A mathematical study of co-dynamics of smoking and alcohol habits

Background Co-addiction to smoking and alcohol poses an important public health concern, as the combined use of these substances can increase health risks, complicate recovery, and contribute to persistent addictive behaviors. Mathematical modeling provides a useful framework for investigating the transmission and persistence of addictive behaviors. This study develops a deterministic compartmental model to investigate the dynamics of smoking, alcohol use, and their co-addiction within a unified framework. Methods A deterministic compartmental model was formulated to describe the transitions among susceptible, smoking-addicted, alcohol-addicted, co-addicted, and recovered populations. The basic reproduction number, R 0 , was derived using the next-generation matrix method, and equilibrium points and their stability properties were analyzed. The unknown parameters of the smoking and alcohol sub-models were estimated by fitting available epidemiological data using bounded nonlinear least-squares optimization in Python. The parameter estimation was performed by minimizing the sum of squared residuals between model predictions and observed data, with the model equations numerically solved using the ‘odeint‘ solver. Numerical simulations were subsequently performed in MATLAB using the ‘ode45‘ solver to examine the dynamics of co-addiction and the effects of key parameters. Results The analysis identified an addiction-free equilibrium and an endemic equilibrium. The addiction-free equilibrium is locally asymptotically stable when R 0 < 1, indicating that addictive behaviors decline and for R 0 >1, the model permits the persistence of addictive behaviors through the endemic equilibrium. The parameter-fitting results showed good agreement between the model predictions and the available smoking and alcohol data. Sensitivity analysis identified addiction transmission and re-addiction parameters as important determinants of R 0 and the long-term dynamics of the system. Numerical simulations further demonstrated that increased transmission and relapse rates promote the persistence of addiction and co-addiction, whereas increased recovery and intervention rates reduce the addicted population. Discussion The findings demonstrate that the transmission rates β 1 and β 2 play important roles in the dynamics of co-addiction. In particular, reducing the alcohol transmission rate β 2 produces a greater decline in the co-addicted population than reducing the smoking transmission rate β 1 . This suggests that alcohol-related transmission has a greater influence on the persistence of co-addiction within the proposed model. The parameters ϵ 1 , ϵ 2 , and ϵ 3 represent the rates at which individuals transition from the recovered compartments R s , R a , and R sa , respectively, to the susceptible compartment S . Increasing these parameters leads to an increase in the co-addicted population in the numerical simulations, indicating that a higher rate of return to susceptibility following recovery may contribute to the persistence of co-addiction. Consequently, interventions aimed at reducing alcohol-use initiation, together with effective smoking-prevention measures, may help limit the development of co-addiction. Overall, these findings emphasize the importance of controlling transmission and relapse processes while strengthening recovery strategies to reduce the long-term public health burden associated with smoking and alcohol co-addiction.