Study Of A Heroin Epidemic Dynamical Model

In recent decade, illicit drug use has become a social and public health issue aroundthe word and virtually increases the risk of transmission of blood-borne viruses, drug-related criminal activities, family issues and public health medical costs in both developedand developing countries. It is estimated that China is the most injecting drug users ofthe world. It has registered1.33million drug users across the country, however the actualnumber of drug users is much higher. Drug users have a lot of high-risk behaviors, suchas sharing injection equipment and selling sexual medicine, which greatly increase thespread of HIV infection. The main reasons of Human Immunodefciency Virus (HIV)and Hepatitis C Virus (HCV) transmission is sharing injection equipment. While mostdrug abusers use intravenous injection. So, we must control drug use to reduce the AIDSepidemic. Heroin is one of the most important and main drugs in the drug use. Onceusing heroin, it will not only hurt the body, but also afect the family and people around.Therefore, it is very important to study heroin models of drug use to prevent the drugabuse, save drug addicts and control the drug problems. The main contents in this papercan be summarized as follows:In the frst part of this paper, we discus global dynamics of a discretized heroinepidemic model with time delay. We derive a discretized heroin epidemic model withdelay by applying a nonstandard fnite diference scheme. We obtain positivity of thesolution and existence of the unique endemic equilibrium. We show that heroin-using freeequilibrium is globally asymptotically stable when the basic reproduction number R0≤1,and the heroin-using is permanent when the basic reproduction number R0>1.In the second part of this paper, we study a distributed delayed heroin epidemicmodel with diferent conscious stages. The model allows for conscious drug users andunconscious drug users. The threshold property of R0is established. It is shown thatdrug-free equilibrium is globally asymptotically stable when R0<1; When R0>1, it is proved that the drug spread equilibrium of the system is globally asymptotically stable.In the third part of this paper, we construct a heroin epidemic model with saturationincidence rate and two distributed delays. We obtain the basic reproduction number ofthe heroin spread for the stability of equilibria. We show that drug-free equilibrium andthe unique drug spread equilibrium are globally asymptotically stable when the basicreproduction number is less than one and basic reproduction number is greater than onerespectively by using the proper Lyapunov functionals.