Analysis For Two Kinds Of Heroin Spread Models

In recent decade, drug abuse continues to occur at home and abroad, it has attracted the attention of the world. According to statistics, the number of drug users in China is the largest in the world. The abuse of drugs in the first place is heroin. Heroin addicts use drugs by inhalation and intravenous injection, illicit heroin abuse has not only increased the spread of human immunodeficiency virus （HIV） and hepatitis C（HCV） virus, but also increased the medical burden of family and society. At the same time, it causes a number of social problems such as illegal and criminal. Therefore, it is a practical significance to study the spread of heroin. The main contents of in this paper are summarized as follows:In the first part, a heroin spread model with discrete time delay is formulated and the expression of the basic reproduction number of the model is obtained. It is shown that the drug-free equilibrium is globally asymptotically stable if the basic reproduction number is less than one. The conditions for the locally asymptotical stability of heroin transmission equilibrium and the existence of Hopf bifurcation are discussed.In,the second part, according to the characteristic of drug users, drug users are divided into conscious and unconscious use of drugs and a heroin spread ODE model with two kinds of conscious stages is formulated. The expression of the basic reproduction number Ro is given. By using the Lyapunov functionals method, it is shown that the drug-free equilibrium is globally asymptotically stable if R₀ < 1. The drug spread equilibrium is globally asymptotically stable if R0_&gt; 1.