The Dynamics Analysis Of Drug Models

Drugs problem is one of the three social enviromental pollutions in the world,which has aroused widespread international attention.Drugs not only seriously harm people's physical and mental health,but also bring about major threat to human survial and devel-opment.This paper sets up some models for the drugs transmission problem,discusses the dynamical behaviors of drug models such as the stability of equilibria,Hopf bifurcation,Bogdanov-Takens bifurcation,limit cycles and the existence of travelling waves under the condition of diffusion by applying qualitative theory,bifurcation theory of ordinary differ-ential equations and theory of reaction-diffusion equations,and analyzes the dependence of system parameters for the prevalence of drug infectious diseases.First,we propose two heroin transmission models.Because susceptible individual is tempted at least two times before he becomes heroin addict,the nonlinear contact rate ?su₁² is bringed in a model.And a nonlinear recovery rate is considered in another heroin model because of the limited medical resources.By the qualitative and bifurcation analysis,we discuss the types of all equilibria and show the models can undergo saddle-node bifurcation,Hopf bifurcation and Bogdanov-Takens bifurcation of codimension 2,even may emerge Bogdanov-Takens bifurcation of codimension 3 at least.Second,since synthetic drugs are more addictive for psychology,a four dimensional transmission synthetic drugs model with psychological addicts is proposed.The different contact rates are discussed and different basic reproduction numbers are obtained.The local stability of drug free equilibrium is decided by the threshold number,and the global sta-bilities are studied by the method of Lyapunov functions.A drug model with contact rate ?e^-m₁U-m₂H is constructed to investigate the impact of media coverage on the spread and control of drug addiction.The dynamical behavior of the model is studied by using the basic reproduction number R₀.The results demonstrate that the media effect cannot change the stability of equilibria but can affect the number of drug addicts.The sensitivity of parameters is analyzed and numerical simulations are given to support the theoretical results.At last,a SIR epidemic model with diffusion and general nonlinear infection forces is studied.We make an assumption that the nonlinear infection contact rate is satisfied with some monotonicity which is also fit to drug transmission.The local and global stabilities of equilibria are discussed.Using the upper-lower solutions method and Schauder's fixed point theorem,we establish the existence of traveling wave and obtain an explicit expression of the minimum wave speed.These results suggest that the spread rate of the individual plays an important role in the transmission of infectious diseases or drugs,in the meanwhile,they also provide a theoretical basis for the formulation of preventive measures.