Stability Analysis Of The Epidemic Model With Public Health Education

Alcoholism has been identified as a chronically disease due to its potential adverse health and social effect In an attempt to account for the influence of public health education in alcohol control and gain some insight into alcoholism dynamics,it is imperative to formulate mathematical models to study its prevalence and transmission.In this thesis,we investigate the dynamic behaviors of alcoholism models with forms of continuous,discrete and delay differential equations respectively.Firstly,based on the propagation mechanism of alcoholism,a continuous ordinary differential model with general nonlinear incidence is proposed.A qualitative analysis about the positiveness and boundedness of the solution for the system is carried out,and the stability of the equilibria is proved by constructing Lyapunov functions.In addition,the discretization of the corresponding continuous model is presented by applying Mickens' nonstandard finite difference scheme.The theoretical and numerical results reveal that the discrete system can effectively preserve the dynamical behaviors of the continuous system.Secondly,in consideration of the heterogeneity of spatial environment,a discrete differential equation model of multi-group population is established.The basic reproductive number R0 is defined and the threshold dynamics of the system are analyzed by employing Lyapunov functional theory and Lasalle invariable principle.The results show,that alcohol-free equilibrium is globally asymptotically stable if R0?1,implying that the disease is eliminated eventually,and conversely,the alcohol-present equilibrium is globally asymptotically stable if R0>1,implying that the disease will be persistent.Finally,considering that latent exists objectively in the course of epidemic transmission of some diseases,a continuous multi-group differential model with time delay is established and studied.The basic reproduction number of the system is defined which completely determine the global dynamics of the model.The existence of the positive equilibrium is examined and its global stability is proved by constructing Lyapunov function,which rules out the existence of Hopf bifurcation induced by a time delay.