A Model of Chaotic Drug Markets and Their Control

Drug markets are often described informally as being chaotic, and there is a tendency to believe that control efforts can make things worse, not better, at least in some circumstances. This paper explores the idea that such statements might be literally true in a mathematical sense by considering a discrete-time model of populations of drug users and drug sellers for which initiation into either population is a function of relative numbers of both populations. The structure of the system follows that considered in an arms control context by Behrens et al. (1997). In this context, the model suggests that depending on the market parameter values, the uncontrolled system may or may not be chaotic. Static application of either treatment or enforcement applied to a system that is not initially chaotic can make it chaotic and vice versa, but even if static control would create chaos, dynamic controls can be crafted that avoid it. Socalled OGY controls seem to work well for this example

A Model of Moderation: Finding Skiba Points on a Slippery Slope

A simple model is considered that rewards ”moderation” - finding the right balance between sliding down either of two ”slippery slopes”. Optimal solutions are computed as a function of two key parameters: (1) the cost of resisting the underlying uncontrolled dynamics and (2) the discount rate. Analytical expressions are derived for bifurcation lines separating regions where it is optimal to fight to stay balanced, to give in to the attraction of the ”left” or the ”right”, or to decide based on one’s initial state. The latter case includes situations both with and without so-called Dechert- Nishimura-Skiba (DNS) points defining optimal solution strategies. The model is unusual for having two DNS points in a one-state model, having a single DNS point that bifurcates into two DNS points, and for the ability to explicitly graph regions within which DNS points occur in the 2-D parameter space. The latter helps give intuition and insight concerning conditions under which these interesting points occur

An Age-Structured Single-State Initiation Model -- Cycles of Drug Epidemics and Optimal Prevention Programs

This paper introduces a model for drug initiation that extends traditional dynamic models by considering explicitly the age distribution of the users. On the basis of a 2-groups model in which the population in split into a user and a non-user group the advantage of a continuous age distribution is shown by considering more details and by yielding new results. Neglecting death rates reduces the model to a single state (1-group) descriptive model which can still simulate some of the complex behavior of drug epidemics such as repeated cycles. Furthermore, prevention programs - especially school-based programs - can eb targeted to certain age classes. So in order to discover how best to allocate resources to prevention programs over different age classes we formulate and solve optimal control models

High and Low Frequency Oscillations in Drug Epidemics

We extend the two-dimensional model of drug use introduced in Behrens et al. [1999, 2000, 2002] by introducing two additional states that model in more detail newly initiated (“light”) users’ response to the drug experience. Those who dislike the drug quickly “quit” and briefly suppress initiation by others. Those who like the drug progress to ongoing (“moderate”) use, from which they may or may not escalate to “heavy” or dependent use. Initiation is spread contagiously by light and moderate users, but is moderated by the drug’s reputation, which is a function of the number of unhappy users (recent quitters + heavy users). The model reproduces recent prevalence data from the U.S. cocaine epidemic reasonably well, with one pronounced peak followed by decay toward a steady state. However, minor variation in parameter values yields both long-run periodicity with a period akin to the gap between the first U.S. cocaine epidemic (peak ~1910) and the current one (peak ~1980), as well as short-run periodicity akin to that observed in data on youthful use for a variety of substances. The combination of short- and long-run periodicity is reminiscent of the elliptical burstors described by Rubin and Terman [2002]. The existence of such complex behavior including cycles, quasi periodic solutions, and chaos is proven by means of bifurcation analysis

Bifurcating DNS Thresholds in a Model of Organizational Bridge Building

A simple optimal control model is introduced, where “bridge building” positions are rewarded. The optimal solutions can be classified in regards of the two extern parameters, (1) costs for the control staying at such an exposed position and (2) the discount rate. A complete analytical description of the bifurcation lines in parameter space is derived, which separates regions with different optimal behavior. These are resisting the influence from inner and outer forces, always fall off from the boundaries or decide based on one’s initial state. This latter case gives rise to the emergence of so-called Dechert-Nishimura-Skiba (DNS) points describing optimal solution strategies. Furthermore the bifurcation from a single DNS point into two DNS points has been analyzed in parameter space. All these strategies have a funded interpretation within the limits of the model

Optimizing Counter-Terror Operations: Should One Fight Fire with "Fire" or "Water"?

This paper deals dynamically with the question of how recruitment to terror organizations is influenced by counter-terror operations. This is done within a optimal control model, where the key state is the (relative) number of terrorists and the key controls are two types of counter-terror tactics, one (“water”) that does not one (“fire”) that does provoke recruitment of new terrorists. The model is nonlinear and does not admit analytical solutions, but an efficient numerical implementation of Pontryagin’s Minimum Principle allows for solution with base case parameters and considerable sensitivity analysis. Generally this model yields two different steady states, one where the terror-organization is nearly eradicated and one with a high number of terrorists. Whereas water strategies are used at almost any time, it can be optimal not to use fire strategies if the number of terrorists is below a certain threshold

Cycles of Violence: A Dynamic Control Analysis (or Model?)

This paper introduce and analyze a simple model of cycle of violence in which oscillations are generated when surges in lethal violence shrink the pool of active violent offenders. Models with such endogenously induced variation may help explain why historically observed trends in violence are generally not well correlated with exogenous forcing functions, such as changes in the state of the economy. The analysis includes finding the optimal dynamic trajectory of incarceration and violence prevention inteverventions. Those trajectories yield some surprising results, including situations in which myopic decision makers will invest more in prevention than will far-sighted decision makers

Incentive Stackelberg Strategies for a Dynamic Game on Terrorism

This paper presents a dynamic game model of international terrorism. The time horizon is finite, about the size of one presidency, or infinite. Quantitative and qualitative analysis of incentive Stackelberg strategies for both decision-makers of the game (“The West” and “International Terror Organization”) allows statements about the possibilities and limitations of terror control interventions. Recurrent behavior is excluded with monotonic variation in the frequency of terror attacks whose direction depends on when the terror organization launches its terror war. Even optimal pacing of terror control operations does not greatly alter the equilibrium of the infinite horizon game, but outcomes from the West’s perspective can be greatly improved if the game is only “played” for brief periods of time and if certain parameters could be influenced, notably those pertaining to the terror organization’s ability to recruit replacements

High and Low Frequency Oscillations in Drug Epidemics

We extend the two-dimensional model of drug use introduced in Behrens et al. [1999, 2000, 2002] by introducing two additional states that model in more detail newly initiated (“light”) users’ response to the drug experience. Those who dislike the drug quickly “quit” and briefly suppress initiation by others. Those who like the drug progress to ongoing (“moderate”) use, from which they may or may not escalate to “heavy” or dependent use. Initiation is spread contagiously by light and moderate users, but is moderated by the drug’s reputation, which is a function of the number of unhappy users (recent quitters + heavy users). The model reproduces recent prevalence data from the U.S. cocaine epidemic reasonably well, with one pronounced peak followed by decay toward a steady state. However, minor variation in parameter values yields both long-run periodicity with a period akin to the gap between the first U.S. cocaine epidemic (peak ~1910) and the current one (peak ~1980), as well as short-run periodicity akin to that observed in data on youthful use for a variety of substances. The combination of short- and long-run periodicity is reminiscent of the elliptical burstors described by Rubin and Terman [2002]. The existence of such complex behavior including cycles, quasi periodic solutions, and chaos is proven by means of bifurcation analysis

Cycles of Violence: A Dynamic Control Analysis (or Model?)

We introduce and analyze a simple model of cycle of violence in which oscillations are generated when surges in lethal violence shrink the pool of active violent offenders. Models with such endogenously induced variation may help explain why historically observed trends in violence are generally not well correlated with exogenous forcing functions, such as changes in the state of the economy. The analysis includes finding the optimal dynamic trajectory of incarceration and violence prevention inteverventions. Those trajectories yield some surprising results, including situations in which myopic decision makers will invest more in prevention than will far-sighted decision makers