final paper redacted

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( ) 0S tSL L

P t Pe

P P

= +

(1)

ClosedFormAnalyticalSolutionsforthePersonnel

SubsystemPopulationStocks

usinganOpenDynamicSystem

ModelingApproach

Thesubjectstudyfocusesonthedevelopmentofaclosedformanalyticalsolution

tothemodelingofthepersonnelsubsystemofanygeneralviolentnonstateactor

(VNSA). Themodelingapproachfocusesontheincorporationofthesystem

independentparameters,consistingofenvironmentalmetricsdescribingthe

social,economic,resourcescarcityandmoral/religiousfactors,basedupona

weightedlinearformulation,intoasingularmetricthatproportionallyweightsthe

flowofindividualsfromcivilsocietyintoasympathizerstock. Theflowof

individualsfromthesympathizerstockintotheVNSAmembershipstockis

weightedbyasecondindependentparameter,arecruitmentfactor. Analytical

solutionsareprovidedforthetwodependentoutputs,thesympathizerstock

membershipandVNSAmembership,normalizedtothecivilsocietystock

membership. Theregionsofapplicabilityofthedevelopedmodelandits

limitationsarediscussedthroughparametricevaluation.

2009

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I. IntroductionOrganizations that utilize terrorism1 as a method represent a pervasive and persistent threat to

domesticandinternationalsecurity. Asubstantialamountofefforthasthuslybeenexpended,onboth

the part of academic researchers as well as government affiliated agencies, in understanding the

formation of such organizations, the methods that they employ2 and the means by which successful

counterterrorism (CT) strategies can be developed and implemented to mitigate or reduce the risk

presented by such organizations. One important aspect in this relatively broad endeavor is

understanding the mechanisms of recruitment into such organizations, the presence of patterns of

recruitmentandtheroleofenvironmentalfactorsintheprocess. Theprocessisclearlycomplicatedand

aspecificrecruitmentparadigmforanyoneparticulargroupunderconsiderationmaynotnecessarilybe

applicabletoanothergroupunderstudy. Thiscomplexity,however,doesnotprecludetheclassification

andcategorizationofabroadsetofplausiblefactorsthatunderpintheaidingoftherecruitmentprocess

ofindividualsfromnormativecivilsociety intoorganizationsthatutilizeterrorismasamethod(herein

referred to as violent nonstateactor orVNSA). Two generalparadigmshavearisen in the research

literature toexplaintheenvironmentalunderpinningsthat facilitaterecruitment. The firstofthese is

the socioeconomic model in which extant relative social and economic deprivation create an

environment that fosters the recruitment of individuals into VNSAs.3 A similar consideration is the

developmentofcleavagesinsocietalstructurebetweenincumbentgroupsasbeingthemechanismfor

1 There exists no singular definition of terrorism that is ubiquitously accepted. For the purposes of the subject study, the

definitionpromulgatedbyPillar,consistingofthequadripartiteelementsofpremeditation,politicalmotivation,targetingof

noncombatants and subnational or clandestine origin is adopted. See: Pillar, Paul. Terrorism and US Foreign Policy(Washington,DC:BrookingsInstitutionPress,2001),1314.

2Libicki,Martin,PeterChalkandMelanieSisson. ExploringTerroristTargetingPreferences. SantaMonica,California,USA:

RANDCorporation. 2007.3 Gurr,Ted. EconomicFactors. InTheRootsofTerrorism,ed.LouiseRichardson(NewYork:Routledge,2006),85102.


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fosteringVNSAdevelopmentandrecruitment4. Organizations,particularly thosewhosepoliticalgoals

aremotivatedbysecularideologiessuchasMarxismandthatincludeanalterationofanextantnational

orderalongethniclines,suchastheSenderoLuminosoinPeru5,theEuskaditaAskatasunainSpain

6and

the formerLiberationTigersofTamilEelam inSriLanka7exemplify thisview. Thesecond,andmore

recentview,isonthereligiousunderpinningsofVNSAs8. Thisviewhasbecomeparticularlyacuteover

the last three decades with the rise of the fundamentalist Shiite state in Iran coupled with state

sponsorshipofgroupssuchasHizballahandwithanumberofbothnationalandtransnationalgroups

espousingafundamentalistSunni ideology. Oftentimes,thedistinctionbetweenthetwoparadigms is

blurredasbothsocioeconomicfactorsofrelativedeprivationandreligiousunderpinningsareinvolved.

Suchmayevenbethecasewithinanorganizationitselfbasedontheroleofaparticularsetofmembers

withintheorganizationortheirgeographicalsourceoforigin. Anexampleofthisscenarioisthatofthe

globalSalafijihadasdemonstratedbythegroupalQaeda9. Memberswithintheleadershiptiertendtobefromrelativelyaffluentbackgroundsandarerelativelywelleducatedwhileoperatives,dependingon

geographical source of origin, either had similar backgrounds as the leadership tier or came from

backgroundsof

relative

socioeconomic

deprivation.

Anumberofmethodologieshavebeenpromulgatedintheliteraturetonotonlyunderstandbutalsoto

predicttheinception,developmentandterminusofVNSAsaswellasthemethodsthatcanbeemployed

duringthislifecyclebynationalandsupranationalentitiestomitigatetheviolenceaspect,asamethod,

4 Piazza,James. RootedinPoverty?:Terrorism,PoorEconomicDevelopmentandSocialCleavages. TerrorismandPolitical

Violence18,159177,2006.5 Choy, Shawn. In the Spotlight: Sendero Luminoso. Center for Defense Information. July 1, 2002. Online:

http://www.cdi.org/terrorism/sendero.cfm(accessed:October21,2009).6 Bhattacharji,Preeti. BasqueFatherlandandLiberty(ETA)(Spain,Separatists,EuskaditaAskatasuna). CouncilonForeign

Relations. 2008. Online:http://www.cfr.org/publication/9271/(accessedSeptember20,2009).7 Bhattacharji, Preeti. Liberation Tigers of Tamil Eelam (aka Tamil Tigers, Sri Lanka, separatists). Council on Foreign

Relations. 2009. Online:http://www.cfr.org/publication/9242/(accessedSeptember20,2009).8 Fine,Jonathan. ContrastingSecularandReligiousTerrorism. MiddleEastQuarterly15,no.1,5969(2008).

9 Sageman,Marc. UnderstandingTerrorNetworks(Philadelphia,Pennsylvania,USA:UniversityofPennsylvaniaPress,2004),

6980.


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regardstotheindividualmetrics. Finally,thedefinitionincludestheconceptofdiscretizingVNSAsinto

componentsystemsorsubsystems. Onemethodthathasshownpromise inevaluatingsuchcomplex

systemshasbeenthatofdynamicsystemmodeling12. Inthisapproach,thequantityoftheitemunder

consideration ispartitioned intoaseriesof stocksandthetimedependentmovementof itemsfrom

onestocktoanotheraremodeledbyaseriesofflows. Bartolomeietal.13discussed,conceptually,the

view of VNSAs as a system of the subject type. The authors identified four subsystems support,

maintenance,cognitiveandconversionprocessesandmodeledenvironmentalfactorsthroughaseries

offourquotientsthatmeasuredthesocial,economic,resourcescarcityandmoral/religiousfactors. The

authorspresentedamodelforevaluatingtheflowofindividualsfromcivilsocietyintothesympathizer

stockandfromthesympathizerstockintoVNSAmembership. Usingajackknifeddataset,theauthors

predicted the number of members in the Sendero Luminoso organization. Leweling and Sieber14

developedasimilarmodel,buildingonthemodelofBartolomeietal.,toevaluatetheflowofindividuals

leaving VNSA membership. These authors developed three stocks individuals in civil society,

individualsaswardsof thestateand individuals inVNSAmembershipaswellasa setofsix flows to

modelthe

flow

of

individuals

into

and

out

of

the

three

stocks.

Leweling

and

Sieber

15applied

their

model

in analyzing the flow of individuals out of theSalafistGroup for PreachingandCombat (GSPC). The

financialsubsystemofthesamegrouphasalsobeensubjecttoanalysisusingthesamemethodological

approach16. AkcamandAsal17used thesystemdynamicsmethodology tosketchout theprimaryand

secondary (feedback) relationships in modeling groups based on ethnicity that utilize terrorism as a

12Hannon,BruceandMatthiasRuth. DynamicModeling(NewYork:Springer,2001),2742.

13Bartolomei, Jason,WilliamCasebeerandTroyThomas. ModelingViolentNonStateActors:ASummaryofConceptsand

Methods. Institute for Information Technology Applications. United States Air Force Academy, Research Publication 4.November,2004.

14Leweling,TaraandOttoSieber. UsingSystemsDynamicstoExploreEffectsofCounterterrorismPolicy. Proceedings:40thHawaiiInternationalConferenceonSystemSciences. 2007.

15Leweling,TaraandOttoSieber. CalibratingaFieldlevel,SystemsDynamicsModelofTerrorismsHumanCapitalSubsystem:

GSPCasaCaseStudy. StrategicInsights5,no.8. November,2006.16Grynkewich, Alex and Chris Reifel. Modeling Jihad: A System Dynamics Model of the Salafist Group for Preaching and

CombatFinancialSubsystem. StrategicInsights5,no8. November2006.17Akcam, Bahadir and Victor Asal. The Dynamics of Ethnic Terrorism. Proceedings: 2005 System Dynamics SocietyConference. July1721,2005.


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method. The importance of evaluating national security issues, which would include the analysis of

VNSAactivity,ascomplexsystemshasbeennotedbyCarafanoandWeitz18.

II. ObjectivesWiththisbackgroundandintroduction,onecannowconsidertheobjectivesofthesubjectstudy. From

thegeneralperspective,thefollowingresearchquestionisaddressed:Cantherecruitmentsubsystemof

aVNSA,viewed in theparadigmaticconstructsense,bemodeledanalyticallyusing theopensystems

dynamics approach? In this regard, the term analytical references the reduction of the recruitment

subsystemintoamathematicalrelationshipbetweenthesystemindependentanddependentvariables.

Theapproachtakenherein is inpartdescriptiveand inpartexplanatoryandfinallyprovidesageneral

frameworkforpredictiveanalysis. This isexemplifiedbythefollowingspecificresearchquestion:Will

theBartolomeietal.frameworkofutilizingthefourenvironmentalfactors(social,economic,resource

scarcity and moral/religious)anda measureof recruitmentefficacyproduceananalytically reducible

modelfortheanalysisofVNSAs? Inansweringthisquestion,thefollowinghypothesesareproposed:

Hypothesis1: An analytically reducible result exists for both the sympathizer stock and VNSAmembership stock when the effect of social, economic, resource scarcity and

moral/religious metrics are modeled as a linear weighted combination producing a

single metric and the recruitment factor is modeled as a proportional weighting of

individualsmovingfromthesympathizerstockintotheVNSAmembershipstock.

18Carafano,JamesandRichardWeitz. ComplexSystemsAnalysisANecessaryToolforHomelandSecurity. BackgrounderNo. 2261. The Heritage Foundation. April 16, 2009. Online:

http://www.heritage.org/Research/HomelandSecurity/bg2261.cfm(accessedApril20,2009).


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Hypothesis2: A linear modeling approach will produce closed form analytical solutions that will besubjecttocertainlimitationsbaseduponthemodelingapproach.

III. AnalyticalMethodsThe previously cited literature, in which the systems dynamics modeling approach was utilized to

evaluate the personnel subsystem, contained certain ambiguities. The analytical method used to

generateasingularmetricfromtheincipientcomponentsofeachenvironmentalfactorwasnotdefined.

The definition of the terms population disaffection and recruiting factor in terms of what each

meantinregardstotherelevantstockofoperationremainedundefined. Finally,thelimitationsofthe

model,arisingfromtheconstructionofthemodel,werenotdiscussed. Instead,modelvalidationwas

basedondata fitting. Theapproach taken in the subject study was to developamodel forwhicha

closed form analytical solution could be determined. This, in turn, required an explicit statement

regardingtherelationshipsbetween incipientenvironmentalmetricsandtheresultingglobalmetricof

populationdisaffection. This termaswellas therecruiting factorweredefined in regards to the

populationstocksuponwhichtheyoperated. Modelgenerationwasaided,conceptually,withtheuse

of the Berkley Madonna (v.8.3.1.4; Berkley, California, USA) system dynamics software and certain

derivationswereeitherconductedorvalidatedusingtheMathematica(v.5.0;Champaign,Illinois,USA)

symbolicmathematicssoftwarepackage.

Eachofthefourquotients,aspresentedbyBartolomeietal.arepositivelyvaluednotonlyintermsof

theirmagnitudebutalsointermsoftheirunderlyingrepresentation. Forexample,theDeweyquotient,

ameasureofthesocialenvironment,isbasedonthetimehistoriesoffreedomofmovement,liberalism,

freedomof speechand freedom of the press. The aprioriexpectation is that societies with greatermagnitudes,undertheperiodunderstudy,foreachofthesevariableswouldbelesslikelytoproducean


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environmentthatfostersthegrowthofVNSAs. Fromthemodelingperspective,andonethatisclearly

dependentonthemodeler,apositivecorrelationissoughtbetweentheenvironmentalquotientsunder

consideration and the scaling of the population at large. Thusly, it is proposed that each quotient

represent the relative negative factors that foster VNSA development. For the Smith quotient, one

would be concerned with the opposite of the Free Market index, a lack of trade, unemployment,

excessivegovernment,a lackof infrastructureandthepresenceofaninformationinfrastructure. This

last input is included intheformproposedtoaccountforthespreadofVNSAideologythroughmedia

sourcesandespeciallytheinternet. FortheMaslowquotient,onewouldbeconcernedwiththeinfant

mortalityrate,lackofpolicemaintenance,alackofsocialcapitol,alackofeducationallevel,foodwater

unavailability,thelevelofalackofmedicalcareandthecrimerateindex. FortheDeweyQuotient,one

wouldbeconcernedwithalackoffreedomofmovement,alackofsocialliberalism,alackoffreedomof

speechandalackoffreedomofthepress. Itisrecognizedthatthefreedomofmovement,speechand

thepressmayalsopositivelycorrelatewiththespreadofVNSAideologyandoperationalcapability. The

Camusquotientisparticularlyinterestingandrepresentsanaspectofthemodelthatisclearlysubjectto

stratification

(as

are

all

of

the

quotients).

Specifically,

one

would

be

interested

in

the

level

of

governmentcorruption,thenumberofreligiousgroups,culturalviolence,thelackofthevalueassigned

tohumanlifeandculturalinhomogeneity. Itshouldbereiteratedthatthechoiceofpositiveornegative

definitions of each variable is solely at the discretion of the modeler and that the forms are readily

interchangeable. Consistency,however,inthemodelingendeavorshouldbemaintained.

Becauseeachquotient,aspresentedistimeinvariant,theimplicitassumptionisthattheenvironmental

factors for a particular temporal period of study, over a time t: ta t tb, remain constant. The

determinationofeachquotient ispredicatedoncalculatinganunderlyingvaluebasedontherelevant

environmental inputs, the determination of themaximum value that theenvironmental inputs could


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produceandfinallytheratioofthetwoterms. Thusly,underlyingvalueisdefinedasI{i:1 i 4},the

maximumvalueasim{i:1 i 4}andthequotientasiq{i:1 i 4}. Thusly,eachquotientisnaturally

constrainedbythefollowingrelationship:

0 1iiqim

= (1)

The relationship between eachof the quotientsand the scaling factorof populationdisaffection will

dependontheparticularsubjectunderstudy. Forthepurposesofthesubjectmodel,itisproposedthat

eachquotientbelinearlyscaledbyaconstantfactori. Linearscalingwithaconstantfactorisproposed

fortworeasons. Thefirstisthatalinearmodelisthesimplestmodelthatcanbeconsideredandthusly

appropriate for the subject first order environmental system model. The second is that a constant

factorischosentobeconsistentwiththetimeinvarianceofeachquotientovertheparticulartemporal

periodofstudy. Thetotaleffectofeachquotientcanbeexpressedmathematicallyas:

ii i iq i

im

= = (2)

The weighting factors are subject to the constraint of summing to unity under the consideration of

mutualexclusivityandencompassmentofthetotalityofenvironmentalfactors. Also,becausethesum

of all quotients multiplied by their respective weighting factors represents a multiplier of the total

population,itmustbebetween0and1. Theserelationshipsareexpressedmathematically,asshownby

equationsError!Referencesourcenotfound.andError!Referencesourcenotfound.,respectively.


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4

1

1ii

=

= (3)

4 4 4

1 1 1

0 1i

i i iq i imi i i

= = =

= = =

(4)

Thetermthusrepresentsthepopulationdisaffection. Thepopulationdisaffection isthemeasureofthenumberofsympathizerstothenumberofthetotalpopulationperunitoftimeunderconsideration.

Theterm{:0 1}isintroducedtoreferencetherecruitmentfactor,whichisthemultiplierofthe

population of sympathizers that are recruited into the VNSA. The recruitment factor represents the

numberofVNSAmemberstothenumberofsympathizersperunittime.

Themodelcontainsfourflowsofindividuals. ThesearePI(showninFigure1asPopulationin)whichis

the time rate of movement of individuals into the general population, PSI (shown in Figure 1 as

Sympathyin),whichisthetimerateofmovementofindividualsintothesympathizerstock,PMI(which

isshowninFigure1asMembersin),whichisthetimerateofmovementintotheVNSAmembership

stockandPMO(which isshown inFigure1asMembersout),which isthetimerateofmovementof

VNSA membership or the VNSA membership rate. The model contains three stocks of individuals.

ThesearePL(showninFigure1asPopulationatLarge),PS(whichisshownFigure1asSympathizers)

and PV (shown in Figure 1 as VNSA Membership). These three stocks represent the number of

individuals in the total population at large, the number of individuals that are sympathizers and the

numberofindividualsthatareVNSAmembers,respectively.


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Figure1. Graphicalrepresentationofthepersonnelsubsystemunderstudy.

The system is characterized by three lineardifferentialequations and twoequations thatgovern the

flows.

LI

dPP

dt= (5)

SSI MI

dPP P

dt

= (6)

VMI MO

dPP P

dt= (7)

4 4 4

1 1 1

iSI L L i L i iq L i

imi i i

P P P P P

= = =

= = = = (8)

MI SP P= (9)

Substitutionofequations(8)and(9)intoequation(6)resultsinthefollowing:


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( )S L SdP

P Pdt

= (10)

Equation(10)isafirstorderlineardifferentialequationinPS. Thesolutionforequation(10)isgivenby

equation(11)andisderivedinAppendixA.

( ) 0t

S L S LP t P P P e

= +

(11)

In arriving at the solution to the number of sympathizers, as shown in equation (11), the total

populationatlargeistakenasbeingaconstant. Inequation(7),theflowofmovementoutoftheVNSA,

forthepurposesofthesubjectmodel,isnonexistentquantityandincludedforbookkeepingpurposes

only. Thusly, PMO is zero. Substitution of equation (11) into equation (9) and substitution of the

resultantintoequation(7)resultsinthefollowing:

0tV

L S L

dPP P P e

dt

= +

(12)

Integratingequation(12)directlyoverthetimedomain:

( ) 0t

V L S LP t P t P P e C

= +

(13)


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If the number of individuals comprising the VNSA is known at the start of the time frame under

considerationsuchthatPV(0)=PV0thensubstitutionintoequation(13)allowsforthefollowingsolution

tobedevelopedfortheconstantofintegrationC.

0 0V S LC P P P

= + (14)

Substitutionofequation(14)intoequation(13):

( ) 0 0 0t

V V S L L S LP t P P P P t P P e

= + +

(15)

Equations (11)and (15)canbenormalized toprovide the time rateofsympathizerdevelopmentand

VNSAmembership,respectively,tothepopulationatlarge.

( ) 0S tS

L L

P t Pe

P P

= +

(16)

( )( ) 00 0

1V tSV S

L L L

P t PP P t e

P P P

= + +

(17)

Equations (16) and (17) thusly represent the governing equations for evaluating the recruitment

subsystemofanyVNSAforwhichthemodelingassumptionsaremet. Thederivationofequation(17)

and(18)servetoshowtheacceptanceofthefirstofthefirsthypothesis.

Because the model does not have an explicit negative feedback loop via a cooption and/or CT

mechanism and because of the modeling assumption of constant environmental factors, the time

duration for application of equations (17) and (18) should be appropriately limited for the modeling


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endeavor under consideration. In examining equation (17)and taking the limit of the equationas

approacheszerorevealsthefollowingsolution:

( ) 0 0lim lim

0 0

S tS S

L L L

P t P Pe t

P P P

= + = + (18)

Equation(18)representsthecaseInwhichthenumberofsympathizerscontinuestoincreaseinalinear

mannerfromtheinitialrelativesympathizertototalcivilsocietypopulation. Examiningequation(18)in

asimilarmannerrevealsthefollowingsolution:

( )( ) 0 00 0

lim lim 1

0 0

V tS VV S

L L L L

P t P PP P t e

P P P P

= + + = (19)

Thisisanexpectedresultinthatitstates,asonewouldexpect,thatwithazerorecruitmentfactorthat

relativenumberofindividualswithVNSAmembershiptothatoftotalcivilsocietyisaconstantequalto

the initial relativevalueand isalso timeinvariant. In furtherexaminingequations (17)and (18),one

seekstodeterminethelimitationsofeachmodelaswellastherelevantconclusionsthatcanbedrawn

fromeachmodel. Itisimportant,onceagain,tonotethatthesubjectmodelisasubsystemmodelalone

and does not include the negative feedback loops that would be associated with even a general CT

subsystemmodeladdition. Withthiscaveatinmind,onefirstconsidersequation(17)ingreaterdetail.

Itisclearfromtheformofequation(17)thatcasesinwhich>,willprovideforaphysicallyincorrect

answer ifthesystem isevaluatedforasufficientperiodof time. Therateofchangepredicted in the

relativenumberofsympathizerswillalsobegreaterthanexpected. Anexampleofthiscaseisshownas

Figure2a. Thisconclusionisindependentoftheinitialrelativenumberofsympathizersandtheactual

valuesofeitherthepopulationdisaffectionorrecruitmentfactor. Asecondlimitingcaseisthatinwhich


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=. Inthiscase,withsufficientenoughtime,thepredictedrelativenumberofsympathizerswillreach

unity and remain constant. This conclusion is also independent of the initial relative number of

sympathizersandtheactualvaluesofeitherthepopulationdisaffectionorrecruitmentfactor.

Figure2. Thetimehistoryplotoftherelativenumberofsympathizersshowninplot2Aisanunrealistic

casebasedon>(PSO/PL=0.00001,=0.15,=0.11). Thetimehistoryplotoftherelativenumber of sympathizers shown in plot 2B is another limiting case based on > (PSO/PL =0.00001,=0.15,=0.15).

The model does produce reasonable results, without consideration for negative feedback, operating

undertheaforeindicatedassumptions,andovera limited timeframe(withnonegativefeedbackthe

modelwillreachanasymptoticvaluegivensufficientenoughtime)when


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Figure3. Thetimehistoryplotof therelativenumberofsympathizersunder theconditionsof>(PSO/PL=0.00001,=0.11,=0.15).

There isanothermannerofviewingtheresultspredictedbyequation(17). Rewritingtheequation in

thefollowingmanner:

( )( )1 2 1

S t

L

P tc c c e

P

= + (20)

( )1 3

S t

L

P tc c e

P

= + (21)

Becausethephysicallimitsonc1are0 c1 1andthelimitsonc2are0 c2 1thelimitsonc3are 1 c3

1. Whenoneaccountsforthelimitsbasedonthepreviousanalysis,thefunctionallimitsforc3based

onavalueofc1arec1 c3 0. Thismethodofpresentingtheresultsnotonlyaids inexhibitingthe

fundamentalrelationshipsbetweenthetermsinvolvedbutalsoreducesthesolutiontoonethatcanbe

visualizedinthreedimensionalformbaseduponthedomainofandt. Anexampleofthisisshownin

Figure4.

20 40 60 80 100Time Hyears L

0.1

0.2

0.3

0.4

0.5

0.6

0.7

PsHtLPL


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Figure4. Plotoftherelativenumberofsympathizersasafunctionof{0 1}andt:{0 t 100}for

c1=0.05andc3= 0.03.

In evaluating equation (18) one does not see the same set of limitations in predicting the relative

populationofVNSAmembersaswithequation(17) inregardstopredictingtherelativepopulationof

sympathizers. However,becausetheanalysisissequential,thelimitationsonequation(17)areimplicit

indefiningtheappropriaterangeofvaluesonequation(18). AnexampleofthisisshowninFigure5.

Figure5. Timehistoryplotof therelativenumberofsympathizersunder theconditionsof>(PL=106,PSO=0.5,PVO=0.5,=0.005,=0.0005).

0

0.25

0.5

0.75

1

0

20

40

60

80

100

Time Hyears L

0.02

0.03

0.04

0.05

PsHtLPL

0

0.25

0.5

0.75

20 40 60 80 100

Time Hyears L

0.05

0.1

0.15

0.2

0.25

PVHtLPL


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Amoregeneralevaluationofequation(18) issomewhatcomplicatedgiventhatthesingularequation

containsfoursystemvariablesandtimeasthefifthvariable. Thisevaluationisapproachedbylookingat

the limits of each parameter to determine a reasonable range over which the subject model can

reasonablybeapplied. TheconstraintsonthenormalizedinitialpopulationofsympathizersandVNSA

membersisreadilyapparent.

0 1SO

L

P

P (22)

0 1VO

L

P

P (23)

0 1VO SO

L

P P

P

+ (24)

Bytakingthe limitofequation(18),asboththe initialrelativepopulationtermsgotowardszero,and

theVNSArecruitmentrate(i.e.thepercentageofthosethataresympathizersthatarerecruitedintothe

VNSA)approachesunity,therelativenumberofVNSAmemberstothepopulationatlarge,asafunction

oftime,reducestothefollowing:

( ) ( )1V tL

P tt e

P = + (25)

Thisequationcanbeevaluatedforthefulldomainof{0


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Figure6. TimehistoryplotforrelativenumberofVNSAmembersundertheconditionsof>(relativesympathizerandVNSAmembershipstartingatzeroand=1).

Takingthesamelimitsontheinitialrelativepopulationtermsandevaluatingthelimitoftherecruitment

factorasitgoestozeroproducestheexpectedresultofzeroVNSAmembership. Theupperlimitofthe

normalizedinitialsympathizergroupcanalsobeevaluated. Asthisvaluereachesunity,thevalueofthe

normalizedinitialVNSAmembershipgroupgoestowardzeroasperequation(25). Withtherecruitment

factorset tozero,equation(18)reduces tounityanunrealisticsolution. Alsowith therecruitment

factorsetto>0,anunrealisticsolutionispredictedwiththenormalizedVNSAmembershipexceeding

unity. When the upper limit of the normalized initial sympathizer population is evaluated, the

normalizedinitialnumberofsympathizersiszerobyequation(25). Therecruitmentfactor,atitslower

limitofzero,producesazeronormalizedVNSAmembershipvalueinunrealisticsolution. Attheupper

limit,equation(18)producesanincorrectresult,undertheseconditions,fortimet=0. Outsideofthese

limits,themodeldoesproduceareasonableprediction,onaprimafaciebasis,forthenormalizedVNSAmembershippopulation. AnexampleofthisisshowninFigure7.

0

0.25

0.5

0.75

1

Time Hyears L

0

0.25

0.5

0.75

1

0

0.1

0.2

0.3

PVHtLPL

0

0.25

0.5

0.75Time Hyears L


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Figure7. TimehistoryplotfortherelativenumberofVNSAmembersundertheconditionsofPVO/PL=

0.003,PSO/PL=0.0005.=0.04,0


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membershipincreaseasafunctionofthesystemindependentparameterstheenvironmentalfactors

describing the social, economic, resource scarcity and religious/moral conditions within an area of

operationandthedegreeofrecruitmentefficacy. Bydevelopingananalyticalframework,itwasreadily

determinablethatthemodelhadcertainphysical limitations. Namely,themodelproducedphysically

infeasible results near the extremes and that the steady state predictions were infeasible when the

recruiting factor () exceeded the population disaffection metric (). An understanding of such

limitations is crucial, beyondjust the case of simple data fitting, in evaluating the credibility of any

modelingendeavor.

Thesubjectapproachisclearlyafirststepinthedevelopmentofacomprehensivemodelingapproach

forVNSAs. ItbroachedjustoneofthefoursubsystemsneededtomodeltheVNSA itselfandmadea

numberofsimplifyingassumptionsregardingtheinteractionoftheVNSApersonnelsubsystemandthe

environment. Onesuchsimplificationwasthetemporalinvarianceoftheenvironmentalfactors. Such

temporalinvariancewillonlybevalidovershortperiodsoftimeandcanalsobesubjecttomodification

by the VNSA itself if the VNSA is utilizing a multidimensional approach to increase sympathy for its

cause19. Thecurrentmodelingapproach,however,readilyallowsfortheaccountingofsuchfeedback

andmodificationloops. ForthepreviouslynotedSenderoLuminosomodelofBartolomeietal.andthe

GSPCmodelofLewelingandSieber,itwasnotedthat inputdata,whenprovided,wasgenerallyinthe

formoftabulartographicdataorintheformofanormalstatisticaldistributionwithanonzeromean

and standard deviation. One of the benefits of using extant software (e.g. the Berkley Madonna

software package or the STELLA software package) is that the governing equations are evaluated

numerically. The governing differential equations form an initial value problem (a mathematical

constructubiquitouslypresentacrossmanyfieldsofstudy)thatareintegratednumericallyateachtime

19Magouirk, Justin. The Nefarious Helping Hand: AntiCorruption Campaigns, Social Service Provision, and Terrorism.

TerrorismandPoliticalViolence20,356375,2008.


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21|P a g e

step,typicallyusingafourthorderRungeKuttaprocedure,toarriveatthesolutionsforthenexttime

step. Whilethisapproachallowsfortheaccountingofinputvariablesinafarmorediverseformthan

that which can be accounted for in the closed form analytical procedure, the latter can be used to

evaluate the system at the mean input values and furthermore allows for a parametric analysis

procedure,asundertakeninthesubjectstudy,todeterminethemathematicalvalidityandlimitationsof

the modeling approach. Another salient issue of concern, when modeling a specific VNSA, is the

availability of actual data for populating the model inputs. This concern, as noted by Leweling and

Sieber20,however,isminimalinthatnationallevelagenciesarelikelytoalreadypossesssuchdata. Full

knowledgeofthevaluationofthesysteminputparametersisnotneededasthedegreeofuncertainty

canbedirectlyencodedintotheanalysis21orthemodelcanberefinedorbroadenedtothedegreefor

which particular subsystem knowledge exists. Regression analysis can serve as a useful tool for

determining the strength or weakness of particular pathways, based upon a general model, that are

mostsalientforthespecificVNSAunderstudy22.

Thecurrent

modeling

approach

also

shows

promise

in

being

able

to

incorporate

parts

or

all

aspects

of

other analytical approaches utilized for modeling the various subsystems of VNSAs. This includes

previousapproachesbaseduponagentnetworkanalysis23

aparticularly importantaspectofkinship

andfriendshiprelationsintherecruitmentprocessfortheglobalSalafijihadmovement24,gametheory

25

20Ibid14.

21Carley, Kathleen. Dynamic Network Analysis. Institute for Software Research International. In Summary of the NRCWorkshoponSocialNetworkModelingandAnalysis,eds.RonBreigerandKathleenCarley. NationalResearchCouncil. 2003.22Robinson,Kristopher,EdwardCrenshawandJCraigJenkins. IdeologiesofViolence:TheSocialOriginsofIslamistandLeftist

TransnationalTerrorism. SocialForces84,no.4,20092026,2006.23Epstein,Joshua,JohnStreinbrunerandMilesParker. ModelingCivilViolence:AnAgentBasedComputational

Approach. CenteronSocialandEconomicDynamics. WorkingPaperNo.20. 2001. AlsoBerry,Nina,TeresaKo,Marinna Lee, Marc Pickett, Ben Wu, Timothy Moy, Julianne Smrcka and Jessica Turnley. Computational Social Dynamic

ModelingofGroupRecruitment. SandiaNationalLaboratories. 2004.24Ibid8.

25 de Mesquita, Ethan. Conciliation, Counterterrorism and Patterns of Terrorist Violence. InternationalOrganization59,145176,2005. AlsoChen,YiMing,DachrahnWuandChengKuangWu. AGameTheoryApproachfor


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andothermodelswritteninadifferentialequationsframework26. Themethodalsoallowsfortheready

translationofqualitativeassessments27intoananalyticalmodelingframework.

There is substantial room for futureworkon this topicand modeling approach. The removal of the

systemparameter limitationsdiscussedabove represent the firstapproachasdoes the development

andincorporationoftheothersubsystems. Withtheextantmodelitself,theincorporationofexisting

numericalandanalyticalmodelingapproachesfromeconomic,socialandpoliticalanalyses is feasible.

Also,themodelingapproachcanbereadilystratifiedtoaccountforthedifferencesintheenvironmental

factors of leaders, members and support personnel as well as to account for the environmental

differences that are present for transnational organizations based on geographical location of

leadership,geographicallocationofrecruitmentandgeographicallocationoftargetselection.

Evaluating Terrorist Threats and Deploying Response Agents in Urban Environments. Journal of Homeland Security andEmergencyManagement6,no.1,125,2009.

26Faria,Joao. TerroristInnovationsandAntiTerroristPolicies. TerrorismandPoliticalViolence18,4756,2006. AlsoFaria,JoaoandDanielArce. TerrorSupportandRecruitment. DefenseandPeaceEconomics16,no.4,263273,2005.

27 Haimes, Yacov and Thomas Longstaff. The Role of Risk Analysis in the Protection of Critical Infrastructures Against

Terrorism. RiskAnalysis22,no.3,439444,2002. AlsoPost, Jerrold,KevinRubyandEricShaw. TheRadicalGroup inContext:1. AnIntegratedFrameworkfortheAnalysisofGroupRiskforTerrorism. Studies inConflict&Terrorism25:73100,2002. AlsoPost,Jerrold,KevinRubyandEricShaw. TheRadicalGroupinContext:2. IdentificationofCriticalElements

intheAnalysisofRiskforTerrorismbyRadicalGroupType. StudiesinConflict&Terrorism25,101126,2002.


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V. AppendixARepeatingequation(10)fromthebodyofthepaperasequation(A1)ofthesubjectappendix:

SS L

dPP P

dt + = (A1)

Because equation (A1) is first order linear in Ps, the equation can be solved analytically28. The

integratingfactorforequation(A1)canbedefinedas:

dt te e = (A2)

Multiplyingequation(A1)byequation(A2):

t t tSS L

dPe e P e P

dt

+ = (A3)

The terms on the left side of the equality simply represent the differentiation by parts of the term

shownontheleftsideofequation(A4).

( )t tS Ld

e P e Pdt

= (A4)

Integratingbothsidesoverthetimedomain:

28Zill, Dennis. A First Course in Differential Equations withApplications (Boston, Massachusetts, USA: PWSKent PublishingCompany,1989),6668.


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( )t t tS S Ld

e P dt e P e P dt dt

= = (A5)

Integratingtherightsideoftheequality(notingthattheintegralisimplicit):

t tS Le P P e C

= + (A6)

ThetermCrepresentsaconstantofintegrationsecondarytotheimplicitnatureoftheintegral. Ifthe

initialpopulationof sympathizers isknown such that PS(0)= PS0 then substitution into equation (A6)

resultsinthefollowingsolutionforC.

0S LC P P

= (A7)

Substitutionofequation(A7)intoequation(A6)andsolvingforPS:

0t t

S L S Le P P e P P

= + (A8)

( ) 0t

S L S LP t P P P e

= +

(A9)

final paper redacted

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