Modeling And Research Of Heroin Transmission Dynamics

Infectious diseases are a major problem that endangers human health.Social problems related to infectious diseases,such as drug abuse,also concerned.A survey shows that heroin is the main drug taken among drug users in China.Therefore,it is of great value to study the mathematical model of heroin for preventing drug abuse,saving drug users and controlling drug transmission.This paper is divided into two parts for studying the mathematical model of heroin transmissionIn the first part,we study a mathematical model with the influence of human behavior on heroin transmission.Firstly,we obtain the nonnegativity and boundedness of the solutions by using comparison principle and the counter proof method.Secondly,the global stability of the equilibrium point is proved by using LaSalle's invariant principle and the second additive compound matrix.Finally,the sensitivity of each parameter to the threshold value is analyzed by means of the partial rank correlation coefficients(PRCCs)and numerical simulations,and the reasonable measures to control the spread of heroin drugs are given.The results show that human behavior can reduce the number of heroin transmissionIn the second part,we study a stochastic heroin transmission model with psychological effects.Firstly,the existence and uniqueness of the positive solution of the model are analyzed by using Stop-Time theory.Secondly,the asymptotic behavior of solutions of the stochastic model near the equilibrium point of the deterministic model is obtained by Ito formula and martingale's powerful number theorem,and the sufficient conditions for the persistence and extinction of the stochastic model are established.Finally,the numerical simulations with the help of Matlab shows the consistency between the numerical simulation results and the theoretical results,and the deterministic model and the stochastic model are compared intuitively.The results show that the effect of blocking the spread of heroin drugs can be achieved by enhancing the degree of noise in the stochastic model.