The Dynamic Behavior Of Three Types Of Stochastic Drug Epidemic Models

As we all know,drug can make people addictive,it not only directly endangers physical and mental health of people,but also brings tremendous pressures and damages to social and public health system due to its prevalence all over the world.Increased use of drug is an issue of concern in many parts of the world.In this thesis,we establish three types stochastic drug epidemic models in order to provide the theoretical basis for the control of drug spreading.By using the Lyapunov function,Ito's formula,the strong law of large numbers,numerical simulation and the LaSalle invariant principle,we discuss the effects of random disturbances,media coverage and relapse on the spread of drug,moreover,we get the law of the spread of drug.The main contents of this thesis are summarized as follows:In the first chapter,we introduce the background and significance of the drug epidemic model.The research status of drug epidemic model are summarized.The definitions and theorems used in this thesis are reviewed.And the frame of this thesis are listed.In the second chapter,based on randomness and stochasticity in real life,we analyze the dynamic behavior of a stochastic heroin epidemic model with bilinear incidence and varying population size,discuss the effects of random disturbance on the spread of drug.By using the random Lyapunov function,Ito's formula and the strong law of large numbers,we show the existence and uniqueness of the global positive solution,in addition,we obtain the conditions for the extinction of the drug addict and the stationary distribution of the drug addict and the ergodic of the drug addict.The sensitivity analysis indicates that prevention would be more important than cure.In the third chapter,based on the importance of media campaigns in our lives,we analyze the dynamic behavior of a stochastic drug epidemic model with media coverage,discuss the effects of random disturbance and media coverage on the spread of drug.By using the random Lyapunov function,Ito's formula and the strong law of large numbers,we show the existence and uniqueness of the global positive solution,in addition,we obtain the conditions for the extinction of the drug addict and the stationary distribution of the drug addict and the ergodic of the drug addict.Numerical simulations are carried out to confirm the analytical results.In the fourth chapter,based on the high relapse rate of people with a history of synthetic drug abuse,we analyze the dynamic behavior of synthetic drug epidemic models with relapse,discuss the effects of random disturbance and relapse on the spread of drug.The global asymptotic stability of the drug-free equilibrium and the drug-addiction equilibrium of the deterministic synthetic drug epidemic model are presented by using the LaSalle invariant principle.Then we show that the solution of the stochastic model is going around each of the steady states of the corresponding deterministic model under certain parametric conditions by constructing some stochastic Lyapunov functions.The sensitive analysis of the basic reproduction number indicate that it is helpful to reduce the relapse rate of people who have a history of drug abuse in the control of synthetic drug spreading.