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Stanislaw Raczynski (2004)

Simulation of The Dynamic Interactions Between Terror and Anti-Terror Organizational Structures

Journal of Artificial Societies and Social Simulation vol. 7, no. 2
<https://www.jasss.org/7/2/8.html>

To cite articles published in the Journal of Artificial Societies and Social Simulation, reference the above information and include paragraph numbers if necessary

Received: 12-Dec-2003    Accepted: 17-Mar-2004    Published: 31-Mar-2004

* Abstract

A discrete-event model of the dynamics of certain social structures is presented. The structures include terrorist organizations, anti-terrorism and terrorism-supporting structures. The simulation shows the process of creating the structures and their interactions. As a result, we can see how the structure size changes and how the interactions work, and the process of destroying terrorist organization links by the anti-terrorist agents. The simulation is agent-oriented and uses the PASION simulation system.

Keywords:
Simulation, Modeling, Terrorism, Discrete Event, Agent-Oriented, Social Simulation, Soft Systems

* Introduction

1.1
The US State Department defines terrorism as "premeditated, politically motivated violence perpetrated against noncombatant targets by sub-national groups or clandestine agents, usually intended to influence an audience". There are other definitions and opinions, but all of them coincide in the elements of premeditation, advance planning and politically motivated action. Undoubtedly, such actions need to be supported by some kind of organization with a well-defined structure and operation rules. Human societies have always been developing social structures. The difference between human-created structures and those formed by animal populations lies only in their degree of sophistication. Our organizations can support religions, scientific development and politics, as well as destruction, crime and racial or religious extermination.

1.2
A social organizational structure is not just a set of individuals interacting with each other. Such a structure should be treated as a system. This means that it may have (and almost always has) an objective or a "superior" goal, and intends to satisfy it. This objective may even be in opposition to the objectives of each one of the structure members, but in many well-known cases in history, it cannot be influenced or changed by the members. A social structure acts as a new agent, using its members as nothing more than a medium to achieve its goal.

1.3
For example, one goal of any political party is always to obtain power and nothing more. Many trade union organizations have lost sight of their original goal (defending the interests of workers) and have also become power-seeking structures. The interaction between different social structures is an interesting problem and can be simulated - to some extent, of course. Regarding this subject, the reader may consult an article I wrote more than 20 years ago (Raczynski 1980), on the simulation of interactions between political, administrative and trade-union structures in the communist totalitarian system.

1.4
Many existing models of social organization dynamics are of an agent-oriented type (see the next section for more details about agents). Smith (2001, 2002) points out the necessity of developing a new simulation technology tool that could be applied directly to the war against terrorism. These tools should address the main components of a terrorist network - the command nucleus, the field cell, communications, the national host, legal, political and cultural aspects, supporters, etc. The "cross-domain" interactions between all of these elements should be simulated to obtain the behavior of the system as a whole and to predict its possible activities. As a part of such simulations, different tools are proposed, creating a kind of system dynamics model.

1.5
An interesting agent-oriented model, called the BC model, can be found in the article by Krause (2000). In that model, the agent attributes include "opinions", and the interaction between agents depend on the distance between their opinions in a non-linear way. These interactions can result in an action being taken by the agent. Other examples of models of social structures based on the concept of opinion interactions can be found in Latane and Nowak (1997), and Galam and Wonczak (2000). A similar approach is taken by Chatterjee and Seneta (1977) and Cohen, Kejnal and Newman (1986). But the BC model and the above works refer to the dynamics of forming social groups in accordance with the existing agents' attributes (opinions), rather than to events such as the destruction of a part of a tree-like social structure by another (adversary) structure. Some quite interesting results, more closely related to the terrorism problem, are described by Deffuant, Amblard, Weisbuch and Faure (2002).

1.6
Another agent-oriented approach is used by Lustick (2000), in which the agents interact on a landscape. It is shown that macro-patterns emerge from micro-interactions between agents. An interesting conclusion is that such effects are more likely when a small number of exclusivist identities are present in the population. The simulation of other mechanisms of clustering in agent-oriented models is described by Younger (2003). That article deals with the creation of social structures in the process of food and material storage.

1.7
Some more general concepts of computational sociology and agent-based modeling can be found in the article of Macy and Willer (2002). Other general recommended readings in the field are: Bak (1997), Cioffi-Revilla (1998), Gotts, Polhill and Law (2003), Axelrod (1997), Epstein and Axtell (1996), and Holland (1998). Many other sources can be found on the Internet. An interesting contribution regarding the modeling of the OBL organization's structure is included in a Vitech Corporation page (Long 2002).

1.8
Many of the works mentioned above deal with the clustering in artificial societies due to the agents' opinions and activities; some of them are simulated on a landscape.

1.9
It should be noted that to look for a model that simulates real human behavior is utopian. Nobody has ever simulated a human in its complete (mental, emotional, physical etc.) behavior. All that can be done is to choose some little part of this complex system in order to simulate its possible actions. In any case, in looking at the models used in the field of system dynamics, soft system simulation and social simulation, one can hardly ever (or never) find any proof that the model used is valid.

1.10
As for the recent war on terrorism, it is a real and serious war. By serious, we mean that the aim of at least one of the parties is extermination, not domination or territorial expansion. Note that the conventional (rather medieval) concept of a "battlefield" has no sense in this war. The war is distributed over the entire globe and the structures involved are always global. The aim of the present study is to experiment with certain hierarchical structures and their interactions. The rules for creating the structures are rather simple, but their behavior is not easy to predict without carrying out simulation experiments. The task of modeling and simulation of the terrorist phenomenon (as that of any other complex social system) is difficult and cannot be resolved by just one person, or even by a small team. An important effort is being made by many researchers and organizations. As an example, we should mention the Los Alamos National Infrastructure Simulation and Analysis Center, where a large amount of simulation software has been developed to support national security.

1.11
The present study is a small contribution to the global effort. The simulation tool presented here is open and should be treated as the core of a possible, larger and more sophisticated model. Two aspects seem to be important here: first, the widely-distributed (in space) character of the simulated structures, which are located in and move over a landscape, but which are not tied to any geographical location. Second, the clustering of the agents, accomplished by creating hierarchical structures with global and local "leaders". Such structures are very comfortable for hidden groups or clusters, because they can hardly be completely destroyed. A member of a hierarchical structure has only the information about his nearest collaborators and not about the structure as a whole, so even if an anti-terrorism agent infiltrates the organizations, the information he can gather is always local and insufficient. In the present model, what is somewhat different from other works on social clustering and group behavior is that we simulate both the process of creating and that of destroying the hierarchical structures by means of other hostile structures formed by the model's agents. Generally speaking, this model does not respond to the question "How to fight terrorism?", but rather, gives some clues on how some hierarchical structures can grow and destroy each other.

* The model

2.1
Modeling and simulation of soft systems where the human factor is crucial to the system behavior is a difficult and risky task.

2.2
As for the general concepts of model validation, these can be defined briefly as follows. We have a real system with the real state X. Let S be the operation of passing from the real system to its model (modeling). So, S converts X(t) (the state of the real system in the time instant t) into XS (t)(the corresponding model state). Let D be the operation of passing from t to (t+h), h being a time increment. This operator maps X(t) into X(t+h), or XS(t) into XS(t+h). Consequently, to pass from X(t) to XS(t+h), we must apply both operations. The model is said to be valid if the two operations commute. In other words, the result of passing from X(t) to XS(t+h) cannot depend on the order in which the operations S and D are applied. This is the fundamental definition of model validity, provided by Zeigler (1976). Zeigler's book has several new editions and continues to be the fundamental book on modeling methodology. However, there is certain confusion among specialists in the field: almost every one of them defines the validation concepts in his or her own way. Many of their works can be found on the Internet. See, for example, http://www.casos.ece.cmu.edu/casos_working_paper/howtoanalyze.pdf.

2.3
The models can hardly be validated using the above, basic definition of validity. Another way to validate them is to apply "input-output validation" or "external validation", which consists of testing the resulting simulator to see how the results match up to the historical data. This is also difficult unless we have sufficient historical data. Even if we can prove that the model behavior is valid for past data, this does not mean that it will be valid in future trajectories. Thus, the use of invalid models of soft systems is a common practice (in fact, the authors of such models usually do not care about the model validity). This does not mean that the model, even if it is invalid, cannot provide relevant and interesting information. What we should take into account is that the numerical results can be false. The model user must be aware of this, and should look, rather, for behavioral properties that are repeated with different data sets and model parameters, and should use his or her experience and intuition.

2.4
Our model is agent-oriented. Let us recall what object- and agent-oriented simulation is. In object-oriented simulation, the model components are represented by objects in the machine's memory. An object is a set formed by data structures and methods. The data structures are defined in accordance with the programming language being used. For example, if the object is a car, the data may contain the car type, the model, the year of production, its weight, motor type, color etc. The methods associated with the object are procedures related to this particular data set and to the rules of the car's behavior; for example, the complete set of equations of the car's movement, the procedures that display the required data, or even an interface with virtual reality tools that can show the animated car's image.

2.5
In agent-oriented simulation, the model components are also objects, called agents. But, unlike conventional objects, the agents are equipped with some kind of intelligence. They can make decisions, communicate with other agents and negotiate, if their local goals are different and they need some compromise to be made. In object- and agent-oriented simulation languages, the user defines one or more agent (object) types. These are type declarations, not agents. The agents are being created during the program run. The resulting agents are called instances of the corresponding type. After being created, the agent may be activated and execute its methods (also called events in simulation terminology). In fact, the agents simulated here are very simple. They are not very "intelligent", and their goals are rather simple. However, their goals do exist. For example, the goals of a "terrorist" is to form a part of a hierarchical organizational structure, and to then commit an act of terrorism. The goal of an anti-terrorist agent is to form a part of the anti-terrorist structure, to infiltrate a terrorist organization and to destroy a part of the infiltrated structure.

2.6
Our model is open; this means it can be easily modified and expanded. Here we present the current version, which is merely the seed of possible future developments. ("Structure" and "organization" are synonyms in the following).

Structures

2.7
All structures in this model are hierarchical. Each structure has a leader (head) and a hierarchical "tree" of subordinates (or "collaborators"). The hierarchical nature of the organizations is the main focus of this model. Of course, organizations that are not exactly hierarchical may exist. However, most of the known organizational structures are of this kind, or have at least some hierarchical features. After all, it is rather common that a group or organization driven by a common aim or idea has one or more leaders (usually there is only one of them), which means that it has at least one or two hierarchical levels. It is impossible to create an "absolutely" valid model of a human being, so the modeling of organizations created by humans is even more difficult. The reader should bear in mind that here we deal only with hierarchical organizations and their interactions. He or she should also observe that the emphasis of this research is placed on how one structure destroys another, and not only on how the organizations are formed.

2.8
Three kinds of structures can be created: Terrorist structures, anti-terrorist structures and terrorist-supporting structures. Terrorist structures (there may be one or more) are those whose members are able to carry out a terrorist act. The members of the anti-terrorist structure are able to neutralize members of a terrorist structure, destroying a part (one or more branches) of the terrorist structure. Note the fact (also mentioned in what follows, and very important in this model) that a model agent can belong to more than one structure. This occurs when a member of the anti-terrorist structure becomes an infiltrating agent of one of the terrorist structures. The terrorist-supporting structure is not a terrorist one, and it does not participate directly in the acts of terrorism. This structure looks for a possible links with terrorist structures. If such a link is established, the structure which it is linked to receives a support. This means its power grows (this is explained further on), and the ability to carry out a terrorist act also grows.

2.9
The main component is an agent. It is an entity that can move over a plain region. The agent has a type attribute. Agent types are as follows.

2.10
In this model, one anti-terrorist, one terrorist-supporting and multiple terrorist structures are considered. The agents move inside a rectangular region. The movements are random, but there exist some attracting points in the region (cities). The attracting forces are strong enough to cause the agents to concentrate, after some model time, around the cities (five cities are created in this model). Note that the "cities" are only attraction points, they are not the centers where the modeled organizations grow. As the entity movement is random, the structures observed in the simulation experiments are not located within the cities. Some of the structures, which at the initial stage are local, become global (extended to the entire simulated area). The common case is that the structure has many local branches, but also, several links to clusters located in other areas ("cities").

2.11
Structure creation is spontaneous, and is a result of the rules of agent behavior. If an agent is of type 2 (potential terrorist), then it constantly looks for other agents of the same type in order to capture them as its subordinates, until it has enough subordinates (the limit of 10 is fixed). For each of the subordinates, the agent who has captured it becomes its superior. This forms a hierarchical, tree-like structure. If an agent has one or more subordinates and no superiors, it is the head (leader) of the organization. But, even as the structure head, it can be captured as a subordinate by a member or leader of another terrorist structure. This causes the structures to merge into larger organizations. Any capturing (linking) event may occur when the two agents involved are close enough to each other (the "contact" distance). Each structure has a power parameter. In the actual model, the power depends on the number of the structure's members. However, it can be increased if there exists a link between the terrorist structure the terrorist-supporting structure. The power of the structure influences the ability to capture new members, and the ability to carry out a terrorist act.

Structure interactions

2.12
As stated before, the interaction between terrorist structures consists only of merging. The interaction between anti-terrorist and terrorist structures is as follows. The members of the anti-terrorist structure have no information about terrorist structures, until a terrorist structure is infiltrated by anti-terrorist agents or until some of the members of a terrorist structure collaborate with anti-terrorist agents. The event of infiltration occurs when an anti-terrorist agent is captured as a subordinate member of a terrorist structure. There is a small probability of this occurring, namely, when the capturing agent recognizes, by mistake, an anti-terrorist agent as a potential terrorist. On the other hand, any member of a terrorist structure can become a collaborator (this does not mean that it becomes a member of the anti-terrorist organization). If there exist collaborators or infiltrating agents, the terrorist structure may be attacked. The attack does not necessarily signify the structure's destruction. It consists of a "local" destruction; the immediate superior of the infiltrated member and all its subordinates are neutralized. This means that their type is changed to type 1 (neutral agent) and all their links in the structure disappear. If the structure is small, this may result in the structure's destruction. In the case of larger structures, this causes considerable damage to the them which usually breaks into two ore more unlinked organizations. The terrorist-supporting structure looks for possible links between its members and members of terrorist structures. If a link is established, the power of the linked terrorist structure grows. The terrorist-supporting structure cannot be attacked nor destroyed.

2.13
All the above actions, like member capturing, collaborating, infiltrating, linking and so on, depend on the corresponding necessary conditions and on a random factor. This random factor, which influences the probability of the event, is controlled by a corresponding model parameter. This allows the disabling of some events or the carrying out of simulation experiments with different levels of probability factors.

2.14
Finally, terrorist acts are simulated. However, these events are not the most important ones in our model. No explosions, casualties or deaths are simulated. If a terrorist act occurs, it is only indicated graphically on the screen and counted. Note that the main purpose of the present simulations are the interactions between the social structures involved, and not the terrorist acts. In future developments, the terrorist acts will be simulated with greater detail, taking into account not only their probabilities but also their magnitude. This can be related to the availability of weapons of mass destruction and the advances in mass destruction technology, which sooner or later will be available to everyone, including terrorists.

* The Tool and the Model Implementation

3.1
All of the mechanisms included in our model and described above in general terms can be implemented in any simulation language that supports object-oriented discrete event simulation. The present implementation has been achieved using the PASION (PAScal simulatION) (Raczynski 1980, 1988, 2003). Let us make some comments about PASION, because it is not a very popular and well-known tool, though the number of users of this package is growing.

3.2
Recall that the PASION program consists of a sequence of process declarations. Each process contains a set of events. This process/event structure is ideal for describing the behavior of model agents. The main process, called AGENT, defines the behavior of our agents. In the main program segment, several hundreds of agents are created as instances of the AGENT process. Each agent is given attributes. They are as follows.

3.3
Initially, all links are cleared, so no structures exist in the set of agents. Once activated, the agents begin to move and look for other agents to link themselves to, and execute other events.

3.4
The events consist of the following actions: agent movement, creating or entering a hierarchical structure, committing a terrorist act, destroying a terrorist structure (by an anti-terrorist agent), creating links between the terrorist-supporting organization and a terrorist organization.

3.5
There is also another process, named WORLD, with only one instance which manages the graphical display of the model state, recalculates the power of each organization and gathers the model statistics. The WORLD process also creates links between terrorist-supporting and terrorist structures, and brings the TOP attribute of all agents up to date.

Fig 1
Figure 1. Graphical symbols

3.6
The program generates an animated image of the current situation, showing the movements of all agents and the structure of each organization. The graphical symbols are shown in figure 1. The links that form hierarchical structures are shown as lines that connect the agents. The last item in figure 1 shows the event of destroying a part of a terrorist organization. As stated before, we do not simulate any birth-and-death processes. The agent population is fixed, with a given percentage of potential terrorists, anti-terrorist agents and terrorist-supporting agents. The main rule is that an agent cannot take any action if it is not a member of one of the organizational structures.

3.7
It should be noted that the model is open to any changes and extensions. The currently included mechanisms should be treated as basic and simple rules needed at the very beginning of the research. Tools such as PASION allow us to easily add new events or processes to the model.

3.8
The structures are created due to the structure-creating events in the agent process. Each agent is permanently looking for other ones that can be captured as its subordinate. The agent being captured cannot have already been captured by another agent (it cannot have a "superior" in the corresponding structure). However, the agent that has one or more subordinates and no superiors (a leader) can be captured as a subordinate. In this way, the structures grow. This mechanism is implemented in a very simple manner. First, the agents move. During the movement, the agent checks its distance from each of his neighbors. If the distance is sufficiently small and the neighbor is of a desired type, the agent captures the neighbor as its subordinate. However, even if all necessary conditions are satisfied, this event only occurs with some probability, which is controlled by one of the model parameters. The other factor that influences the probability of capturing a subordinate is the power of the organization as a whole which, in turn, depends on its size and on the links with the terrorist-supporting structure. There is no room here to give a detailed algorithm for each model event. The above example explains, in general terms, the way the model has been constructed. As mentioned before, we can have only one anti-terrorist structure and one terrorist-supporting structure. However, multiple terrorist organizations can exist. The relationship between a terrorist and a terrorist-supporting organization is defined by the links between the members of the two organizations. If such links exist, then the linked terrorist organization receives additional power from the terrorist-supporting organization. This power parameter influences the ability to commit an act of terrorism and to capture new members. Thus, the more powerful organizations grow faster.

3.9
The struggle between anti-terrorist and terrorist organizations consists of destroying a part of a terrorist organization and erasing the corresponding links. The unlinked agents become "neutral" (recall that no killings or deaths are simulated). A neutral agent does nothing. However, there is the probability that such an agent can become a terrorist again and can be incorporated into another terrorist organization. The event of destroying terrorist links can be created only if there exist anti-terrorist agents that have infiltrated a terrorist organization, or if there exists a collaborating terrorist. In other words, this activity is possible only if some information about a terrorist organization is available by means of intelligence activities. If a terrorist structure is being attacked, it normally does not disappear. Only some links - those linked to a collaborator or an infiltrating agent - are destroyed, and the linked agents are neutralized. In our model, neither the anti-terrorist organization nor the terrorist-supporting structure can be attacked. The attacked structure may end up being damaged. This consists of erasing several links (not the entire structure). This erasing process starts with the superior of the infiltrating or collaborating agent and may continue, with certain probability, along to the last (lowest) level in the structure. However, the attempt to damage the structure may fail. The probability of failure depends on the power of the attacked structure. Note that the power of a terrorist organization is a function of the size of the organization and of the number of links to the terrorist-supporting structure, if such links have been created

* Simulation Experiments

4.1
In all the experiments, the model was completely abstract, created without using any real data. Thus, no quantitative conclusions can be made from the results. However, by changing the model parameters, one can observe the interesting behavioral properties of the model, and carry out many "what-if" experiments.

4.2
The experiments have been done with a total of 500 agents, approximately one third of them being potential terrorists, one third of them anti-terrorist, and one third of them potential anti-terrorist agents.

4.3
The model as a whole is controlled by a number of parameters, most of them fixed in the code. The parameters define the model behavior, mainly by means of the probabilities of certain events' occurrence. In what follows, the parameters denominated as "rate" are not exactly the rates or probabilities, but they can control the corresponding probabilities. In general, if a parameter is set at zero, the corresponding event or model behavior is disabled. The parameters that can be defined by the user at run time are as follows.

4.4
Even such a limited number of interaction rules and controlling parameters allows us to carrying out many simulation experiments. First of all, we can investigate how each parameter influences the model behavior. By changing the rules of interaction (this requires an intervention in the model code), one can see how relevant certain model events are, and what the result of the changes is. What we show here is merely a small example of a simulation experiment.

4.5
The experiment consists of enabling and disabling the actions of the terrorist-supporting structure. Disabling can be achieved simply by setting the "Terrorist-supporting creation rate" to zero.

4.6
Our simulations start with all the structures cleared. This is not a realistic situation. However, it can be interpreted as a situation in which we start with a non-existent terrorism problem, and then the problem suddenly appears. In any case, other experiments can be carried out, skipping the early "warm-up" period and examining the model behavior for a greater simulation time. As stated before, the model is rather abstract, so the time unit is not defined. In this program, the time interval for the agent's movements was set to 7, which can be interpreted as 7 days, if one wants to situate the events in real time. The final simulation time was equal to 3650 time units.

Fig 2
Figure 2. A screen shot of an early stage in the model run

4.7
Let us start with the terrorist-supporting structure enabled. Figure 2 shows an early stage of structure creation. There are several small terrorist organizations, one small anti-terrorist organization and one terrorist-supporting organization. The agents move on the screen, and structure links are being created. In figure 3 we can see the event of destroying a part of a terrorist structure. The bold red lines are the links being destroyed. Figure 4 shows the average trajectory for the number of active terrorists, taken from 50 repetitions of the simulation run. By active terrorist, we mean a terrorist who belongs to a terrorist organization.

Fig 3
Figure 3. Destroying a terrorist organization. Bold lines show links being destroyed

4.8
The shadowed area is the region where the number may be situated, with probability equal to 0.9. Figure 5 depicts the same trajectory, but with the terrorist-supporting organization disabled. What is interesting is that these curves nearly always have a maximal point. It is also interesting to note that the organizations are not necessarily located inside or near the "cities". Some of them are local, but there are also structures that expand to other regions ("far" links). This is due to the constant movement of the agents. After reaching the maximum, the number of terrorists diminishes. The numerical results were as follows.

4.9
But these are only average values. We must remember that the average value taken from simulation experiments with random factors is not very informative. Note that, for example, for the trajectory of the number of terrorist acts (with terrorist-supporting enabled), the confidence interval with a probability of 0.9 is between 6 and 50 (see figure 7), with an average of 31.

Fig 4
Figure 4. Average and confidence intervals for the number of active terrorists. Terrorist-supporting structure enabled

4.10
Figure 6 is a 3-dimensional image (a standard PASION output) of the probability density function for the number of accumulated terrorist acts. The horizontal axes are the time and the number of terrorist acts, and the vertical axis is the probability density function.

Fig 5
Figure 5. Average and confidence intervals for the number of active terrorists. Terrorist-supporting structure disabled

Fig 6
Figure 6. 3D image of the probability density function for terrorist acts. Terrorist-supporting structure enabled

4.11
Figure 7 shows the same trajectory as a 2D plot, indicating the limits of the confidence intervals for a probability level of 0.9 (shadowed region).

Fig 7
Figure 7. Average and confidence intervals for the number of terrorist acts. Terrorist-supporting structure enabled

* Conclusions

5.1
The present simulation is a first stage of research that should be continued and which may result in a much more detailed and realistic approach to the modeling of the dynamics of terrorism. Similar models could perhaps be developed using the System Dynamics approach, with a more global view, but with System Dynamics we would not be able to see the structure creation and the agent's movements. Note that this geometrical approach, which includes attraction centers (cities), can be very important and can hardly be introduced into System Dynamics models. Our experiments have been done on a PC. Obviously, by using multi-processing and distributed simulation, one can work with similar models but which are hundreds of times faster and greater. Note that one of the aims of this model is to simulate the process of creating, as well as that of destroying, a hierarchical structure while possessing only partial information about the structure as a whole. An "infiltrating" or "collaborating" agent has only partial information and is only able to see his superior and subordinates, so his action cannot result in destroying the entire structure.

5.2
The model presented here should be treated as a small contribution to the problem of terrorism modeling and simulation, rather than a weapon in the war on terrorism. The study is focused both on the model and the simulator. The emphasis of our model has been placed on the hierarchical structures' interactions and on the widely-distributed character of this kind of war. In future research, these two aspects of this war should always be taken into account. In other words, in order to combat a widely-distributed enemy, it is necessary to develop widely-distributed strategies. It seems that the actual defense systems, based on territorial and nation-oriented strategies, are rather obsolete and cannot be used in such a new kind of war.

5.3
As PASION models are open to changes, we can easily add new processes and events. In particular, in order to simulate the model dynamics for longer model time intervals, a birth-and-death process should be added to renew the involved human resources. Also, a process of improvement and proliferation of the weapons of mass destruction should be added to control the impact of terrorist acts.

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