Optimal prevention and treatment in a dynamic drug model with diverse feedbacks and relapse

Abstract

This thesis aims at modelling and analysing the implications of potential relapse in a dynamic drug model. More precisely, an optimal control model incorporating the two states "drug users" and "teetotallers" is formulated. The main feature of the model is the fact that individuals quitting drug use do not simply leave the system but rather end up in the precarious state of a teetotaller with a certain risk of relapse. Relapsing teetotallers constitute a second inflow of drug users in addition to ordinary initiation. However, teetotallers are also assumed to have a dissuasive effect on initiation. Consequently, the number of teetotallers has both a positive and a negative effect on the overall drug problem. Moreover, the dynamical system is influenced by the controls "prevention" and "Treatment". Especially the use of treatment in consideration of high relapse rates is of substantial interest. After an introductory exploration of the underlying uncontrolled dynamics, Pontryagin's Maximum Principle is applied in order to solve the optimal control model. As a consequence of the model's complexity, the optimal solution and stable paths are calculated numerically using the Matlab-toolbox OCMat. Furthermore, a sensitivity analysis, describing the impact of variations of model parameters on the long-run solution, is conducted.