A Theoretical Model of Public Response to the Homeland Security Advisory System

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1 A Thortical Modl of Public Rspons to th Homland Scurity Advisory Systm Amy (Wnxuan) Ding Dpartmnt of Information and Dcision Scincs Univrsity of Illinois Chicago, IL Using a diffrntial quation modling approach, this papr xplors th issu of public rspons to, and confidnc in, anti-thrat warnings Th ffcts of anti-thrat warnings and thir associatd public confidnc lvls ar modld as a group of nonlinar diffrntial quations Th analytical solutions of ths nonlinar diffrntial quations ar drivd to show how warning frquncy and th duration of a warning affct public confidnc, and how th ffcts of anti-thrat warnings ar constraind by th dgr of public concrn as th thrat lvl changs Phas plan analysis suggsts that th numbr of warnings for a particular typ of thrat has a thrshold lvl Blow this thrshold, incrasing th numbr of rliabl warnings can improv th crdibility and ffctivnss of th warning systm Howvr, onc th numbr of warnings xcds th thrshold, th gratr th numbr of warnings issud th lss th public rsponds and th lowr public confidnc bcoms Th rsulting graphic rprsntation is an asy-to-undrstand mthod for authoritis to us to issu advisory warnings whil maintaining th public s confidnc in th systm Kywords: Warning thory, nonlinar diffrntial quation, public confidnc, homland scurity scinc 1 Introduction Currntly trrorist attacks ar significant thrats to US homland scurity To protct th public and infrastructur, authoritis usually issu a thrat warning advisory to th public whn thr is a potntial thrat [, 3, 8] Th warning advisory is issud through a fiv color codd systm (s Figur 1) which rprsnts lvls of risk rlatd to a potntial trror attack Each thrat lvl has a corrsponding list of rcommndd actions that th public should tak in ordr to rduc th liklihood or impact of a potntial attack Thrfor, whn a warning advisory is issud, authoritis hop th public will follow th advisoris listd on th DHS (Dpartmnt of Homland Scurity) Citizn Guidanc on th Homland Scurity Advisory Systm wb pag and tak th rcommndd actions [1, 4] For xampl, as of March 8, 006 th country rmains at an lvatd risk, cod yllow, for a trrorist attack Givn this thrat lvl, th public is rcommndd to tak th following twlv actions: JDMS, Volum 3, Issu 1, January 006 Pags Th Socity for Modling and Simulation Intrnational 1) ) 3) 4) 5) 6) 7) 8) 9) 10) Dvlop a family mrgncy plan Shar it with family and frinds, and practic th plan Visit wwwradygov for hlp crating a plan Crat an Emrgncy Supply Kit for your houshold B informd Visit wwwradygov or obtain a copy of Prparing Maks Sns, Gt Rady Now by calling BE-READY Know how to shltr-in-plac and how to turn off utilitis (powr, gas, and watr) to your hom Examin voluntr opportunitis in your community, such as Citizn Corps, Voluntrs in Polic Srvic, Nighborhood Watch or othrs, and donat your tim Considr complting an Amrican Rd Cross first aid or CPR cours, or Community Emrgncy Rspons Tam (CERT) cours Rviw stord disastr supplis and rplac itms that ar outdatd B alrt to suspicious activity and rport it to propr authoritis Ensur disastr supply kit is stockd and rady Chck tlphon numbrs in family mrgncy plan and updat as ncssary

2 Ding Figur 1 Color-codd thrat lvl systm (Sourc: Homland Scurity Advisory Systm, dhspublic) 11) 1) Dvlop altrnat routs to/from work or school and practic thm Continu to b alrt for suspicious activity and rport it to authoritis If th cod yllow alrt (lvatd) appars vry day, it implis th warning advisory is issud vry day This suggsts that th public nds to prform ths twlv actions vry day until th warning is off Th qustion that ariss is whthr ths twlv actions ar bing rviwd and rpatd as ncssary by th public and ar takn vry day givn th continuation of th cod yllow alrt Rcntly w randomly intrviwd 5 housholds living in th Chicago ara and askd 1) whthr thy knw th currnt thrat lvl of th warning advisory, and ) whthr thy prformd all th rquird rcommndations corrsponding to th currnt thrat lvl For qustion 1, nin housholds answrd no, sixtn said thy blivd th alrt was in cod yllow sinc thy did not har of any chang in th lvl of th alrt Rgarding qustion, non of thm prformd th twlv rcommndations listd abov vry day Whn askd why, thy mntiond thy did follow th rcommndations th first tim th warning was issud Howvr, aftr svral months, thy bcam accustomd to th warning and flt that thr was no diffrnc whthr thy compltd all rcommndations or only som of thm Gradually, thy stoppd following th twlv rcommndations vn though th cod yllow alrt was still on Suppos th ffct of a warning advisory is masurd by xamining whthr th public rsponds to th issud warning by taking th rcommndd actions corrsponding to a particular thrat lvl Thn our small survy data sms to suggst that th continuous warnings may not gnrat th dsird ffct in trms of stimulating th public s rspons W know that anti-thrat warnings can hlp sav livs and rduc th costs of potntial disastrs Howvr, warning about trrorist thrats is diffrnt from th familiar warnings about svr wathr Warnings about svr wathr will not chang th occurrnc of th wathr vnt That is, th svr wathr will still occur no mattr whthr a warning is issud or not But th warnings about trrorist thrats may allow trrorist to altr targts, thrby scaping lgal justic whil still causing grav harm Thrfor, issuing an anti-thrat warning may rsult in a chang in th occurrnc of a potntial thrat Additionally, if th potntial thrat dos not matrializ ach tim th warning is issud and no public notic that th warning is ovr is givn, th public may gradually los attntion and ignor ths warnings rsulting in a failur to prform th rquird rcommndations If this occurs frquntly, it may gradually rod th crdibility of th warning advisory systm and public confidnc [7] In this papr, w dvlop diffrntial quations to modl th rlationship btwn warning frquncis and thir associatd public rspons and confidnc Doing so will hlp us undrstand how to prsrv public confidnc in th warning advisory systm whil maintaining its ffctivnss This papr is organizd as follows In sction, w dscrib th problm stting and th modl formulation which quantifis th intraction of th warning rat and th rsulting lvl of public rspons and confidnc For our mathmatical modl, our philosophy is to xamin th qualitativ bhavior of th modl through phas analysis, and to invstigat th quantitativ bhavior through finding an analytical solution of th modl Thrfor, w analyz th qualitativ bhavior of th modl using th prturbation mthod in sction 3 and prsnt analytic solution dvlopmnt in sction 4 In sction 5, w discuss major contributions, limitations of this rsarch and avnus for futur rsarch Problm Stting and Modl Formulation According to DHS, th goal of issuing a warning for th authoritis is to inform diffrnt lvls of govrnmnt agncis to tak appropriat protctiv masurs on on hand, and on th othr hand to alrt th public that thr may b som typ of thrat to th Unitd Stats and that th public nds to tak informd actions [, 46 JDMS Volum 3, Numbr 1

3 A Thortical Modl of Public Rspons to th Homland Scurity Advisory Systm 3, 5] In this papr, w focus on th public s rsponss That is, w do not discuss how diffrnt lvls of govrnmnt agncis ract to a thrat warning Th trm rspons usd in this papr is dfind as th public taking thos rcommndd actions listd in DHS Citizn Guidanc whn a warning is issud If th public dos somthing othr than ths rquird actions, thir rspons is countd as zro bcaus th purpos of th Homland Scurity Advisory Systm (HSAS) is to inform th public and suggst thy prform spcific tasks [] W dfin a thrat vnt as an vnt that can caus th authoritis to issu warning(s) Whn a dcision to issu a warning is mad, it normally covrs th ntir nation or a targt rgion for a priod of tim Th targt rgion is thn in a thrat stat For xampl, sinc its stablishmnt in March 00, th Homland Scurity Advisory Systm (HSAS) national thrat lvl has rmaind at lvatd alrt stat, a cod yllow warning has bn on, xcpt for fiv priods during which th administration raisd it to high alrt whr a cod orang warning was issud W also assum that th public consists of popl who can undrstand th languag usd in th warning mssag aftr haring or rading th warning and that th public is awar that ach thrat lvl has an associatd list of rcommndd actions Our rsarch problm can thrfor b dscribd as follows Givn a thrat stat (i, E, whr E dnots a st of thrat stats; a thrat stat corrsponds to a thrat lvl in HSAS) undr which warnings would b issud for a targt rgion r (i, r R, whr R dnots a st of targt rgions), w modl how th warning rat influncs its ffctivnss in trms of public rspons and confidnc in th targt rgion r 1 Modl Formulation For a particular thrat stat, thr ar a corrsponding maximum numbr of possibl action itms rcommndd by DHS (S Appndix A for a aild list of rcommndations) Th public may incur tim, labor, or montary costs to prform ach rcommndd action itm W us th trm task-load to labl such costs Lik many studis in conomics, w can us mony or tim to masur task-load Bcaus th task-load rquird to complt vry particular action itm may b diffrnt for ach individual, w lt P dnot th avrag taskload ndd for th targtd population to complt all th action itms rcommndd in stat by DHS Not that th subscript max in P rfrs to th maximum numbr of possibl action itms listd for th corrsponding thrat stat W dfin p ( t ) as th amount of th task-load th public complts in rspons at tim t in stat (i, quivalntly th numbr of action itms th public taks) Whn popl undrstand and trust a warning, thy ar mor likly to tak ths rcommndd protctiv actions In othr words, if popl tak action to rspond, it must imply that thy trust th warning information From this point of viw, p ( t ) also indicats public confidnc in th warning systm Also lt th duration of a warning b τ Th unit for τ is any convnint unit of tim such as an hour, day, wk, and so on If th duration of a warning can b tratd as continuously issuing th sam typ of warning signal, w can lt w ( t ) t /t, rprsnting th numbr of warnings issud at tim t in stat Sinc th authoritis control th warning frquncy, th warning is an outsid stimulus to th public Thortically, th authoritis can issu ndlss warnings in stat But intuitivly th duration of warnings cannot last forvr in stat, and th public s tolranc is limitd Sinc thr xists a maximum possibl duration tim of warnings that th public can tolrat in stat, w lt W rprsnt this maximum valu In othr words, it rflcts th maximum numbr of warnings can b issud in stat without roding th ffctivnss of th warning systm and public confidnc Usually whn a warning is issud, it should captur popl s attntion If popl do not raliz thr is a warning about an impnding thrat, thy will not rspond Upon rspons, th amount of th task-load th public can actually complt (i, quivalntly th numbr of action itms th public can actually tak) is also subjct to th public s capability Sinc humans hav physical and mntal limitations, a chang in th amount of th task-load compltd pr unit tim in stat (i, a chang in th numbr of action itms prformd pr unit tim in stat ) is proportional to th availabl capabilitis of th targtd population Lt α b a positiv proportionality constant indicating th dgr to which th public prcivs th warning information; thn w can hav th following diffrntial quation: dp α [ P p ] whr α is labld as th attntion cofficint Now w add to th modl by assuming that th authoritis may incras warning frquncy to indicat th svrity of th thrat stat or to rach thos who ignord th arlir warnings Thn th avrag of th marginal incrmnt in stimulating th public to prform rcommndd actions is also proportional to how many warnings that th public can tolrat in stat That is, Volum 3, Numbr 1 JDMS 47

4 Ding dp w [ W w ] Combining th two quations abov lads to dp P α [ p ] + b [ W w ] w whr β is a positiv constant In addition, to th public a warning is an outsid stimulus Thrfor, whn facing such a stimulus, popl will dcid whthr th impnding thrat is rlvant to thm or if thy ar at risk If rlvant, thy will chck what action itms thy nd to prform and whthr thy hav th ability to complt th rquird task-load Thus, a chang in th numbr of warnings issud pr unit tim in stat is affctd by th availabl capability th public posssss to complt th rquird task-load, as wll as th xtnt to which th thrat is gard to th public s immdiat concrn That is, if popl think thy ar not at risk, thy may not rspond If w lt th nonngativ constant γ b a masur of prcivd thrat risk th xtnt to which th warning is gard to th public s immdiat and rlvant concrns (i, gographical proximity of th thrat to narby rsidnts vrsus thos living in anothr town), thn w hav anothr diffrntial quation: dw γ ( P p ( t ) ) Hr, γ is also trmd as th concrn cofficint Thos at risk ar mor concrnd than thos who ar not Combining all ths quations abov togthr, w hav th following diffrntial quation systm capturing th rlationship btwn warning rat and th public rspons and confidnc dp α [ P p ( t )] + b w ( t )[ W w ( t )] dw γ [ P p ( t )] subjct to th initial conditions w (0) 0, p (0) 0; and nonngativ constants α, β, and γ W would lik to chck th way our modl systm bhavs to undrstand th rlationship btwn w and p with tim t Our gnral approach to achiving this goal is to xamin th qualitativ bhavior of th modl through phas analysis, and to invstigat th quantitativ bhavior through finding an analytic solution of th modl systm (1) () 3 Examining th Qualitativ Bhavior of th Modl Notic that our modl systm is nonlinar du to th prsnc of th w trm in quation (1) It would b intrsting to know whthr th systm bhavs priodically W thn us th prturbation mthod to study th qualitativ bhavior of th systm This tchniqu is spcially usful to invstigat a nonlinar autonomous systm whr it may b vry difficult or prhaps vn impossibl to find analytical solutions [6] 31 Priodic or Nonpriodic Bhavior? dp dw Sinc ( ) + ( ) p w [ α ( P p ) + β w ( W w )] p [ γ ( p )] + P w α < 0, thrfor, w conclud that th systm of quations (1) and () dfins a family of non-closd and nonpriodic curvs for p ( t ) and w 0 This proprty is vry important bcaus it indicats that th public rspons and confidnc, p ( t ), dos not bhav priodically with w ( t ), th numbr of warnings issud 3 Finding Critical Points To xamin th qualitativ bhavior of th modl systm, w nd to find th critical points of th modl systm Th critical points of th modl systm occur whr p ( t ) 0 and w ( t ) 0 Thus, stting th right-hand sids of both quations (1) and () qual to 0, w hav α [ P p ] + β w [ W w ] 0 γ [ P p ] 0 Solving this pair of quations simultanously for p and w ( t ), w obtain two critical points ( p, w ) ( P ) and ( p, w ) ( P, 0) Point ( P, 0) indicats that th public has compltd th rquird task-load in stat whn no warning is issud This may imply that th targtd population is wll traind and alrady at a hightnd stat of radinss Thus no warning is ncssary Howvr, currntly th public obtains official thrat alrts about trrorist attacks only from HSAS That is, (3) (4) 48 JDMS Volum 3, Numbr 1

5 A Thortical Modl of Public Rspons to th Homland Scurity Advisory Systm HSAS is th only mans by which thrat and advisory information is dissminatd From this point of viw, vn if th public is alrady at a hightnd stat of radinss, at last on warning nds to b issud for th purpos of notification Othrwis, th public will not know of th xistnc of a potntial impnding thrat Du to this rason, w will not discuss th cas of ( P, 0) in this papr Thrfor w focus our phas plan analysis on point ( P ) as th singl point of intrst for th systm (1) and () 33 Phas Plan of th Systm Givn th systm of (1) and (), w can rmin th slop of th systm s solution through any point ( p, w ) Graphically, w can draw a short lin of th propr slop through ach of many points ( p, w ) in th p w plan This is calld a dirction fild/map; s Figur A solution curv of th systm is thn tangnt to th dirction lin at ach point through which th curv passs Thus, th dirction fild givs a visual rprsntation of what th family of possibl solution curvs to th systm of (1) and () looks lik Th construction of th dirction fild for a spcific diffrntial quation can b quit tdious to carry out Blow w giv just a brif dscription of how w sktch th phas graph of th systm in th first quadrant in th vicinity of th critical point ( P ) (bcaus w ar only intrstd in th ara whr p 0 and w 0 ) Th p w plan can b dividd into four aras basd on th rlationship btwn p and P, max, as wll as th rlationship btwn w and W ; s Figur a 331 Ara 1 whr p < P, max and w < W In this ara, w hav ( P p ) 0, ( W w ) 0, and w 0 Whn t incrass, w will gt Combining ths two dirctions togthr, w will know that with tim t incrasing, th combination dirction of p and w will b as shown in a singl arrow hr (s Figur b) 33 Ara whr p < P, w > W, p > 0, and w > 0 If th authoritis continuously issu a warning in stat, thn w hav t > 0 and th warning incrmnt > 0 Thrfor, w hav w dw w t Thus, with tim t incrasing, Sinc > 0 th changing dirction of dw > 0, w t is from quation () w can obtain that γ ( P p ( t ) ) > 0 This implis that ( P p ( t ) ) > 0 must xist bcaus γ is positiv Suppos that w ( T ) W at tim t T, w hav p ( T ) P This mans that popl compltd th rquird task-load whn t T Now whn t T + t, w hav w ( T + t) W + w, which is gratr than zro Sinc dw w ( T + t) w ( T ) w t t > 0, from quation () w can infr that γ [ P p ( T + t) ] > 0 holds This implis that it must b [ P p ( T + t) ] > 0 bcaus γ is positiv Thus, w hav p t [ α +β ( P p ) w ( W w ) ] 0 dp p ( T + t) p ( T ) t p ( T + t) P t < 0 So, p will incras with tim t and th dirction of p t will b Similarly, w can hav w t [ γ ( P p ) ] 0, indicating that w also incrass with tim t Thrfor, th dirction of w t will b Thrfor, with tim incrasing th changing dirction of p is Th combination dirction of p and w will b as shown in a singl arrow (s Figur b) Figur c displays th trajctoris in th vicinity of th critical point ( P ), whr arrows indicat th dirctions that solutions to p mov as w incrass Sinc th trajctoris ar nonpriodic and non-closd curvs, thy show that, undr th Volum 3, Numbr 1 JDMS 49

6 Ding (a) Th First Quadrant: Four possibl situations (b) Short lin sgmnts and dirction fild (c) Phas graph Figur Phas portrait of th systm Th phas graph shows how p bhavs with varying w with tim incrasing givn a particular stat p ( t ) incrass whn w incrass in rgion I of Figur c This indicats that incrasing th numbr of warnings will stimulat th public to tak th rcommndd actions, which in turn hlps build public confidnc in th warning systm But, whn w xcds W, p ( t ) will dcras with w( t ) (s rgion II) That is, onc w ( t ) > W, p will chang dirction immdiatly and ntr into rgion II Whn this happns, th public s nthusiasm for rspons dcrass as th numbr of alrts incrass This implis that th public rspons will gradually dcras to th point of inaction if too many similar alrts of th sam typ of thrat ar issud Similarly, in rgion III of Figur c, incrasing w ( t ) will lad to th dcras of p ( t ) In othr words, if th numbr of warnings is abov W, th public rspons and confidnc dcrass with tim t assumption of th systm (i, p and w 0, p P, max ), p ( t ) incrass whn w incrass in rgion I of Figur c This indicats that incrasing th numbr of warnings will stimulat th public to tak th rcommndd actions, which in turn hlps build public confidnc in th warning systm But, whn w xcds W, p ( t ) will dcras with w( t ) (s rgion II) That is, onc w ( t ) > W, p will chang dirction immdiatly and ntr into rgion II Whn this happns, th public s nthusiasm for rspons dcrass as th numbr of alrts incrass This implis that th public rspons will gradually dcras to th point of inaction if too many similar alrts of th sam typ of thrat ar issud and it will rsult in dcrasd public confidnc in th warning systm Similarly, in rgion III of Figur c, incrasing w ( t ) will lad to th dcras of p ( t ) In othr words, if th numbr of warnings is abov W, th public rspons and confidnc dcrass with tim t Th solution trajctoris of th systm support th vidnc from our small sampl survy which indicats that continuously issuing th sam 50 JDMS Volum 3, Numbr 1

7 A Thortical Modl of Public Rspons to th Homland Scurity Advisory Systm typ of thrat warnings may not gnrat th dsird ffct in trms of stimulating th public rspons Th phas plan suggsts that W for ach vnt stat plays an important rol in public rspons and confidnc W ar intrstd in how an appropriat W for ach vnt stat can b rmind so that th propr numbr of warnings can b issud without roding th ffctivnss of th warning systm or th lvl of public confidnc in it To answr this, w invstigat th quantitativ bhavior of th modl blow 4 Invstigating th Quantitativ Bhavior of th Modl W can rwrit quations (1) and () as dp dw b W w b[ w ] γ[ P p ] α γ Obviously, quation (5) is nonhomognous bcaus th right-hand sid is not qual to zro Sinc th gnral solution to th nonhomognous quation is th sum of th complmntary solution and any particular solution, w nd to find th complmntary solution to th associatd homognous quation of quation (5) Th associatd homognous quation of quation (5) can b obtaind by stting th lft-hand sid of quation (5) qual to zro That is, dp dw b W w b[ w ] γ[ P p ] 0 (5) (6) 3 γ[ p ] 6 γ P [ p ] Solving for p(t ), p ( t ) P 3 b[ w ] 3 b W [ w ] b b ± P + [ w ( t )] W [ w ( t )] 3γ γ 3 Sinc 0 p P, w tak a sign whn rmining to hav a fasibl solution Thrfor, p ( t ) P, max b 3 b P + [ w ( t )] W [ w ( t )] (9) 3γ γ Equation (9) dfins th solution trajctoris of quation (6) in phas plan Lt S c dnot quation (9) W borrow mthods for solving constant-cofficint non-homognous linar quations (s Giordano and Wir [6]) and assum that this approach can b applid in a nonlinar situation Thn, th gnral solution to th systm (5) will b th sum of th gnral solution of quation (6) and any particular solution of quations (1) and () Sinc th two critical points ar also two particular solutions to th systm of (1) and (), a format of th gnral solution to our modl systm can b writtn as Or p w S c + p w S c + P W P 0 (8) (10) (10 ) Extnding quation (6) givs us Notic that Sc is givn implicitly, and thr is no gnral solution procdur for solving nonlinar γ[ P p ] dp quations Thus, in appndix w provid an (7) approximation approach to find th closd form of th [ b W w ] dw b [ w ] dw gnral solution to our modl systm Solving quations (9) and (10) in th vicinity of th Intgrating both sid of quation (7) from 0 to t yilds critical point ( P ), that is, lt p ( t ) P, w will gt γ γ p (0) [ p ] + γ P [ p ] + [ γ P p (0)] 3 P b b + [ w ( t )] W [ w ( t )] 0 3 b bw 3 b w (0) bw 3γ γ [ w ] + [ w ] + [ w (0)] 3 3 Bcaus w 3 ( t ) W at th point ( P ), b bw 3 b w (0) bw [ w ] + [ w ] + [ w (0)] w substitut it into th abov quation and obtain th 3 3 following constraint rlation: Substituting th initial conditions w (0) 0 and p (0) 0 into th abov quation givs us 3 W 3γ b P (11) Volum 3, Numbr 1 JDMS 51

8 Ding Thortically, aftr th authoritis dcid th dgr of risk (γ) rgarding th thrat vnt and th suggstd rcommndations thy xpct th public to tak in stat, th authoritis can us quation (11) to calculat th corrsponding W, th maximum numbr of allowabl warnings that can b issud in stat 5 Discussion and Conclusions Effctiv anti-thrat warnings can hlp sav livs and rduc th costs of potntial disastrs Howvr, if th potntial thrat dos not matrializ ach tim th warning is issud and no public notic that th warning is ovr is givn, if th warnings ar issud too frquntly, or if a warning lasts forvr, th public may gradually ignor ths warnings rsulting in no prformanc of th rquird rcommndations as suggstd by our small survy data If this occurs frquntly, it may rsult in failur to rspond in ral mrgncis, lik th boy who crid wolf In this papr w construct a nonlinar diffrntial quation modl to undrstand how warning frquncy impacts its ffctivnss in trms of public rspons and confidnc W modl th ffctivnss of a warning advisory in trms of whthr th warning (1) capturs th public s attntion, () is gard to th public s immdiat and rlvant concrns (i, dgr of risk), and (3) prompts popl to follow th advisoris listd in th guidlins corrsponding to ach particular thrat lvl and tak th suggstd actions A warning should stimulat popl to tak informd actions So whn a warning is issud, if popl tak action to prform th rcommndations suggstd by DHS, it must imply that thy trust th warning Th qualitativ bhavior of our modl systm prdicts that th numbr of warnings for a particular thrat stat issud without roding th ffctivnss of th warning systm and public confidnc has a thrshold lvl Blow this thrshold valu, incrasing th numbr of rliabl warnings can improv th crdibility and ffctivnss of th warning systm Howvr, onc th numbr of warnings xcds th thrshold, th gratr th numbr of warnings issud th lss th public rsponds and th lowr public confidnc bcoms Whn this happns, th impact of warnings will hav th opposit ffct of th on xpctd If too many alrts in th sam thrat stat ar issud or if a singl warning is rpatd for a long tim, th public will gradually los attntion and ignor thm Our modl s prdiction is consistnt with our small survy rsult In addition, a formula for calculating a thrshold of warnings for ach thrat stat is drivd It suggsts that th thrshold valu is constraind by th prcivd risk of thrat and th public s capability to rspond Bcaus thrats do not always matrializ, it is xpctd that th authoritis giv aild xplanations for any warning for which th thrat provs fals in ordr to prvnt th dvlopmnt of public distrust This incrass flxibility in th numbr of ffctiv warnings th authoritis can issu Not that our modl is formulatd basd on th stting dscribd in sction givn a stat (i, a thrat lvl) undr which warnings ar issud That is, th modl is thrat-stat spcific For xampl, in an lvatd risk stat, th authoritis issu a cod yllow warning Howvr, whn in a high risk lvl, a cod orang warning is issud So, cod yllow and cod orang rprsnt two warning signals and ar in two diffrnt stats: th lvatd risk stat and th high risk stat, rspctivly W us a subscript to indicat diffrnt individual stats (or thrat lvls) Thrfor, can b lvatd risk stat, high risk stat, svr risk stat, and so on As in th study of mor gnral dynamic systms (i, in conomic, biological, or lctrical aras, and so on), w us a continuous systm approach (i, th systm is in th form of a st of diffrntial quations) to study human social bhavior in this papr Howvr, a continuous social bhavior somtims may b intrruptd by a suddn catastrophic non-continuous vnt If this occurs, our modl systm would undrgo a suddn switch from on quilibrium stat to anothr, as shown in Figur 3 Hr, an outsid forc may caus th systm to mak such a suddn jump For xampl, suppos that our modl systm is in a particular stat (g, th thrat lvl is low risk) Now if th warning issuing authoritis suddnly rais th thrat lvl from low risk to lvatd risk du to th vnts of Sptmbr 11 (whr cod yllow is issud), th modl systm would switch to a nw stat labld as lvatd risk or cod yllow In this nw stat (i, lvatd risk stat), th systm of quations (1) and () displays how th public rspons bhavs with varying warning rats Hr, th authoritis raisd thrat lvl, as an outsid forc to th public, maks th systm jump from th low risk stat to th lvatd risk stat Sinc th public dos not know how th authoritis rmin th lvl of th risk stat, if w assum such a rmination is basd on a numbr of factors unknown to th public but known by th authoritis, w can us a function G ( v 1, v,, v m ) to rprsnt such a dcision, whr v1, v,, v m ar various factors Thn w can conclud that our modl systm would hav a suddn stat jump/switch whn th function G is suddnly addd to th systm In this papr w focus on public rspons and confidnc Futur rsarch could b xtndd to study how govrnmnt agncis rmin: (1) th chang of thrat stat, that is, th xplicit quantitativ 5 JDMS Volum 3, Numbr 1

9 A Thortical Modl of Public Rspons to th Homland Scurity Advisory Systm Figur 3 Th systm s switching bhavior du to a suddn catastroph Figur 3a shows that th systm is in a particular stat (g, th thrat lvl is stat 1) Now, th authoritis suddnly rais th thrat lvl to stat, this maks th systm undrgo a suddn switch from stat 1 to stat as shown in Figur 3b xprssion of th function G ( v 1, v,, v m ) as suggstd in [5]; () what typ of action itms and how many action itms th public in diffrnt targt aras nd to prform in ach thrat stat sinc popl living in diffrnt gographic aras may hav diffrnt rquirmnts; and (3) whn a public notic that a warning is ovr should b issud to improv th dsign of th homland scurity advisory systm itslf Our modl also has som limitations Th attntion cofficint, α, is tratd as a constant in this papr In th ral world, it may vary with tim t bcaus th human body and mind may bcom fatigud ovr tim Thus, th attntion cofficint α can b modld as a function of tim Also constants β and γ may b rmind through a sampling survy or historical data Our rsarch could provid thortical foundations for such mpirical studis 6 Rfrncs [1] Citizn Guidanc Citizn guidanc on th Homland Scurity Advisory Systm; 001 Availabl from: gov/dhspublic/ [] CRS Homland Scurity Advisory Systm: possibl issus for congrssional ovrsight Washington, DC: Congrssional Rsarch Srvics; 004 Jan 9 [3] DHS Th Homland Scurity Advisory Systm; 001 Availabl from: [4] Fdral Guidanc Guidanc for fdral dpartmnts and agncis; 001 Availabl from: http//wwwdhsgov/ dhspublic/ [5] GAO Homland Scurity risk communication principls may assist in rfinmnt of th Homland Scurity Advisory Systm Th US Gnral Accounting Offic rport on th Homland Scurity Advisory Systm; 004 Availabl from: wwwgaogov/cgi-bin/gtrpt!gao t [6] Giordano FRir MD Diffrntial quations: a modling approach Rprintd with corrction Addison-Wsly Publishing Company; 1994 [7] McCarthy M Building an nduring capability for homland scurity scinc and tchnology Rmarks in th first DHS Tchnology Confrnc; Boston, MA; 005 [8] PPW, Th Homland Scurity Advisory Systm: thrat cods & public rsponss PPW tstimony bfor th Hous Subcommitt on National Scurity, Emrging Thrats and Intrnational Rlations; 004 Availabl from: www partnrshipforpublicwarningorg Volum 3, Numbr 1 JDMS 53

10 Ding Appndix 1 DHS Citizn Guidanc (Availabl onlin at Citizn Guidanc on th Homland Scurity Advisory Systm 54 JDMS Volum 3, Numbr 1

11 A Thortical Modl of Public Rspons to th Homland Scurity Advisory Systm Appndix A Form of th Gnral Solution Using Linar Approximation Sinc th critical point ( P ) is a particular solution to th systm of (1) and () and it is th on that w ar intrstd in this papr, lt x p P, and y w W Doing this givs Thrfor, th gnral solution to th linar approximatd systm of th original (1) and () is thn, p k 1 λ1t w k m t m λ P + W 1 + dx x dp dp [ p P ] p p whr k1 k c1 a1 c1 b 1 and m1 m c a c b and α [ x] +b [ y + W ] [ y] dy y dw [ w W ] γ[ P p ] w w γ[ x] Rwriting thm, w hav x t + α x t +b W y t b y t ( ) ( ) ( ) ( ) and y + γ x 0 Sinc (A1) is a nonlinar diffrntial quation, w approximat it with its associatd homognous linar quation Now th systm of (A1) and (A) bcoms That is, x ( t ) α x bw y y γ x max (A1) (A) Acknowldgmnts Th author thanks th ditor and thr anonymous rfrs for valuabl commnts Author Biography Amy (Wnxuan) Ding is an Assistant Profssor in Dpartmnt of Information and Dcision Scincs and Dpartmnt of Computr Scincs at Univrsity of Illinois, Chicago, USA Sh spcializs in mathmatical modling of complx problms such as using diffrntial dynamics to modl issus in businss, nginring, and social scinc Dr Ding also works on analytical philosophy, particularly focusing on mathmatical dscription of natural intllignc and ida gnration in scintific discovry Sh arnd BS and MS dgrs in computr scinc from National Univrsity of Singapor, a MPhil in public polic and managmnt, and a PhD in information tchnology and cognitiv scinc from Carngi Mllon Univrsity x α bwmax x y γ 0 y (A3) Th ignvalus for (A3) ar λ 1, α α ± ( ) +b γ W (A4) Thn th gnral solution to (A3) is x a 1 1 c λ 1 y ( t ) b 1 t a λ t + c b ; whr a1, b1, a, b, c1, and c ar constants Volum 3, Numbr 1 JDMS 55

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