A dynamical model of drinking and smoking with optimal control analysis

The intake of alcohol is dangerous, and the smoking of tobacco is savage, but it is life-threatening to practice both smoking and drinking. According to the World Health Organisation, the world loses about 8.5 million people each year as a result of smoking tobacco and drinking alcohol. To study this, we present a mathematical model that investigates the co-dynamics of alcohol drinking and tobacco smoking, as well as some control strategies. In contrast, many studies focus solely on the dynamics of alcohol consumption or tobacco smoking. Also, these studies assume that an individual who may recover from both alcohol drinking and tobacco smoking may relapse. We determined the basic reproductive number by employing the next-generation matrix approach. We conducted local and global stability analyses for the drinking, smoking-free, and endemic states. We then conducted extensive research into secondary infections related to smoking and drinking. We then performed numerical simulations and analysis using the parameter values from the literature. The study further examined the influence of some key parameters on secondary co-dependence infections, which occur when one infected individual enters the population and recovers from both over time. For example, in this study, it was shown that the contact rates a1 and a2 have a direct relationship to the spread of drinking and smoking. In contrast, recovery rates δ1, δ2 showed an inverse relationship. In addition, we conducted an optimal control analysis by suggesting the following: drinking prevention efforts, smoking prevention efforts, recovery efforts on the co-dependence of drinking and smoking, recovery efforts on drinking, and recovery efforts on smoking. The simulations indicated that using these controls can help reduce the number of smokers and drinkers within eight weeks.