Abstract
Modeling terrorism is both necessary and difficult. While the necessity comes from the all too obvious real-world pressures our society is facing, the difficulty stems from the underlying complexity of the phenomena itself – there are many variables to account for, they are hard to measure, and the relationships between them are confounding. Since modeling terrorism is at its most onerous when it comes to predicting specific attacks, their timing and scale, we opted to work around this using observed probabilistic distribution and integrate power laws into our system dynamics model. After evaluating thousands of simulations runs, this allows us to replicate historical data as well as produce prognostic scenarios, while maintaining what we believe to be authentic behavior. Compromises need to be made, but we believe that this approach can be useful for systems highly dependent on events or parameters which we are unable to predict but whose distributions are known.
Funding source: Strategic Support Program for Security Research in the Czech Republic, Ministry of the Interior http://dx.doi.org/10.13039/100009532
Award Identifier / Grant number: VJ01010122
Funding source: Charles University institutional funding http://dx.doi.org/10.13039/100007397
Award Identifier / Grant number: 260 453
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Research funding: This research was supported by the research grant “Resiliency of the armed forces and armed security corps to hybrid threats” (VJ01010122) financed by the Ministry of the Interior of the CR/Strategic Support Program for Security Research in the Czech Republic 2015–2020 (IMPAKT 1).
List of Historical Attacks
| Date | Day n | Killed | Injured | Casualties | Perps | Country | Place |
|---|---|---|---|---|---|---|---|
| 2007-06-30 | 180 | 0 | 2 | 0.2 | 2 | United Kingdom | Abbotsinch |
| 2007-11-12 | 315 | 0 | 1 | 0.1 | 1 | Switzerland | Crissier |
| 2008-12-31 | 730 | 0 | 2 | 0.2 | 1 | Denmark | Odense |
| 2009-03-20 | 809 | 0 | 10 | 1 | 1 | France | Lyon |
| 2009-06-01 | 882 | 1 | 1 | 1.1 | 1 | USA | Little Rock, AR |
| 2009-10-12 | 1015 | 0 | 1 | 0.1 | 1 | Italy | Milan |
| 2009-11-05 | 1039 | 13 | 32 | 16.2 | 1 | USA | Fort Hood, TX |
| 2010-05-14 | 1229 | 0 | 1 | 0.1 | 1 | United Kingdom | London |
| 2010-12-11 | 1440 | 0 | 2 | 0.2 | 1 | Sweden | Stockholm |
| 2011-03-02 | 1521 | 2 | 2 | 2.2 | 1 | Germany | Frankfurt |
| 2012-03-11 | 1896 | 1 | 0 | 1 | 1 | France | Toulouse |
| 2012-03-12 | 1897 | 4 | 1 | 4.1 | 2 | Belgium | Brussels |
| 2012-03-15 | 1900 | 3 | 0 | 3 | 1 | France | Montauban |
| 2012-03-19 | 1904 | 4 | 0 | 4 | 1 | France | Toulouse |
| 2012-07-18 | 2025 | 6 | 30 | 9 | 3 | Bulgaria | Burgas |
| 2013-04-15 | 2296 | 3 | 264 | 29.4 | 2 | USA | Boston, MA |
| 2013-05-22 | 2333 | 1 | 0 | 1 | 2 | United Kingdom | London |
| 2013-05-25 | 2336 | 0 | 1 | 0.1 | 1 | France | Paris |
| 2013-05-26 | 2337 | 0 | 2 | 0.2 | 3 | United Kingdom | Full Sutton |
| 2013-11-18 | 2513 | 0 | 1 | 0.1 | 1 | France | Paris |
| 2014-05-24 | 2700 | 4 | 0 | 4 | 1 | Belgium | Brussels |
| 2014-09-24 | 2823 | 1 | 1 | 1.1 | 1 | USA | Moore, OK |
| 2014-10-20 | 2849 | 1 | 1 | 1.1 | 1 | Canada | Saint -Jean -sur-Richelieu |
| 2014-10-22 | 2851 | 1 | 3 | 1.3 | 1 | Canada | Ottawa |
| 2014-10-23 | 2852 | 0 | 3 | 0.3 | 1 | USA | New York City |
| 2014-12-15 | 2905 | 2 | 4 | 2.4 | 1 | Australia | Sydney |
| 2014-12-20 | 2910 | 0 | 3 | 0.3 | 1 | France | Joue-Jes-Tours |
| 2014-12-21 | 2911 | 0 | 11 | 1.1 | 1 | France | Dijon |
| 2015-01-07 | 2928 | 17 | 19 | 18.9 | 3 | France | Paris |
| 2015-02-03 | 2955 | 0 | 2 | 0.2 | 1 | France | Nice |
| 2015-02-14 | 2966 | 2 | 5 | 2.5 | 1 | Denmark | Copenhagen |
| 2015-04-19 | 3030 | 1 | 0 | 1 | 1 | France | Paris |
| 2015-05-03 | 3044 | 0 | 1 | 0.1 | 2 | USA | Gariand, TX |
| 2015-06-26 | 3098 | 1 | 2 | 1.2 | 1 | France | Saint-Quentin-Fallaveir |
| 2015-07-16 | 3118 | 5 | 2 | 5.2 | 1 | USA | Chattanooga, TN |
| 2015-08-21 | 3154 | 0 | 3 | 0.3 | 1 | France | Arras |
| 2015-09-17 | 3181 | 0 | 1 | 0.1 | 1 | Germany | Berlin |
| 2015-10-02 | 3196 | 1 | 0 | 1 | 1 | Australia | Parramatta |
| 2015-11-04 | 3229 | 0 | 4 | 0.4 | 1 | USA | Merced, CA |
| 2015-11-13 | 3238 | 128 | 129 | 140.9 | 9 | France | Paris |
| 2015-12-02 | 3257 | 14 | 17 | 15.7 | 2 | USA | San Bernardino, CA |
| 2015-12-05 | 3260 | 0 | 3 | 0.3 | 1 | United Kingdom | London |
| 2016-01-01 | 3287 | 0 | 2 | 0.2 | 1 | France | Valence |
| 2016-01-07 | 3293 | 0 | 1 | 0.1 | 1 | France | Paris |
| 2016-02-11 | 3328 | 0 | 4 | 0.4 | 1 | USA | Columbus, OH |
| 2016-02-26 | 3343 | 0 | 1 | 0.1 | 1 | Germany | Hanover |
| 2016-03-22 | 3368 | 32 | 236 | 55.6 | 5 | Belgium | Brussels |
| 2016-04-16 | 3393 | 0 | 3 | 0.3 | 3 | Germany | Essen |
| 2016-06-12 | 3450 | 49 | 53 | 54.3 | 1 | USA | Orlando, FL |
| 2016-06-13 | 3451 | 2 | 0 | 2 | 1 | France | Magnanville |
| 2016-07-14 | 3482 | 85 | 303 | 115.3 | 1 | France | Nice |
| 2016-07-18 | 3486 | 0 | 5 | 0.5 | 1 | Germany | Wurzburg |
| 2016-07-24 | 3492 | 0 | 15 | 1.5 | 1 | Germany | Ansbach |
| 2016-07-26 | 3494 | 1 | 1 | 1.1 | 2 | France | Normandy |
| 2016-08-06 | 3505 | 0 | 2 | 0.2 | 1 | France | Charleroi |
| 2016-08-19 | 3518 | 0 | 1 | 0.1 | 1 | France | Strasbourg |
| 2016-08-30 | 3529 | 0 | 1 | 0.1 | 1 | France | Toulouse |
| 2016-09-17 | 3547 | 0 | 39 | 3.9 | 2 | USA | NYC, NJ, MN |
| 2016-10-05 | 3565 | 0 | 3 | 0.3 | 1 | Belgium | Brussels |
| 2016-11-28 | 3619 | 0 | 13 | 1.3 | 1 | USA | Columbus, OH |
| 2016-12-19 | 3640 | 12 | 49 | 16.9 | 1 | Germany | Berlin |
Model Equations
| {INITIALIZATION EQUATIONS} |
| : s Attack_Counter = 0 |
| : s Directed_Terrorists = 0 |
| : s Eliminated_Terrorists = 0 |
| : c INITIAL_TOTAL_POPULATION = 885e6 |
| : c BASE_SYMPATHIZERS_FRACTION = 0.0013 |
| : c INITIAL_VISIBILITY = 0.05 |
| : S Visibility = INITIAL_VISIBILITY |
| : c CAUSE_APPEAL = 1 |
| : s Sympathizers = INITIAL_TOTAL_POPULATION*BASE_SYMPATHIZERS_FRACTION*Visibility* CAUSE_APPEAL |
| : c TERRORISTS_FRACTION = 0.0003 |
| : s Inspired_Terrorists = Sympathizers*TERRORISTS_FRACTION |
| : s Total_Casualties = 0 |
| : s Total_Population = INITIAL_TOTAL_POPULATION |
| : c ATTACK_SEED_D = RANDOM(0, 1000) + 25 |
| : c ATTACK_SEED_I = RANDOM(0, 1000) |
| : c SCALE_SEED_D = ROUND(1/RANDOM(0, 1000)ˆ(1/2)*42) |
| : c SCALE_SEED_I = ROUND(1/RANDOM(0, 1000)ˆ(1/8)*7)-2 |
| : c SEVERITY_SEED_D = RANDOM(0, 1) |
| : c SEVERITY_SEED_I = RANDOM(0, 1) |
| : c ENFORCEMENT_RESPONSE = 0.05 |
| : c ATTACK_FREQUENCY_THRESHOLD = 990 |
| : c COPYCAT_STRENGTH = 0.01 |
| : c copycat_effect = Visibility*COPYCAT_STRENGTH |
| : c AT1 = 1 |
| : c inspired_attack_scale = IF ATTACK_SEED_I>(ATTACK_FREQUENCY_THRESHOLD-copycat_effect*Inspired_Terrorists/SCALE_SEED_I) THEN MIN(SCALE_SEED_I, Inspired_Terrorists)/AT1 ELSE 0 |
| : c directed_attack_scale = IF (ATTACK_SEED_D>ATTACK_FREQUENCY_THRESHOLD) THEN MIN(SCALE_SEED_D, Directed_Terrorists)/AT1 ELSE 0 |
| : c regime_enforcement = SMTH3((inspired_attack_scale + directed_attack_scale)*ENFORCEMENT_RESPONSE, ENFORCEMENT_DELAY) |
| : c POLICY_RESPONSE = 0.1 |
| : c regime_policies = SMTH3(POLICY_RESPONSE*Visibility, POLICY_DELAY) |
| : c VISIBILITY_PER_CASUALTIES = 1 |
| : c HISTORICAL_CASUALTIES = TIME |
| : c SWITCH = 1 |
| : c SCALING_EXPONENT = 2.02 |
| : c CASUALTIES_PER_SEVERITY = 1 |
| : c SEVERITY_PER_DIRECTED_TERRORIST = 9.8 |
| : c directed_casualties = SEVERITY_PER_DIRECTED_TERRORIST*directed_attack_scale *CASUALTIES_PER_SEVERITY* (1/SEVERITY_SEED_D)ˆ(1/SCALING_EXPONENT) |
| : c SEVERITY_PER_INSPIRED_TERRORIST = 1.4 |
| : c inspired_casualties = SEVERITY_PER_INSPIRED_TERRORIST*inspired_attack_scale *CASUALTIES_PER_SEVERITY* (1/SEVERITY_SEED_I)ˆ(1/SCALING_EXPONENT) |
| : f casualties_per_day = IF (SWITCH=9 AND TIME < 3653) THEN (HISTORICAL_CASUALTIES) ELSE (directed_casualties + inspired_casualties) |
| : c visibility_generated = casualties_per_day*VISIBILITY_PER_CASUALTIES |
| : c MAX_VIS = 1 |
| : c AT_VIS = 90 |
| : f change_in_visibility = MIN((MAX_VIS-Visibility)/AT1, (visibility_generated-Visibility) *(Visibility*(MAX_VIS-Visibility))/MAX_VISˆ2/AT_VIS) |
| : c AT_POP = 13*7 |
| : f sympathizers_change = (CAUSE_APPEAL*Total_Population*Visibility*BASE_SYMPATHIZERS_FRACTION-Sympathizers)/AT_POP |
| : c GROWTH_RATE = 0.0022/365.25 |
| : c population_growth = Total_Population*GROWTH_RATE |
| : f total_population_change = population_growth − casualties_per_day |
| : f attack_occurences = IF (casualties_per_day > 0) THEN 1 ELSE 0 |
| : c ELIMINATIONS_PER_PERP = 0.8 |
| : c INSERTION = STEP(21, 2920)–STEP(21, 2921) + (STEP(21, 624*7)–STEP(21, 624*7 + 1))*1 |
| : f direction = IF SWITCH > 0 THEN INSERTION ELSE 0 |
| : f directed_elimination = regime_enforcement*Directed_Terrorists/AT1 + directed_attack_scale*ELIMINATIONS_PER_PERP |
| : c PERSUASION = 4e-6 |
| : c INS_AT = 1 |
| : f inspiration = Sympathizers*Visibility*PERSUASION/(1 + regime_policies)/INS_AT |
| : f inspired_elimination = regime_enforcement*Inspired_Terrorists/AT1 + inspired_attack_scale*ELIMINATIONS_PER_PERP |
| : c ENFORCEMENT_DELAY = 365 |
| : c POLICY_DELAY = 208*7 |
| : c “S-VAR” = 0 |
| { RUNTIME EQUATIONS } |
| : s Attack_Counter(t) = Attack_Counter(t − dt) + (attack_occurences) * dt |
| : s Directed_Terrorists(t) = Directed_Terrorists(t − dt) + (direction − directed_elimination) * dt |
| : s Eliminated_Terrorists(t) = Eliminated_Terrorists(t − dt) + (directed_elimination + inspired_elimination) * dt |
| : S Visibility(t) = Visibility(t − dt) + (change_in_visibility) * dt |
| : s Sympathizers(t) = Sympathizers(t − dt) + (sympathizers_change − direction − inspiration) * dt |
| : s Inspired_Terrorists(t) = Inspired_Terrorists(t − dt) + (inspiration − inspired_elimination) * dt |
| : s Total_Casualties(t) = Total_Casualties(t − dt) + (casualties_per_day) * dt |
| : s Total_Population(t) = Total_Population(t − dt) + (total_population_change) * dt |
| : c ATTACK_SEED_D = RANDOM(0, 1000) + 25 |
| : c ATTACK_SEED_I = RANDOM(0, 1000) |
| : c SCALE_SEED_D = ROUND(1/RANDOM(0, 1000)ˆ(1/2)*42) |
| : c SCALE_SEED_I = ROUND(1/RANDOM(0, 1000)ˆ(1/8)*7)-2 |
| : c SEVERITY_SEED_D = RANDOM(0, 1) |
| : c SEVERITY_SEED_I = RANDOM(0, 1) |
| : c copycat_effect = Visibility*COPYCAT_STRENGTH |
| : c inspired_attack_scale = IF ATTACK_SEED_I>(ATTACK_FREQUENCY_THRESHOLD-copycat_effect*Inspired_Terrorists/SCALE_SEED_I) THEN MIN(SCALE_SEED_I, Inspired_Terrorists)/AT1 ELSE 0 |
| : c directed_attack_scale = IF (ATTACK_SEED_D>ATTACK_FREQUENCY_THRESHOLD) THEN MIN(SCALE_SEED_D, Directed_Terrorists)/AT1 ELSE 0 |
| : c regime_enforcement = SMTH3((inspired_attack_scale+directed_attack_scale)*ENFORCEMENT_RESPONSE, ENFORCEMENT_DELAY) |
| : c regime_policies = SMTH3(POLICY_RESPONSE*Visibility, POLICY_DELAY) |
| : c HISTORICAL_CASUALTIES = GRAPH(TIME) |
| : c directed_casualties = SEVERITY_PER_DIRECTED_TERRORIST*directed_attack_scale *CASUALTIES_PER_SEVERITY* (1/SEVERITY_SEED_D)ˆ(1/SCALING_EXPONENT) |
| : c inspired_casualties = SEVERITY_PER_INSPIRED_TERRORIST*inspired_attack_scale *CASUALTIES_PER_SEVERITY* (1/SEVERITY_SEED_I)ˆ(1/SCALING_EXPONENT) |
| : f casualties_per_day = IF (SWITCH = 9 AND TIME < 3653) THEN (HISTORICAL_CASUALTIES) ELSE (directed_casualties + inspired_casualties) |
| : c visibility_generated = casualties_per_day*VISIBILITY_PER_CASUALTIES |
| : f change_in_visibility = MIN((MAX_VIS-Visibility)/AT1, (visibility_generated-Visibility) *(Visibility*(MAX_VIS-Visibility))/MAX_VISˆ2/AT_VIS) |
| : c AT_POP = 13*7 |
| : f sympathizers_change = (CAUSE_APPEAL*Total_Population*Visibility*BASE_SYMPATHIZERS_FRACTION-Sympathizers)/AT_POP |
| : c GROWTH_RATE = 0.0022/365.25 |
| : c population_growth = Total_Population*GROWTH_RATE |
| : f total_population_change = population_growth − casualties_per_day |
| : f attack_occurences = IF (casualties_per_day > 0) THEN 1 ELSE 0 |
| : c INSERTION = STEP(21, 2920)–STEP(21, 2921) + (STEP(21, 624*7)–STEP(21, 624*7 + 1))*1 |
| : f direction = IF SWITCH>0 THEN INSERTION ELSE 0 |
| : f directed_elimination = regime_enforcement*Directed_Terrorists/AT1 + directed_attack_scale*ELIMINATIONS_PER_PERP |
| : f inspiration = Sympathizers*Visibility*PERSUASION/(1 + regime_policies)/INS_AT |
| : f inspired_elimination = regime_enforcement*Inspired_Terrorists/AT1 + inspired_attack_scale*ELIMINATIONS_PER_PERP |
| : c POLICY_DELAY = 208*7 |
| { TIME SPECS } |
| STARTTIME = 0 |
| STOPTIME = 3652 |
| DT = 1 |
| INTEGRATION = RK4 |
| RUNMODE = NORMAL |
| PAUSEINTERVAL = 0 |
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Supplementary Material
The online version of this article offers supplementary material (https://doi.org/10.1515/jhsem-2020-0029).
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