# Mathematical Modelling to Evaluate Measures and Control the Spread of Illicit Drug Use

• Afsaneh Bakhtiari
• Alexander Rutherford
Chapter
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 18)

## Abstract

Millions of street-involved-youth worldwide are vulnerable to using and trading illicit drugs, which also place this group at high risk of drug-related criminality and health problems. It is often the case that drug users begin trafficking under the social influences within the drug culture to generate income for supporting their drug habits. The relative merits of behavioural (primary) or law enforcement (secondary) interventions for controlling the spread of drug use are widely debated. In this paper, we develop a network model to evaluate the effectiveness of modelling strategies. A network model with traffickers, current drug users and potential users is constructed. Traffickers exert social influence on current users to deal drugs and on potential users to initiate drug use. Primary intervention prevents potential users from initiating drug use while secondary intervention acts to reduce initiation into trafficking. To accomplish this, we vary the hypothetical social influence parameters in the model. Next, we analyze the properties of this system using dynamical system methods including mean field approximation (MFA), fixed point theory and bifurcation analysis. Furthermore, to evaluate the relative effectiveness of the two interventions, we study the properties of the phase transition between a drug-free and a drug-endemic state at equilibrium mathematically. Drug-free and drug-endemic states are separated by a curved phase transition. Via the shape of the phase transition curve we obtain the optimal intervention. Our findings confirm that a combination of primary and secondary interventions is the optimal intervention strategy. The optimal mixture of the two strategies depends on the relative numbers of drug users and traffickers.

## Keywords

Network Model Drug User Social Influence Bifurcation Curve Stable Node
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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