Mathematical modeling of neurobiological model of drug addiction
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Abstract
Introduction: addiction is considered a central nervous system disease that consists of three stages: intoxication, withdrawal and craving. The neurobiological model of addictions proposed by Volkow et al. (2003) includes the states of control, memory, motivation and reward, however, in order to generate an explicit and universal solution to this disease is necessary to mathematize theoretical models.
Objective: to propose and to develop the mathematization of a stochastic model using Markov Chains in the phenomenon of psychoactive substances and addiction.
Method: mathematization of a stochastic model using Markov Chains and differential equations.
Results: using Markov Chains, models of two brains were compared, one healthy (no consumer of psychoactive substances) against one of a person with addiction and through transition probabilities we observed differences from one model to another. Differential equations were used to estimate the time of the effect of a drug in the body and combined with trigonometric equations we sought to estimate the best function to continue with an addiction and relapse.
Discussion and conclusions: the obtained mathematical modeling indicates that the neurobiological model of addictions may be represented by a Markov Chain inhomogeneous. In the case of a healthy brain, it can pass with equal probability (p = 1/3) from one state to another, in the case of a person with addiction, the transition probabilities depend on time, drug type, dose and route of administration.
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