Drug abuse remains a major problem in the United States. For several years, drug overdoses, primarily involving synthetic fentanyl and methamphetamines, have been the leading cause of death among individuals aged 18 to 45. We first review the progression of the drug overdose epidemic for various subpopulations, stratifying mortality data by age, sex, race, drug type and geographical location. We show how data assimilation methods can be integrated with age-structured population models to forecast future overdose trends. We also propose a mathematical framework to describe the neurobiology of drug addiction that incorporates well known psychiatric and cognitive paradigms. In our model, drug-induced dopamine activates a biphasic reward response whereby a pleasurable, euphoric state is followed by unpleasant, negative cravings and withdrawal symptoms. Neuroadaptive processes triggered by drug use enhance the negative component of the reward response, against which the user compensates by increasing drug dose and/or intake frequency. The resulting positive feedback loop eventually leads to full addiction. Our model gives rise to different pathways to addiction that allow to represent a diverse set of user profiles (such as genetics, age) or drug potencies, and to explore methods to best alleviate withdrawal symptoms.