Optimal control intervention strategies in low- and high-risk problem drinking populations

A mathematical model of drinking that incorporates the impact of relapse, in a rather simple setting, is analyzed primarily under the impact of two time-dependent controls (policies) put in place over a finite time horizon. The model divides the population of interest in three drinking classes: occasional or moderate drinkers (S), problem drinkers (D) and temporarily recovered (R). In this SDR framework, individuals are assumed to mix at random within their shared drinking environments. The transmission process is modeled as a social "contact" process between D, S, and R-individuals within an unchanging shared drinking environment. High relapse rates in this framework, including those resulting from temporarily effective detoxification programs, under some circumstances can do more harm than good. The impact of two intervention strategies aimed at distinct processes is explored in high- and low-risk drinking cultures as defined in this manuscript. The first intervention's goal is to reduce the intensity of "social influence" while the second tries to slow down the rate of peer-induced relapse. The use of time-dependent controls in low-risk drinking environments may remain effective beyond their lifespan (implementation time horizon). However, the effectiveness intervention programs in high-risk environments ends soon after the control efforts have been terminated. That is, the system returns to its previous problem drinking state once the controls have been removed. Control measures will have a long-term effect only when carried out in conjunction with policies that generate dramatic changes in the population's behavioral norms. That is, changes that reduce significantly the risks of abusive drinking in shared drinking environments must be implemented or costly programs put in place if change is going to be lasting. A brief discussion of the relationship between the use of costly measures and the achievement of shifts in drinking norms will be discussed.