)
Figure 1: Comparing Results of Full- and Reduced Scale Models. Results show close agreement for Epsilon .02 and
above; smaller reduced-scale models exhibit discretization artifacts
3.4 Summary
This section has built on the fact that any dimensionally consistent relations for a dependent variable can be
reformulated in a manner that involves only dimensionless parameters, and using a smaller number of parameters
are needed than were present in the original system. This reformulated model implicitly defines equivalence
classes of similar systems such that each dimensionless parameter holds the same value across all members of the
equivalence class. We described a (primarily mechanical) process exploits this property of similarity to define
reduced-scale individual-based models that yield results that can be precisely scaled up to derive output that
would result from a full-scale simulation. This fact is generally used to reason about constructing scale models of
systems in the real world, but we have proposed here a novel use of this approach to create reduced-scale models
of large-scale individual-based models.
14
4 AComment on Heterogeneity
Dimensional analysis has traditionally been applied for continuous physical systems in which a relatively small
number of homogeneous parameters (e.g. Y oung’s modulus, density, temperature, etc.) describe the properties of
a distributed parameter (e.g. spatially continuous) system. There are compelling reasons for modeling fairly
(parameter-wise) homogeneous populations using individual-based models - including the ability to represent
non-memory-less processes, multi-scale phenomena, and detailed percolation across networks. Nonetheless,
given the strong benefits individual-based models offer for understanding the behavior of heterogeneous
populations [1], it is important to consider how the approach presented here extends to such systems.
It is straightforward to apply the approach above for systems exhibiting limited heterogeneity - for example, for
individual-based models in which the governing parameter a differs widely across the population. The absence of
complications from heterogeneity reflects the fact that although the governing parameters of the individuals may
vary in a heterogeneous system, the dimensions of the governing parameters of different individuals are identical.
Adding limited heterogeneity to a system therefore does not affect the number of free governing parameter
(Submatrix A by the conventions introduced in the A ppendix). The only practical difference caused by addition
of limited heterogeneity is that more dimensionless parameters 7; will generally be created. With the exception of
any free governing parameters (whose values are directly chosen), all of the heterogeneous parameters of a
particular dimensionality will be adjusted in an identical fashion when creating a reduced-scale model.
Consider a variant of our example from the last Section. While we previously assumed that all individuals in the
population share values for recovery time (@), assume now that individuals may be associated with any of k
distinct values of a, 1<i<k. These values all share identical dimensionality 1/Time. The dimensional set matrix
for the heterogeneous system would remain very similar to Figure 2; we would simply replace the column for a
with k columns, one for each a. Dimensionless parameters 7; would be created for each such governing
parameter a. Because the dimensions of each a governing parameter are identical to that the dimensions of a,
the dimensionless parameter associated with each of the new governing parameters would have precisely the same
form as did the dimensionless parameter m2 associated with a.
The situation is only superficially more complex if we consider heterogeneity in a free governing parameter such
as . In such a case, just one of heterogeneous parameters (say, 1) would be a free parameter in Submatrix A;
all the others would be placed with the bound parameters in Submatrix B (for example, listed just before time t).
In accordance with the approach described above, the value of the free governing parameter 1,x5 in the reduced
scale model would be freely chosen by the modeler. All other governing parameters ; would he associated with
dimensionless parameters.
At an operational level, introducing limited heterogeneity does not in general complicate the creation of scale
models. The analysis is simply the same as with aggregate parameters; all that is required is systematic scaling of
additional governing parameters to realize the scale model.
Despite the mechanical simplicity of reflecting heterogeneity in the analysis, the discrete nature of individual-
based models imposes an important limitation. Specifically, the approach presented here assumes that the
cardinality of the reduced-scale population is sufficient to maintain the count of heterogeneous parameter classes
that were seen in the original system. (In the notation above, it assumes that the population in the reduced-scale
model is sufficiently large to represent all k heterogeneous values.) For example, this approach should work fine
if there were 100 (k) different values of q in both a full-scale population of 10° and in a reduced-scale population
of 10°. Such straightforward scaling will not be possible if we make use of a reduced-scale model with a
population size that is too small to represent the heterogeneity in the full system (for example, a reduced-scale
population size of 50 if we need to represent 100 different values of parameters), as it will not be possible to
associate an individual with each of the reduced-scale parameters. If we wish to consider multiple dimensions of
heterogeneity, the reduced-size model would need to be of sufficient size to represent combinations of these
values.
15
5 Limitations
This paper has described a rigorous, systematic and surprisingly general approach for deriving a reduced-scale
models whose results can be straightforwardly scaled up to yield results representative of a corresponding large-
scale model. These approaches draw on the theory of similitude and dimensional scaling that have been
extensively exploited in engineering and the applied sciences.
While we believe that the basics of the described approach are sound, the approach is not without some
drawbacks and significant research challenges remain for practical application.
Perhaps the biggest challenge was mentioned in the previous section: The use of dimensional scaling for highly
heterogeneous populations. Given the strong motivations for using individual-based methods for such
populations, it is very important to work to identify systematic means of representing them in reduced-scale
models.
A related issue concems discretization effects. Both theory and empirical observation by the author suggest that
the discrete cardinality of populations poses challenges to achieving full similitude with smaller-scale reduced-
size models. In addition to heterogeneity scaling concerns, other discretization artifacts can also become
pronounced for smaller models. For example, stochastic departures may reflect the fact that as one lowers the size
of the reduced-scale population, discrete probabilistic outcomes for the small counts of individuals in reduced-
scale model may differ from those of the full-scale model.
6 Conclusion
The System Dynamics community is increasingly applying individual-based models for insight. Such models
offer particular value for investigating the effects of targeted interventions and for studying systems with
populations exhibiting high heterogeneity, complex and dynamic network structures. Unfortunately, simulation
of such models for large populations is often extremely expensive. While it is desirable to gain insight into the
behavior of models using reduced-scale populations, naive construction of such reduced-size models can yield
erroneous conclusions. Within this paper, we have described a rigorous, systematic and general technique for
building such models.
While the described technique requires further development to address the full richness of modern modeling
practice, we believe that it has considerable potential.
More broadly, we believe that dimensional analysis offers many inviting avenues for future research.
Specifically, we believe there are high benefits to be gained by applying dimensional scaling theory to model
analysis [6]. We believe these benefits are likely to be particularly substantial for individual-based models which
are less tractable to closed-form analysis. We believe that the concepts and associated tools of incomplete
similitude and intermediate asymptotics [9, 12] hold particular promise for further simplification of dimensional
reasoning for individual-based systems. By building on dimensional approaches directly confronting the issue of
multi-scale analysis and approximation, renormalization group theory may also be of great value.
16
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17
Appendix 1: Derivation of Dimensional Matrix
Section 3.1 described the Buckingham II Theorem. This theorem shows that any dimensionally homogeneous
relation giving the value of a dependent variable x, in terms of other model parameters can be reformulated as an
equation involving only dimensionless parameters I1;. This section presents a systematic matrix approach for
identifying a complete set of dimensionless variables needed for such a reformulation. Interested readers may
wish to refer to the more complete exposition of this approach in [8].
The matrix approach is motivated by the fact that in determining the expressions for a particular variable, we are
simply selecting the exponent for each governing parameter x; in that expression. This exercise amounts to
solving a set of simultaneous linear equations specifying constraints on the exponents of each governing
parameter x;, The equations are weighted sums of these exponents, reflecting the fact that multiplying variables
sums exponents for each dimension, and that raising a governing parameter to a power multiplies all dimensional
exponents for that governing parameter by that power. The constraints enforced by these equations reflect the fact
that in order to yield dimensionless products, the sum of the exponents associated with each dimension must total
to 0.
Following [8], we define 4 matrices, arranged as follows. The dimensions of each matrices are enclosed in
parentheses following the matrix name.
Governing Parameters (xi)
X1 x x3 we XD XD we X
dy
B Matrix (Dx(n-D)) A Matrix (DxD)
dp
q TH
i sé D Matrix (Identity; (n-D)x(n-D)) C Matrix ((n-D)D)
i T™-D
Figure 2: Dimensional Set Matrix Structure
Matrices A and B within Figure 2 are to be specified by the modeler. Each entry (i,j) in these matrices specifies
the exponent associated with dimension d; within the dimension product of powers of x;. Thus, a given column j
of matrices A or B represents the exponents associated with each successive dimensions d, within the dimension
of the corresponding governing parameter x;. For example, if governing parameter x; were of dimension
Person’/Time = Person*Time'', the entry in the Person dimension row for x; would be 2, while that in the Time
dimension row would be -1. The entries for any other rows would be 0.
The A and B matrices can be populated by means of a simple dimensional analysis on each governing parameter
x; in turn - where for each such x; (row) we simply read off the exponent for each dimension (row). The
governing parameters chosen for matrix A must be dimensionally independent (in the sense that no product of
powers of these governing parameters can be dimensionless), but may be arranged in any order. These
parameters are generally those which the modeler believes will have the greatest influence on the dependent
variable and those that we wish to explicitly specify for the reduced-scale model. The other parameters can be
determined from them.
Entries for matrices C and D have very different meaning than do the entries in A and B. An entry in matrices C
and D specifies the exponent associated with a particular governing parameter (that associated with the column)
within a particular dimensionless variable (that associated with the row). Thus a given row of Matrices C and D
18
specifies the exponents of each governing parameter within the specific dimensionless variable associated with
2
%
that row. For example, if z, = 77 then the entry for row 1 of C and D would have a “2” in the first column
Xs Xa
(corresponding to x, a -1 in the 3“ column (corresponding to x;) and a -1/2 in the 4" column (corresponding to x4
Matrix D is of fixed structure (the (n-D)x(n-D) Identity matrix) and thus requires no modeler specification.
Matrices A, B and D are all populated directly by the modeler, using previously-known information. By contrast,
matrix C must be calculated in order to determine the definitions of the dimensionless variables I; in terms of the
x; Matrix C is calculated from of Matrices A and B. In what amounts to a solving of the implicit set of
simultaneous linear equations governing the exponents of the governing parameters, we determine the contents of
Cas
Cc =-(A"B)!
It is notable that the A needs to be invertible - hence the requirement for the dimensional independence of the
governing parameters of A.
Once matrix C is determined, the definition of each dimensionless variable as a product of powers of the
governing parameters x, can be read off from rows of C and D.
It bears emphasizing that the dimensionless variables identified through this analysis are specific to both the set of
independent governing parameters chosen (those associated with the columns of matrix A) as well as to the
dimensionality of the chosen dependent variable. The choice of the dimensionally independent governing
parameters will in general have a large impact on the dimensional variables identified. By contrast, choosing
dependent variables of differing dimensionality will only affect® the expression for 1
5 Recall that the dimension of the dependent variable is only encoded within the leftmost column of submatrix B. Reflection
on the linear algebra involved reveals that this column only influences the topmost row of submatrix C (the row that
indicates the other independent governing parameters that play a role in x,). Thus, the expressions for all dimensionless
variables other than x; would remain the same. This bears emphasizing, as it means that the value of the ‘Y expression will
remain the same regardless of the dimensionality of the dependent parameter.
19
Appendix 2: Derivation of Scaling Relationships
This appendix provides a straightforward and methodical means of deriving the scaling relationships between the
bound governing parameters of the reduced-scale and full-scale models (such as those manually derived in
3.3.5.2)
Reflecting on the structure of the dimensional matrix (see A ppendix 1, and the example in Table 1), we can
recognize that each dimensionless constant 7; is defined as the product of a unique (bound) governing parameter
(the diagonal entries in Submatrix D) with some product of powers of the r free variables (the columns in
Submatrix C; for the example, c, Py, and P,).
Given this pattem, that an equation establishing 7, ns=1; rs can be transformed directly into an equation giving the
scaling coefficient for the (bound) governing parameter of interest in terms of a product of power of the
coefficients by which each relevant free parameter was reduced.
All we need to do to arrive at such an equation is to divide both sides of the equation by the product of powers of
the free governing parameters for the reduced-scale model, and by the value of the governing parameter of interest
in the full-scale model (all of which we assume to be non-zero). The resulting formula specifies the ratio of the
(bound) governing parameter in the reduced-scale and full-scale model (the scaling coefficient of that bound
parameter) in terms of the product of powers of the ratio of each relevant free governing parameters in the full-
and reduced-scale models (the scaling coefficients for the free parameters).
M,t
For example, consider 7, = 7 . Because we know the value of this dimensionless parameter must be the same
h
in the full- and reduced-scale models, we must have
M nrstas _ Myrgtes
Pyrs Pus
Dividing both sides by the product of powers of the relevant free governing parameters in the reduced-scale
model and by the value of the bound parameter in the full-scale model (Mj,;s) yields
S)
Mars _ tas
My rs (EE
Pas
But each of the ratios above is something familiar. The ratio on the left represents the factor by which the bound
parameter My is scaled in the full- and reduced-size models. The ratios on the right are exactly the ratio between
full- and reduced-scale magnitudes for each of the free governing parameters - the scaling coefficients such as ¢
we chose in section 3.3.5.1. For the example, we chose
Pras = €Phes
Purs = €Py es
trs =trs
And thus the ratios above are as follows:
bes 4
ths
20
and
Pues _ 1
P,
h RS
€
Asa result, we can see that the scaling coefficient for the bound parameter M), is
mon (EJ
a)
Pars
Because of the structure of the dimensional matrix, the process we have used here is general, and is applicable as
long as the free and governing parameters are non-zero.
This approach allows us to write down the scaling coefficient for each bound governing parameter by inspection.
Specifically, we can immediately derive the formula for the ratio of a reduced-scale bound parameter to the
corresponding full-scale bound parameter by starting with the product of powers of free parameters occurring in
on the right hand side of a dimensionless variable and substituting in the reciprocal of the scaling coefficient for
each free parameter wherever that free parameter occurs in that expression.
: ‘ , is se : M,t
Consider, for example, the bound governing parameter M), associated with dimensionless parameter 7, = a
h
h,RS
M 1
By the procedure we have outlined, the ratio of will be given by fan =é,
rs (7)
é
By the same token, consider
cP,t
P,
™ =
Cre
Using the same approach, we can write down by inspection the value of the ratio —** :
Ces
21
