When Two Become One: Optimal Control of Interacting Drug Epidemics

In this paper we set up a model of interacting drug epidemics by taking the ”direct-product” of variations on a one-dimensional optimal control model that exhibits a so-called DNSS or Skiba threshold. Such a threshold reflects in drug policy the well-established paradigm of ”eradication vs. accommodation”; that is, depending on the initial state, stabilizing a drug market at either a low or a high level of use is optimal. We investigate whether and how this ”eradication vs. accommodation” paradigm extends to higher dimensional models and whether and how the presence of a second drug can alter the optimal policy description for the first drug. The main results are, first, the presence of a second drug market can dramatically alter or even reverse the optimal policy prescription for the first drug market. Second, with interacting drug markets ”eradication vs. accommodation” is no longer a binary choice. Rather, we observe a fascinating series of situations with multiple optimal steady states and complex structures of optimal solutions.