What Disaster Response Management Can Learn From Chaos Theory

Conference Proceedings
May 18-19, 1995

Edited by
Gus A. Koehler, PhD.

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Gus Koehler

Over the past eight years I helped manage California state government's medical disaster response. Disaster that I responded to include earthquakes, fires, civil disturbances, massive explosions, and hazardous materials spills both liquid and gas. As a state manager my role was to reinforce the city and county emergency medical response in the field and at hospitals with personnel and supplies so that injuries could be rapidly stabilized.

Managing a state's medical response to a major medical disaster involves tracking the relationships between various dynamic processes that occur between one city's or county's emergency medical service's (EMS) response and an emergent macro or disaster area wide pattern involving many cities and counties. The trick is to "see" what is happening during an event that has occurred at a random moment in time, that generates an unknown number of casualties in unknown locations and with many local EMS systems responding that have damaged components. Having "seen" this, the disaster manager tries to influence the development of the multi-county EMS disaster response system such that more lives are saved.

In what follows I try to show that fractal and path dependent theories are useful for understanding such events, derive hypothesis that may predict the optimal size and cost of an efficient EMS disaster response system, and propose methods for testing them.


Day-to-Day Emergency Medical Systems

An emergency medical system is "a coordinated arrangement of resources (including personnel, equipment, and facilities) which are organized to respond to medical emergencies, regardless of cause" (ASTM, 1988). Typically, an EMS system has the following components:

An emergency medical systems are tightly coordinated to save lives. A failure in any one of the components can cause life threatening delays.

Medical Disasters

A medical disaster occurs when the local EMS system's normal triage standards, transport, medical care and local mutual aid resources are severely damaged and/or overwhelmed by victims.

These conditions, which will be developed more fully below, require substantial changes in the way emergency medical care is delivered. They raise the following fundamental questions:

Macro-level: how do I perceive and understand the emergence of the emergency medical response structure relative to the progression of the forces causing the disaster? That is, is there a pattern that emerges over the entire affected geographical area, including the state's various emergency response systems that feed personnel and material into it, which can be anticipated and used to manage the response? Micro-level: how does the emergence of dynamic microstructure such as fire service establishing localized emergency operation centers, hospitals organizing their internal response, distribution of patients, and rapid spontaneous emergence of response structure effect the macro-pattern, and what can be done to improve or direct this dynamic such that the most efficient emergency medical response macrostructure emerges?


EMS Phase Transitions

There are two types of phase transitions: first-order and second-order (Waldrop, 1992). A first-order phase transition involves a sharp change from one physical state to another. An example is rapid transformation of water to ice. The change is very abrupt and well defined. A second-order phase transition takes more time to accomplish and is less precise. Once the transition starts no clear cut structure remains or immediately emerges but there are lots of little structures coming into and going out of existence. Later, a new structure forms but only after an extended period of time. Langton describes such a second order phase transition as follows:

"Order and chaos intertwine in a complex, ever-changing dance of submicroscopic arms and fractal filaments. The largest ordered structures propagate their fingers across the material for arbitrarily long distances and last for arbitrarily long times" (Waldrop sighting Langton, 1992, p. 230).
I believe that emergency medical services systems go through a second-order phase transition immediately following a major medical disaster. The EMS phase transition occurs at the moment when the day-to-day EMS system is struck by a disaster and just before it self-organizes to form a new "disaster" EMS system (or fails to do so) to care for the injured. Experience shows that this process of establishing a new EMS disaster structure is not orderly and may take many days to accomplish.

For example, an EMS phase transition would follow a Richter magnitude 7.5 earthquake on the Hayward fault in the San Francisco East Bay area. A report by the California Division of Mines and Geology makes some sobering predictions (1987).

"Deaths resulting from this scenario earthquake are estimated to range from 1,500-4,500 depending upon the time and day of occurrence. Hospitalized casualties are estimated to be 3 times the number of deaths [4,500-13,500]; significant non-hospitalized casualties are estimated at 30 times the number of deaths [45,000-135,000]. Eight of the 26 acute care hospitals (99 beds or more) in Alameda and Contra Costa Counties are located within one mile of the Hayward fault. This represents a bed capacity of 2,300 of a total of 6,200 available in these major facilities (about 35%). Almost all buildings at these 8 sites were constructed prior to adoption of more stringent hospital building requirements in 1972."
The proximity of the hospitals, fire stations, communications system, and population to the Hayward fault leads to the prediction that major portions of the EMS system will be destroyed or severely impacted. Field responders will be scattered all over the area trying to deal with the injured, fires, and other problems. It will be difficult to link field responders with each other, to hospitals, or to emergency response management systems. Studies of medical disasters such as the Bhopal chemical disaster in India, and major earthquakes show that standing or damaged health care facilities are inundated by those who are least injured first, followed by those who are more injured (Jacobs, 1983). The vast majority of thousands of earthquake victims will be rescued and waiting for or being cared for within the first 24 to 72 hours of the event. Hospitals will probably continue to see a considerable number of casualties for an additional six days, (De Vill de Goyet, 1976) and may continue to perform earthquake related operative procedures for four of five weeks after the event. The death toll for an earthquake peaks about the same time that the majority of injured have been rescued. Clearly the elements of the EMS system resemble a second-order phase transition as they try to reorganize.


EMS Phase Transition, Chaos, and Fractals

When we examine the organizational dynamics of this error making process we are talking about chaos theory. The structural forms which a chaotic process leaves in its wake are fractals. In summary, chaos is the process; fractals are the resulting structure (Pietgen, Jurgens, and Saupe, 1993).

Fractals are geometric objects that demonstrate self similarity across several size scales. For example, a cauliflower head contains branches or parts, which when removed and compared with the whole are very much the same, only smaller. These clusters can be broken up into even smaller clusters and they too will appear to be similar to the original cluster. This concept of self-similarity can be applied to a geographical area such as that occupied by a mountain range or river system. Pictures taken from space shown self-similarity across several scales (see Plates 4 and 14 in Peitgen, Jurgens, and Saupe, 1993). Viewed from the macro perspective of the multi-county of the entire disaster, we see a large number of local EMS responders and organizations all reaching out toward, across, and around one another. I propose that at a certain point a percolation fractal emerges integrating all of these components. This fractal is a the macro-structure that links all of the local EMS responders and organizations together into an EMS disaster response system. The fractal is self-similar in so far as the local EMS structure resembles the entire multi-city and multi-county response structure.

The EMS Phase Transition and the Generation of Errors

According to Feignebaum, a series of ever increasing errors over carries an organized system over the edge of order into a phase transition and on to chaos (Feigenbaum, 1993). A medical disaster destroys a substantial portion of the EMS system's predictability. From the every-day perspective, the decisions and actions that are being made following a disaster are an ever increasing series of errors. It is at the phase transition/chaos boundary that decision "errors" accumulate at a high and self-reinforcing rate. The decisions are driven by the immediate conditions faced at that particular moment in a particular geographic place within the larger disaster area. An EMS second order phase transition occurs when the number of self-reinforcing errors are so great that the components of the day-to-day EMS system are no longer effectively integrated.

Percolation Fractal: the EMS Macrostructure

According to Peitgen, Jurgens, and Saupe:

"When a structure changes from a collection of many disconnected parts into basically one big conglomeration, we say that percolation occurs. The name stems from an interpretation of the solid parts of the structure as open pores. ...Let us pick one of the open pores at random and try to inject a fluid at that point. What happens? If the formation is below the percolation threshold...we expect that the pore is part of a relatively small cluster of open pores. ...Below the threshold we can only inject some finite amount of fluid until the cluster is filled but no more" (Peitgen, Jurgens, and Saupe, 1992, p.464).
The "collection of parts" are all of the existing and emergent elements of the EMS disaster response. The EMS components of the "finite cluster" that are traced out by "injecting a fluid" are a combination of the following "pores" and "connections". "EMS Pores" are defined as:

"All" means hospitals, staging areas, and any other entity involved in the disaster response both in the geographic area of the disaster and outside of it.The connections that extend out from and connect to other EMS pores over time are:

If fluid was injected into the center of an EMS pore it would reveal communications paths, and the road or air corridors being taken to deliver the injured, supplies and personnel. This local pattern is defined as an "EMS cluster".

Immediately following the disaster the pattern of EMS clusters that emerges across the area of the disaster would probably be highly localized to discreet "pores" and their immediate connections. As time goes by, more and more connections are made between EMS pores such as hospitals, and communications centers, with emergency operations centers and staging areas. Ideally, all of the EMS clusters connect to each other and a percolation fractal emerges.

Percolation Threshold

We've already observed that at the moment when one overall structure emerges from all of these disconnected parts percolation occurs. This event is called the "percolation threshold." Figure 1 shows three triangles that clarify what a percolation threshold is. Each triangle is subdivided into a large number of smaller triangles, some shaded and some not. Each sub-triangle is colored black or not according to a random event. This random event occurs with a prescribed probability of not being colored at all (0.0) to a virtual certainty of being colored (1). The overall shape and connectedness of the individual triangles depends dramatically on the probability chosen. When the probability of a triangle being colored is around 0.3 (30%) very few of the triangles are filled in in contrast to when it is closer to 1.0, say 0.7 (70%)(see Figure 1). At a critical value called the "percolation threshold", the number of triangles and their interconnectedness seem to become glued together into one big irregular lump.

Peitgen, Jurgens, and Saupe tell us that:

"Right at the percolation threshold this maximal cluster is a fractal!" (p.465)
Returning to our problem, how close the EMS system is to the percolation threshold is defined by the probability of an EMS cluster being connected to another EMS cluster (the interconnection of the little triangles in Figure 1). This probability has been measured experimentally (the model was an idealized simulation and did not represent an EMS system) using a simulation of probablistically determined points. It is approximately 0.5928 (Peitgen, Jurgens, and Saupe, 1993, p. 466) This suggests that each cluster throughout the disaster area must have almost a 60% chance of being connected to another for a percolation fractal (EMS disaster response system) covering the entire area of the disaster to emerge. At the percolation threshold the probability for establishing an EMS system is high since the maximum interconnectedness of the entire disaster area is achieved as demonstrated by the emergence of a percolation fractal.

Mathematical simulations predict that this occurs very rapidly. However, the same studies predict that the time it takes to get to this point will be longer for a large number of clusters, in fact exponentially longer, than the time it takes for a smaller number of clusters. For example, if it takes ten hours to reach the percolation point for a multi-casualty incident with twenty clusters, it will take ten times as long for 100 clusters, and 100 times as long for 500 clusters (Peitgen, Jurgens, and Saupe, 1993, p. 466). We will return to this point later.


Our discussion has shown that the behavior of the underlying network and the emergence of a percolation fractal is determined almost entirely by the probability of connectedness of each pore, and of each EMS cluster with other EMS pores and EMS clusters. In this section I will try to identify some of major EMS factors that determine how an EMS pore develops and what determines its connection probability. The factors that account for this variation can be grouped into four categories:

The first three factors define how an EMS pore evolves. The last identifies the connection process associated with reaching the percolation threshold. All four factors are interrelated; the rate that the first set of factors change at effects the last which in turn feeds back to affect the first group.

Factors that Affect the Evolution of an EMS Pore Sensitivity to Initial Conditions

How the day-to-day EMS system is disrupted establishes the initial conditions that will govern how the local response self-organizes (Jantsch, 1980). Diagram 2 identifies some of these critical factors. Probably the most important are:

Responders make decisions --or "errors"--based on the initial state these factors occupy following a disaster. This produces an ever increasing decision making error rate leading the response over the day-to-day EMS organizational edge into a phase transition and on to chaos. These distinctions also establish the new EMS pores that may eventually aggregate into an EMS percolation fractal.

The Injured

First, the rate at which the medical disaster itself unfolds depends upon continuation of the mechanism of injury (after shocks, continuous release of a poisonous gas), the rate that victims are being rescued, their injuries (crushing vs. gas inhalation for example) and the type of medical care required. The dynamical relationships of thee conditions drive the response elements.

Unusual oscillatory relationships between injury groups may emerge due to the maldistribution or lack of medical resources. Individuals with life threatening injuries may compete with less injured people, both represented by friends and relatives or directly with each other for care. The epidemiologist May, speaking about infectious diseases notes that "....it is widely accepted that non-seasonable fluctuations arise as a consequence of the dynamical interactions between two or more populations whether they be host and parasite, predator and prey, plant and herbivore, indeed any of these combinations" (May, 1984, p. 587). The differential rates at which the most injured and less injured arrive at a Casualty Collection Point or hospital could create such oscillations. If there is unimpeded access to care the least injured get the most care. They quickly overload available resources and the more injured die from long waiting times. On the other hand, if care is restricted to the most injured, which is very personnel and supply intensive, the result may be fewer deaths but more morbidity. Thus unusual oscillatory behavior in mortality/morbidity statistics could result from triage criteria, arrival rates and times of the two populations, and competition for scarce medical care. Field Responders and Supplies

A second group of rates is driven by how quickly responders can be mobilized, transported to where they are needed, fed, and be relieved by fresh staff. For example, following the start of the Los Angeles civil disturbance, fire fighters were able to quickly organize themselves but were delayed in getting into the field by a lack of law enforcement support. Once in the field, the logistical support structure for food, rest, and personal hygiene was terrible at best (Koehler, 1993). Thus the rate that first responders become exhausted (24 hours is probably the maximum limit for most people) and can be replaced will have an important affect on the response. A third set of rates involves how quickly supplies are used relative to type of injury and the number requiring care. For example, inhalation injuries require different supplies that do crushing injuries. IV solutions must be available in large numbers. Again, oscillatory effects may result from the availability of medical supplies, movement of medical personnel in and out of the area, and the rate that the injured arrive for care (Koehler, 1991). Communications

The ability of the managers to regulated the medical disaster response depends on availability and quality of communications For example, the location, number of injured and their injuries must be effectively communicated to the county if personnel and supply needs are to be met. A runner would be quite slow and probably able to carry relatively little updated information in contrast to digital radio communications. Thayer suspects that it is the blocking of communication, and thus the creation of non-equilibrium between critical response elements that leads to mutations and the emergence of new organizational forms (Thayer, 1975). I suspect that there is more to it than blocking communications; how timely and relevant the information is about how the disaster/response is going is equally important.

Following a disaster there is communications failure and information overload. For example:


California state government exercises have shown that the availability of transport is a key variable regulating how quickly the response system can be organized. Casualties will have to be moved from the site of their injury to emergency care field stations, hospitals and evacuation areas. A significant percentage of the surface transportation in the disaster area, possibly more than half, will be carried out by the public. The remainder will require public and private agency assistance. Depending on the disaster, a very large number of casualties may have to be evacuated. For example, one plan calls for the evacuation of 60,000 casualties from seven counties in Southern California within 96 hours. Such an effort requires very rapid availability and proper configuration of a considerable number of CH-47 and UH-1 helicopters, C5A, and C-130 aircraft (Authority, 1986 and associated working papers). This evacuation rate in turn drives how quickly hospitals and other care centers use up their personnel and supplies.

The rate that supplies are used must be matched by the combined rate of acquisition, volume, handling and supply delivery systems (an estimated 700 tons of material will have to be acquired and moved to the disaster site within 72 hours for a major earthquake medical disaster). Organizational Structuration

"Organizational structuration" is the second major factor that regulates EMS pore formation. "Organizational structuration" refers to the process that formal and informal organizations must go through to self-organize.

The factors that regulate the speed of organizational structuration following a disaster are:

Spontaneous emergence of small scale community organizations

Immediately after a major disaster many families and neighbors spontaneously organize themselves to rescue and care for victims. Within 30 minutes of a major disaster 75% of the health survivors are actually engaged in efficient rescue operations (Drabek, 1989). Sociologist have identified factors that influence emergent group formation:

Local EMS field responders also begin to organize themselves. Typically individual fire departments, and ambulance companies assess the local damage and injuries and begin to act based on limited local information. Dispatching and inter-hospital communications are often spotty, or overloaded. This limits the formation of response structures. If there was a disaster exercise prior to the event, the established lines of authority don't break down. But, if authority is weak, it can completely disappear. Generally, the tendency is for overall authority not to be exercised (Drabek 1989). In fact old jurisdictional disputes between agencies may be exacerbated as was the case in Los Angeles between the Los Angeles City Police Department and Los Angeles County Sheriffs Department (Webster, 1993). Forty-Two Types of Organizational Structuration Process

Established organizations like hospitals and fire departments are impacted in different ways by a disaster. Krebs and his colleges have demonstrated that any one of 42 different organizational types may emerge as these and emergent organizations self-organize (Krebs, 1989). This theory and its supporting empirical evidence is to extensive to review here.

The point is that organizations will take a multitude of different paths as they go through their own structuration process depending on:

Looking at the entire disaster area, all of the various EMS pores would be structurating differently. Differential stress levels for different organizations

Organizational stress can take different forms. Not all public and private organizations will experience the same level of stress. Those who respond to multi-casualty events such as fire and ambulance personnel may experience less stress than health department personnel who have only conducted infrequent disaster drills.

Increased stress on organizations tends to increase the rate of decision making, particularly at lower levels. An emergency department nurse may make critical organizational decisions that, under normal circumstances are made by the department chief. More autonomy and less coordinated decision making results in the rapid commitment of organizational personnel and resources. Such a commitment may be adaptive in the short run but lead to serious problems in the long run. For example, over commitment of staff quickly burns them out.

Delays between Availability of Information, Decision-making and Action

The third factor that regulates EMS pore formation is delays between when a decision is made and the time it is carried out. The basic premiss of time delayed control theory is that the consequences of an action taken now tend to be delayed into the future (Waldrop, 1992). The process is analogous to trying to adjust the hot water in the shower. It takes a few moments for the effect of the adjustment to actually reach the spray hitting your skin.

Disaster management has similar problems. The delay between the availability of information, decision-making, and the final action can take a number of forms. It can be:

These delays in turn affect the evacuation, supply, transportation and other rates already identified. Delays in decision-making can also create various unintended oscillations. Both increase the error rate driving the system even further into chaos.

Our brief discussion of organizational structuration shows that there will be considerable variation in how any one EMS pore connects. These dynamic variations are multi-dimensional and very hard to control.

Principal of increasing returns

Once an EMS pore begins to form it automatically reaches out for support. At the same time state, county, and city Emergency Operations Centers are reaching towards some of them. Brian Author's work may help understand this connecting up process.

Arthur developed the "principal of increasing returns" or path dependent processes to account for the outcome of self-reinforcing structures that typically possess a multiplicity of possible outcomes (Arthur, Ermoliev, and Kaniovski, 1987). Basically, the theory holds that once a point begins to aggregate it is likely to aggregate further.

What happens...where [business] firms' locational choices depend on part upon the numbers of firms in each region at the time of choosing? Here increments to the regions are not independent of previous locational choices. ...We now have a path-dependent process, where the probability...of an addition to a [particular region over another] becomes a function of the numbers of firms, or equivalently, of the proportions of the industry, in each region at each time of choice (Arthur, Ermoliev, Kaniovski, 1987, p. 295)
Generally, more and more firms tend to come where the previous firms have already located (Arthur, 1990). The principal of increasing returns also appears to describe the dynamic process of how a single EMS pore becomes an EMS cluster by connecting to the response structure through path dependant processes. In our case path dependant processes are the factors that shape the creation of supply, communications and other constantly reinforced paths that link EMS pores. The more EMS pores there are in a particular area, the greater the chance of their being connected.

Sociologist have identified various inter-organizational linking behaviors that may regulate path dependant processes (Drabek, 1989). Some of them are:


The theoretical steps that lead to the formation of a EMS disaster response percolation fractal are:

  1. An EMS phase transition occurs due to a progressive or sudden break down of the day-to-day EMS system. More and more decisions errors are made driving the EMS system over the organizational edge into a second-order phase transition and on to chaos.

  2. Unpredictable initial conditions, organizational structuration processes, differential stress, and other factors provide seed points for the evolution of EMS pores.

  3. The delay between decision making and actions creates unexpected variations and oscillations of critical rates that affect EMS pore formation and connections.

  4. EMS pore connectedness varies due to sociological conditions and decision making activities, and the principal of increasing returns. Decisions made very early on and the number of EMS pores and clusters in an area may have a decisive affect on whether the percolation threshold can be reached.

  5. If the percolation threshold is reached fractal percolation occurs resulting in an area wide EMS disaster system percolation fractal.

  6. The percolation threshold may not be reached because the probability of connecting all of the EMS pores does not reach 60%. The result is a clumpy structure with a high mortality and morbidity rate then would have been the case if a percolation fractal had emerged.

Some Implications for Managing the Formation of EMS Disaster Medical Systems

The discussion of EMS connection making has been very oversimplified. In reality multiple connections have to be made to any one EMS pore for it to function. The connection probability of 60% applies to making a critical connection with multiple other forming EMS clusters. We already know that the time to make these connections increases expediential with increased numbers of connections. This can be changed by increasing the number of hardened communications linkages that cannot be disrupted by the disaster and are capable of handling the information load. Satellite communications and geolocators for emergency response vehicles is critical. Procuring the number of transport vehicles, supplies, and personnel on very short notice posses a different and more challenging problem.

The experimental percolation threshold findings suggest that the number of ground and air transport, as well as materials and personnel needed to achieve the 60% connection rate must increase exponentially to significantly improve the probability of any one cluster being connected. This is a hard task for disaster managers who typically have few "connection" resources and large amounts of personnel and supplies at their immediate disposal.

If EMS cluster growth is determined by the principal of increasing returns, than the amount of resources must be very high indeed since clusters that are close to many other clusters will have a much high probability of being reinforced while more isolated ones don't. This leads to the prediction that large highly visible urban counties with good communication and low numbers of casualties will quickly receive aid; small, rural counties with poor communications are more likely to be ignored even if they have many casualties relative to their population and EMS system's capacity. This occurred during the California Loma Prieta earthquake. San Francisco and Alameda Counties received the most attention early on even though Santa Cruz and San Bonito Counties had relatively more damage.

The most interesting conclusion suggested by this theory is that it should be possible to determine how long it will take for a predicted number of EMS clusters for a particular disaster (an earthquake on the Hayward fault for example) to reach the percolation threshold. This estimate of how long it will take to establish an EMS disaster response system can then be compared with the time available for saving the vast majority of lives which, as noted earlier, is between 24 to 72 hours following an earthquake. (The length of this standard will probably vary by the profile of disaster related injuries.) More optimal EMS systems could be designed by reducing the number of clusters or increasing the connection rate. This could be accomplished by, among other things, improving communications and transportation and increasing hospital carrying capacity so that the percolation threshold can be quickly achieved. Data on system alternatives could be used to estimate costs. To my knowledge, no means is currently available to estimate possible EMS response costs for large disasters.

Testing the Theory

Computer simulation is probably the best method for testing the theory. EMS simulations already exist for field care (Koehler, Foley, and Jones, 1991), and for hospitals and for emergency medical systems (Fletcher and Delfosse, 1979; 1981). These simulations look at single units or a single path of interactions rather than at the entire disaster area as would be required for a test of this theory. Some kind of connectionist model would have to programmed that would tie all of these elements together (Waldrop).

Key concepts and assertions that should be experimentally investigated are:

  1. Determine if the day-to-day EMS system goes through a second order phase transition and mathematically define it.

  2. Prove that the EMS disaster system percolation threshold is approximately 0.5928.

  3. Demonstrate that path dependant processes do occur during a disaster response and show how the various rates, structuration processes, etc., affect when the paths are established, how they are reinforced, and how plastic they are in terms of redirecting the flow of supplies, communications, etc.

  4. Conduct simulation experiments to determine how many clusters are likely to form after a Hayward Fault Earthquake and the length of time it would take to cross the percolation threshold for an efficient EMS system to emerge. Determine if this data can be used to estimate the cost of the Emergency Medical Services response.


American Society for Testing Materials (ASTM) (1988). "Standard Terminology Relating to Emergency Medical Services (Standard F117-90a)," Philadelphia.

Arthur, B., Krmoliev, Y., and Kaniovski, Y. (1987). "Path-dependent Processes and the Emergence of macro-structure". European Journal of Operational Research, 30.

Arthur, B. (1990). "Positive Feedback in the Economy". Scientific American, February.

Drabek, T. (1986). Human System Response to Disaster. New York: Springer-Verlag.

Feigenbaum, Mitchell (1993). "Foreword" in Peitgen, H., Jurgens, H., and Saupe, D. (1993), Chaos and Fractals. New York: Springer-Verlag.

Koehler, G. (1991). "The Emergency Medical Services Response to the I-5 Interstate Highway Multi-Casualty Incident" a chapter in "Task Force Report to Governor Pete Wilson, "Dust-Related Collisions, Interstate 5, Panoche Junction Overcrossing/Kamm Ave., November 29, 1991".

Koehler, G. (1993). "Medical Care for the Injured: The Emergency Medical Response to the April 1992 Los Angeles Civil Disturbance." California Emergency Medical Services Authority.

May, R. (1984). "Oscillatory Fluctuations in The Incidence of Infectious Disease and the Impact of Vaccination," Journal of Hygiene 93, 587-608.

Peitgen, H., Jurgens, H., and Saupe, D. (1993), Chaos and Fractals. New York: Springer-Verlag.

Waldrop, M. (1992). Complexity: The Emerging Science at the Edge of Order and Chaos. New York: Simon and Schuster.

Webster, W. (1992). The City in Crises. Office of the Special Advisor to the Board of Police Commissioners City of Los Angeles, October 21, 1992.


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