Dynamics of Drug Consumption: a Theoretical Model

A continuous time model is proposed to describe the dynamics of drug consumption in a given country. The model has two state variables, addicts and dealers, and eleven parameters, including the total effort exerted by the State, which is considered as control parameter. The model is highly nonlinear and the analysis shows that it is characterized by a transcritical and a fold bifurcation. This implies that for intermediate values of the State's effort the model has two stable equilibria, one trivial, corresponding to the absence of drugs, and one positive, corresponding to drug consumption. On the contrary, for low and high values of the effort only one of the two equilibria is stable. This suggests a two-step control policy. First, exert a very high effort for a few years, so that the system has the time to approach the trivial equilibrium, and then reduce the effort but maintain it sufficiently high so that drug consumption cannot rise anymore. Interesting results on the role played by the price of the drug and the severity of the punishment inflicted to dealers, as well as on the allocation of the effort between therapy and police, have also been obtained.

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