Modelling of Substance-Abuse and Its Progression Stages

Substance-abuse mathematical model is proposed to study the dynamism of substance-abuse and its various stages of progression from the justification that substance abuse leads to substance addiction with the development of tolerance and dependence. We analyze the model where we prove existence and uniqueness of the solution to the system, positivity of the solution, invariant region of the solution, substance-free equilibrium state of the system and discuss local stability as well as global stability of the model at substance-free stability state. Epidemiological approach is adopted in proposing the mathematical model which enables scientists to study the dynamism of substance-abuse and its various stages of progression.