From Public Outrage to the Burst of Public Violence: An Epidemic-Like Model
aa r X i v : . [ phy s i c s . s o c - ph ] O c t From Public Outrage to the Burst of PublicViolence: An Epidemic-Like Model
Sarwat Nizamani ∗
1, 2 , Nasrullah Memon † , and Serge Galam ‡ The Maersk McKinney Moller Institute,University ofSouthern, Denmark University of Sindh, Pakistan Centre National de la Recherche Scientifique (CNRS), France(October 2, 2013)
Abstract
This study extends classical models of spreading epidemics to de-scribe the phenomenon of contagious public outrage, which eventuallyleads to the spread of violence following a disclosure of some unpop-ular political decisions and/or activity. Accordingly, a mathematicalmodel is proposed to simulate from the start, the internal dynamicsby which an external event is turned into internal violence within apopulation. Five kinds of agents are considered: “Upset” (U), “Vio-lent” (V), “Sensitive” (S), “Immune” (I), and “Relaxed” (R), leadingto a set of ordinary differential equations, which in turn yield thedynamics of spreading of each type of agents among the population.The process is stopped with the deactivation of the associated issue.Conditions coinciding with a twofold spreading of public violence aresingled out. The results shed a new light to understand terror activ-ity and provides some hint on how to curb the spreading of violencewithin population globally sensitive to specific world issues. Recentworld violent events are discussed. ∗ [email protected] † [email protected] ‡ [email protected] eywords: sociophysics, opinion dynamics, epidemic modeling, public vio-lence, minority spreading The topic of epidemic spreading has inspired a great deal of research formany years, leading to the development of a series of models [1],[2] and [3]for which the SIR is a cornerstone [1]. One main focus is the determinationof the existence of a threshold for the spread in parallel to evaluating thetime scale of the epidemic. Indeed, the spreading of an illness obeys simi-lar qualities as the spreading of minority opinions due to the fact that bothare based on local interactions among a few agents starting from at leasttwo. The rumor-spreading phenomenon is most emblematic of the analogy[4], [5] while models of epidemics are continuously using ordinary differentialequations [1], opinion dynamics models use either continuous [6] or discretevariables [7].In this analysis, we extended the application of the mathematical frame ofordinary differential equations with continuous variables to investigate thepropagation of hatred within a subclass of a heterogeneous population. Inparticular, we focused on the conditions by which the spreading of publicoutrage leads to outbreaks of public violence.We focused on delayed reactions of outrage, which occur in one part of theworld, driven by happenings which took place earlier in a different part ofthe world. More precisely, we concentrate on political actions that were per-ceived locally to be harmless and insignificant while perceived by distantpopulations as unbearable offense [8, 9, 10].For instance, we can cite the making of the Anti-Islam movie named “In-nocence of Muslims” by an Egyptian-born US resident . When an Arabic-dubbed version was put on air in September 2012, it took few weeks beforereactions flared outside the US, leading to the death of dozens of people andhundreds of injuries. Another example took place in India; known as “Oper-ation Blue Star 1984,” a military operation on the sacred grounds of one ofSikhs community in India caused around 5000 casualties.Such remote reactions [8, 9, 10] may last for days, weeks, or months, creating http://en.wikipedia.org/wiki/Reactions-to-Innocence-of-Muslims http://en.wikipedia.org/wiki/Nakoula-Basseley-Nakoula http://en.wikipedia.org/wiki/Operation-Blue-Star . Thereare two different states of “contamination” in the model. The first state isan agent being “Upset” (U) by the inciting event and the second step is anagent turning “Violent” (V) to implement revenge. In addition, following theepidemic nomenclature, we introduce Sensitive agents (S), Immune agents (I)and Relaxed agents (R) leading to a total of five kinds of agents.The rest of the paper is organized as follows: Section 2 reviews a series ofrelated works, while Section 3 presents our model combining the five kindsof agents with the rules of interactions. The model equations are solved nu-merically in Section 4 for a series of specific cases, following the variationsof proportions of each kind of agents. Section 5 describes the various stagesof the ending dynamics. Lastly, Section 6 concludes the paper mentioningpossible directions for future work. In this paper, we present a mathematical model for hatred issue awareness,which to the best of our knowledge is the first attempt to mathematicallymodel such issues. We discuss related work in connection to proposed modelin this section, which are epidemic models or rumor-spreading models. Theepidemic model [1] describes a general framework, which is then extended torumor spreading. According to basic framework, a population of N individ-uals is divided into three groups. The process of the epidemic begins whenan individual gets an infection and turns into an infectious (I) agent. Therest of the agents in the population become susceptible (S) to the infectionand are referred as S. The infectious agents may spread the infection amongS by contacting them. The agents I who recover and do not get re-infectednor spread the infection are referred as R agents. The basic rumor spreadingmodel is comprised of the three states of the epidemic model, which wasfirst proposed by Daley and Kendal [4], and is referred to as the DK model. • The agents S are sensitive persons who are un-aware of the issue andbelong to the issue-sensitive population. These agents have a similarrole as ignorant /susceptible in rumor spreading / epidemic models. • The agents U are upset persons who are aware of the issue and areinvolved in spreading it by using some communication media (for ex-ample social media) or social gatherings, but do not take any violentaction against the issue. These are like spreaders/infectious in rumorspreading / epidemic models. • The agents I are immune agents, who have their viewpoint for whichthey do not get upset, like vaccinated agents in SIR model. However,the I agents are contacted by the U agents to get them upset like thefailure of vaccination in epidemic model. • The agents V are the violent persons who are involved in violent reac-tions against the issue. The reactions may be in the form of protests ortargeting specific individuals or locations and have the highest proba-bility of inflicting injury or death. • The agents R are the relaxed, who become relaxed after staying upsetfor ξ unit times and violent for η unit times from the agent U and theagent V, respectively. These agents are also the additional agents inthe model, who neutralize agents U and V; when everyone in U andV groups turn to agent R, the issue vanishes. At the end of the issue,the agents R and I are alike; however, before the end of the issue, theI agents are contacted by the agents U and some of them turn to U.That is why the two agent types are considered separately.5he proposed model is discussed in Section 3, which simulates the dynamicsof drastic issues by answering the following research questions:Q1. Does the mathematical model have the potential to describe the effectof the inciting incidents?Q2.What is the effect of interaction between various population categoriesduring the lifetime of violence-causing events?Q3. What is the final state of the issue at the end?In order to address the research questions, the following hypotheses are de-signed and supported by the proposed model:H1. The results of inciting incidents can be well-described mathematicallyusing mean-field equations.H2. The interaction among various issue sensitive populations causes in-crease in violent and upset agents in the beginning, then decrease with thepassage of time.H3. The issues vanish at the end.In the following section, we discuss the proposed model of hatred issue aware-ness beginning with the basic SIR framework [14] of three types of the agents,then continue to an extended model comprised of four types of agents andleading to a proposed model of five types of agents. The proposed model encompasses five types of agents, an expansion of thebasic epidemic model which leads to hatred contamination, then from hatredcontamination to violence, ushering in a five-type agent model.
As in epidemiology where the epidemic process is initiated when an infectiousagent contaminates other agents, hatred can be passed by exchanging words,whether fact or rumor. However, in both cases, only susceptible agents be-come contaminated. A spreading process is thus initiated which can eitherlead to a large-scale epidemic or simply fade away.Nevertheless, in the case of hatred, besides getting upset, an agent can turnviolent, creating destruction outside the pure process of being outraged. Sucha state seems to be specific to the case of hatred. At the same time, someagents are immune towards the emotional content of the incriminated issue.6o set our model, we started from an analogy with a basic SIR epidemicmodel [1] by considering three types of agents: Sensitive (S), similar to sus-ceptible agents, Upset (U), similar to infectious agents, and Relaxed (R),equivalent to recovered agents. The simplest form of the model is shown inFigure 1 where each square represents an agent type, while an edge showsthe transition between agent types showing the rate at which transition takesplace.
Figure 1: SIR-like simple model
Similarly, to many diffusive phenomena, only a few agents are upset atthe beginning. However, once upset, these agents come into contact withsensitive agents trying to make them upset too, turning on the process ofhatred spreading. The transition from S to U depends on contact rate perunit time, resulting in reproduced U agents. It is given as αP U (cid:0) SP (cid:1) = αU S (where P is the total sensitive population), the number of new U agentsper unit time reproduced. Accordingly, agents S turn to agents U at a rate α ,producing an increase of αU S in the U agents. At the same time, people donot stay upset forever about a specific issue and eventually relax. We assumeit occurs at a constant rate ξ , i.e., on average a U individual is relaxed after ξ time units with relaxing transmission represented as ξU . An associatedprocess is described by the set of ordinary differential equations, dSdt = − αSU (1) dUdt = αSU − ξU (2) dRdt = ξU (3)On this basis, we add the possibility for an S agent in contact with a Uagent not to get upset and instead become immune to the issue, similar to thecase of either natural immunity or vaccination in epidemics. The proportionof I agents produced per unit time is given by βP U (cid:0) SP (cid:1) = βU S leading tothe equations, dSdt = − αSU − βSU (4)7 Idt = βSU (5)as shown in Figure 2. Figure 2: Addition of I (immune) agent type
However, it is known in epidemiology that vaccination does not always pro-duce 100% immunization. The same holds true here with the possibility foran I agent to eventually turn upset while having a subsequent contact withthe U agents as shown in Figure 3. The rate of shift per unit time is denotedby Eqs. 2 and 5 and are modified respectively as, dUdt = αSU − ξU + κU I (6) dIdt = βSU − κU I (7) Figure 3: Addition of a new edge I to U in result of some failed immunization
Now we discuss an important aspect of the hatred issue, in which some of thesensitive agents are extremely sensitive; in addition to becoming upset, theyturn violent in order to implement some revenge for what they consideredas an outrage. They are denoted as violent agents (V). Two paths to create8 V agent are introduced. A direct path results from an encounter betweenU and S agents at a rate γ while the second one emerges from encountersbetween two U agents at a rate σ . In addition, V agents can also turn Sagents into U at a rate µ . V agents also relax after η unit times.A new differential equation is thus obtained for the V agent dynamics inaddition to a modification of Eqs. 3, 4, and 6, which yield, dSdt = − αSU − βSU − γSU − µSV (8) dUdt = αSU − ξU + κSI − σU U + µSV (9) dRdt = ξU + ηV (10) dVdt = γSU − ηV + σU U (11)In the final set, we have a set of five differential equations to describe thedynamics of the respective proportion of each kind of agent, i.e., sensitive(S), upset (U), immune (I), relaxed (R), and violent (V) as illustrated inFigure 4. Figure 4: The model with five agent types
To begin implementing the model, we start with some initial conditions( S , U , I , V , R ) at time t = 0 under the constraint S + U + I + V + R = 1,to study the various possible scenarios of the time evolution of those pro-portions ( S t , U t , I t , V t , R t ) at time t. Indeed, when a bursting event occurs9 U/S contacts per unit time to yield U agents 0.5-5 β U/S contacts per unit time to yield immune agents 0.5 γ U/S contacts per unit time at transmission rate γ from S to V 0.2 µ V/S contacts per unit time at transmission rate µ from S to U 0.1 ξ ξ time units an individual remains spreader 1-30 η η time an individual remains violent 1-20 κ Contact rate that transmits I to U 0.5 σ U/U contact rate transmitting them to V 0.5
Table 1: Model parameters with description and values somewhere in the world, focusing in some specific part of the world wherethe population is sensitive to such an event, at t = 0 we have only a fewsensitive agents who become aware of it. Subsequently, they get either upsetwith a proportion U or immune with proportion I . The rest of the popu-lation stays in the sensitive state with S = 1 − U − I with V = R = 0.Those initial values result from an external effect. Then, internal dynamicsare turned on, driven by a word-of-mouth phenomenon. Hence, the lifetimeinternal dynamics take on a shape of a system of linear equations given inthe matrix form: t t ..t ∞ S U I . . . . .. . . . .. . . . .S ∞ R ∞ I ∞ = The internal dynamics stop at a certain time, t = ∞ , resulting in thesteady state of the system, where hatred and violence are no more active. Theinternal dynamics consequences in the final assumptions S ∞ = 1 − R ∞ − I ∞ .Thus, R ∞ + I ∞ yields the final number who ever was aware of the “bursting”event. In order to numerically analyze the proposed model, we consider the param-eter values and their description given in Table 1.10 .1 From epidemic to hatred contagion
The numerical simulations of this simple model illustrate the behavior ofthe model in the simple case when we have only three types of agents andonly two transitions. As the agent type S is monotonically decreasing and theagent type U is increasing by receiving agents from the S and then decreasingby transmitting the agents to the agent type R. In the beginning, the densityof the U increases, but after reaching a peak value, it begins to decrease.We needed to determine the necessary conditions for U to increase and thento decrease. We also determined the condition for U to outbreak (necessarycondition for which U increase). The basic condition for U to increase is thatwhen S t > ξα , while S t < ξα causes U to decrease. It is a necessary conditionfor outbreak that S > ξα , hence U will increase up until some point. Anothercondition for outbreak is to determine reproduction rate [25] R¸ > R¸ is areproduction rate and it should not be confused with agent type R). Thereproduction rate determines the number of secondary U agents producedby the initial U agents. It is determined as R¸ = αξ , if the condition holds, theissue may be considered serious. Figure 5 (a-e) illustrates the basic model ofhatred contagion by varying parameter values; hence the condition of peakvalue can also be verified.For numerical simulations we have used varying values of the parameters( α and ξ ) in Figure 5 (a-e). In each of the Figure peak value of the U agentshas been given with the necessary condition for outbreak for the given peak. An enhancement to the basic model has been integrated by introducing theI (immune) agent type, which is resulted by the individual’s viewpoint tonot get upset. Figure 6 shows the dynamics of the whole system with newlyintroduced agent type for varying values of parameters. In this enhancedmodel, the reproduction rate depends on more parameters. It is estimatedas R¸ = α + κβ + ξ and if R¸ > α is the rateat which the agents S turn to agent U and κ is the rate at which the I agents(losing their immunization) turn to the U agents; while at the rate of β , theagents S are immunized (do not turn U) and ξ is the rate at which the agentsU turn relaxed. Hence, the reproduction rate R¸ > a) Peak value of U at t = 11 . S t ≈ ξα = 0 .
222 (b) Peak value of U at t = 5 . S t ≈ ξα = 0 . t = 5 .
273 timeunits, and S t ≈ ξα = 0 . t = 20 .
56 timeunits, and S t ≈ ξα = 0 .
208 (e) Peak value of U at t = 22 . S t ≈ ξα = 0 . Figure 5: Basic model with varying range of parameters a) R¸ = 2.4 (b) R¸ = 3.55385(c) R¸ = 2.85 (d) R¸ = 1.32 Figure 6: Numerical simulations of extended model with Immunization illustrated and also the reproduction rate is given for each of the varyingvalues of α and ξ .In Figure 6 (a-d), we presented dynamics of the issue awareness withextended model. In this model we estimated reproduction rate, which de-termines whether there will be outbreak for the U agents or not. In eachillustration this reproduction rate has also been given. It should be observedthat higher is the reproduction rate, the maximum peak the U agents attain. A fully evolved proposed model for hatred issue awareness is comprised of 5types of agents and corresponding transitions. The numerical simulations ofthe final model are illustrated in Figure 7, with varying parameter values. Inthe model, it can be observed that the two important agent types U and Vinitially increase, approach a peak, then decrease and finally reach 0, hence13he issue vanishes. The numerical simulations shown in Figure 7 are also theevidence of the situation. For the dynamics of U i.e., increasing, reachingpeak and decreasing has a relationship with S that has been extracted. Also,it has been determined that for what minimum S value the U increases, i.e.,what is the condition for U to outbreak? As the proposed model is complexand comprised of many parameters, for outbreak of U and V, the regressionrelation has been determined. In Figure 7 (a-e), for various peak values ofU, the necessary conditions for U outbreak have been given.Figure 7 presents the numerical simulations of the proposed five-agent modelwith the conditions of U outbreak with respect to S; while Figure 8 illustratesthe density curves of the agent types S, U, and V and conditions for Voutbreak with respect to U. The overall range of parameter values used inthe simulations are given in Table 1.Figure 8 presents the dynamics of V agents in the proposed model. InFigure 8 we have used varying values of α , ξ and η which affect the peak valueand outbreak of the V. In Figure 7 and Figure 8, we determine the outbreakfor U and V respectively, which is estimated using regression relation and isdescribed in following sub-section. As the value of U is dependent mostly on the value of S, to determine theoutbreak of U, we have considered the state of S. A regression relation hasbeen established between U and S, which shows that how U increases, de-creases or attains the maximum value in relation to S. In order to determineregression relation, we first extracted peak values of U for various parametervalues and established a regression relation between the peak of U and S t .The two parameters that strongly affect the value of U are α and ξ , so wehave taken varying values of these two parameters while rest of the parame-ter values have been kept constant as given in Table 1.The condition when U is at peak is given by: S t ≈ − . . (cid:16) ξ (cid:17) + 0 . S > − . . (cid:16) ξ (cid:17) + 0 . a) Peak U at t = 14 . , S t =0 . ≈ − . . ξ +0 . S t = 17 . , S t =0 . ≈ − . . ξ +0 . t = 5 . , S t =0 . ≈ − . . ξ +0 . d ) P eakU att = 5 . , St =0 . − . . / ) +0 . t = 8 . , S t =0 . ≈ − . . ξ +0 . Figure 7: (a-e). Numerical simulations of the proposed model with varying peakvalues of U agents and its relation to S t a) Peak V at t = 10 . U t =0 . ≈ . α − . ξ + 0 . η + 0 . t = 9 . U t = 0 . ≈ . α − . ξ + 0 . η + 0 . t = 12 . U t =0 . ≈ . α − . ξ + 0 . η + 0 . Figure 8: Peak of V with respect to U S < − . . (cid:16) ξ (cid:17) + 0 . CorrelationCoef f icient = n P xy − ( P x ) ( P y ) q n ( P x ) − ( P x ) q n ( P y ) − ( P y ) (12)Where n is the total number of observations, x is observed value and y iscomputed value. The regression relation is considered to be a good approxi-mation of observed values when coefficient correlation is near +1 and greaterthan 0.5. The other measures used for estimating quality of regression rela-tion are mean absolute error and root mean squared error. The values near0 are considered to be good approximations of observed values. Like the U agents, the V agents also increase first, reach a peak, and finallydecrease and vanish. It is therefore needed to find under what conditions theV increases, when it reaches peak, and when it decreases. The V agents aredependent on the U agents; we have determined regression relation to findout condition for outbreak. The first condition for which V reaches peak isgiven by: U t ≈ . α − . ξ + 0 . η + 0 . U > . α − . ξ + 0 . η + 0 . When the hatred “bursting” dynamics cease to end, then only the I, R, andS agents exist. The agents R are those who once were upset or violent butnow have been relaxed, while the agents I are those who were aware aboutthe event but did not take any interest in it. The density of the agents I andR at the end determines the total population who ever became aware of theevent. The final condition S ∞ = 1 − R ∞ − I ∞ is used to compute the finalsize of the hatred “bursting” event aware population. Figure 9 illustrates thecombined density of the agents I and R from the beginning until the end ofthe dynamics.Figure 9 shows the final state at t ∞ when most of the issue-sensitive popula-tion was aware of the issue but everyone had become relaxed or was immu-nized. It may be noted that near t the sum of the R and I agents approaches1. In Figure 9, we have used a varying values of α , for which the contactbetween the U and the S agents resulted in new U agents per unit time. In the present research, we explored the growth of population awareness andviolence against the hatred “bursting” events that cause upset and violenceamong certain groups in a population. We described social behavior of thepopulation using a five-state model by deriving differential equations. Wepresented numerical simulations of the model using different initial condi-tions and various sets of parameters’ values.The outbreak conditions of violence have also been determined. Violent18 igure 9: Final state of issue awareness on varying rates of α reactions to an issue increase as the upset agents have well motivated thepopulation who then become violent. The mathematical model shows howthe upset and violent agents increase at first, then decrease and finally vanishwith time. We also have shown why using a basic three-state model of epi-demic to hatred contagion is insufficient requiring indeed a five-state model.In the future, we plan to further extend the model by predicting the potentialvictims of violence in the midst of such drastic issues with respect to geo-graphical locations. Once the potential victims are determined, the modelwould map the potential victims and violent agents on the 2D lattice usingpercolation theory and determine the likelihood of victimization of the po-tential victims. We also plan to further investigate the hatred events in orderto determine the severity of the issues by keeping in view the issue-sensitivepopulation, the range of population in various geographic locations affectedby the issue and some other parameters.To conclude, our study may shed a new light to comprehend terror activ-ity and provide some hint on how to limit the spreading of violence withinpopulations globally sensitive to specific world issues. Nonetheless, it shouldbe noted that we are dealing with models which are not the reality althoughthey might well to some extent help to grasp the reality.19 cknowledgement This work was supported in part from a convention DGA-2012 60 0013 0047075 01