Mathematical Modeling of Drug Use: The Dynamics of Monosubstance Dependence for Two Addictive Drugs

Abstract: Based on previous work done in this field, we build a dynamical system that describes changes in drug addiction in an isolated population when two addictive substances are available simultaneously. We then use our model to investigate whether the system captures the process of users switching drug habits. One of the motivations for this project is to mathematically check the conjecture that being addicted to a less-addictive substance will effectively lead individuals to become dependent on more addictive and potentially more dangerous drugs. We introduce additional assumptions, under which our model is reduced to a competitive Lotka-Volterra system. This dynamical system has three or four fixed points, stability of which then gives an implication about the outcomes of the competition between addictive substances and, therefore, the fate of individuals in the population. From the analysis of the reduced model, we determine that there actually exist parameter regimes that capture the following dynamics: depending on the initial distribution of the drug preference; either the use of both drugs will die out, the usage of one of the drugs will become prevalent, or addictions to both drugs will coexist.