The logistic model is a commonly used mathematical model to describe the development of addiction. It assumes that the probability of addiction increases as the level of exposure to the addictive substance increases, but at a decreasing rate.

The threshold is the key parameter in the logistic model of addiction. It represents the level of exposure to the addictive substance at which the probability of addiction starts to increase.

The renewal model is a commonly used mathematical model to describe the relapse process in addiction. It assumes that the risk of relapse is highest immediately after quitting the addictive substance and then decreases over time.

The hazard rate is the key parameter in the renewal model of relapse. It represents the instantaneous risk of relapse at any given time.

The epidemic model is a commonly used mathematical model to describe the spread of addiction in a population. It assumes that addiction spreads through contact between individuals, similar to the spread of an infectious disease.

The transmission rate is the key parameter in the epidemic model of addiction. It represents the rate at which addiction spreads from one individual to another.

Cost-benefit analysis is a commonly used mathematical model to describe the economic costs of addiction. It compares the costs of addiction, such as healthcare costs and lost productivity, to the benefits of addiction treatment.

The benefit-cost ratio is the key parameter in the cost-benefit analysis of addiction. It compares the total benefits of addiction treatment to the total costs.

Randomized controlled trials are commonly used mathematical models to describe the effectiveness of addiction treatment. They compare the outcomes of individuals who receive addiction treatment to the outcomes of individuals who do not receive addiction treatment.

The treatment effect is the key parameter in the randomized controlled trial of addiction treatment. It represents the difference in outcomes between individuals who receive addiction treatment and individuals who do not receive addiction treatment.

The hyperbolic discounting model is a commonly used mathematical model to describe the decision-making process in addiction. It assumes that individuals discount the value of future rewards more heavily than the value of immediate rewards.

The discount rate is the key parameter in the hyperbolic discounting model of addiction. It represents the rate at which individuals discount the value of future rewards.

The dopamine reward pathway model is a commonly used mathematical model to describe the neurobiological mechanisms of addiction. It assumes that addiction is caused by the activation of the dopamine reward pathway, which is responsible for feelings of pleasure and reward.

Dopamine release is the key parameter in the dopamine reward pathway model of addiction. It represents the amount of dopamine that is released in the brain when an addictive substance is consumed.

Twin studies are commonly used mathematical models to describe the genetic factors that contribute to addiction. They compare the rates of addiction in identical twins, who share 100% of their genes, to the rates of addiction in fraternal twins, who share only 50% of their genes.