Modeling the Complex Dynamics of Dual Addiction: A Co-Dynamical and Optimal Control Perspective Under Behavioral Uncertainty

In this study, a mathematical model is proposed which analyze the dynamics of co-abuse of cigarette smoking and drugs. Cigarette smoking and illicit drugs use, aggregates the massive public health concern particularly when both substances are abused simultaneously. Numerous studies are present where single-substance dynamics are studied but mathematical investigation to co-abuse is limited. The main objective of this study is to construct a novel mathematical model to study the interplay of cigarette and drugs and their combined dynamics. This co-epidemiological model contains system of non-linear equations which segregates the population into six compartments; susceptible, cigarette users, drug users, co-abusers, under treatment and recovered. Positivity and boundedness are analyzed to explore that the system is well posed. Reproduction number is calculated using next generation method and stability analysis is conducted for the proposed model. Sensitivity analysis is conducted for identification of dominant parameters of the proposed model. It is determined that the parameters representing peer influence rate of smoking and progression of smoking to co-abuse are two most sensitive parameters. Optimal control analysis is performed by yielding three controls on proposed model using Pontryagin's Maximum Principle to prevention strategies of co-abuse. It reveals that early awareness and fast recovery treatment break the chain of transmission. This model helps to understand the initiation and progression of addiction and treatment of co-abusing. MATLAB is used for numerical simulation and visualization of graphs, which will further be helpful for treatment and control of the dual addiction. The model analysis revealed that early prevention and better rehabilitation is helpful in smoking and drug addiction control.