The Qualitative Analysis Of Epidemic Models With Nonlinear Incidence Rate

Abstract:Epidemic have been a major threat of human health since ancient times.In this paper,four kinds of epidemic models with nonlinear incidence are established.The qualitative analysis of the models are carried out by using the dynamics of epidemic,and the purpose of which is mainly to predict and judge the development trend of the epidemic.The existence and the stability of the equilibriums are proved by using the stability theory of ordinary differential equations.The sufficient conditions for global asymptotic stability of the disease-free equilibrium and endemic equilibrium are obtained.By using vaccinate susceptible,treat infected individuals and increase media effect,the epidemic is prevented and controlled.The main contents are as follows:Firstly,the gradual loss of immunity after inoculation of certain epidemic is considered,a class of epidemic models with continuous inoculation and nonlinear incidence are established.The basic reproduction number of the model is obtained by using the method of reproducing matrix.By constructing Liapunov function and using the theory of LaSalle invariance principle,the global asymptotic stability of the model is proved.The conclusion is verified by numerical simulation.Secondly,the latent period of epidemic is taken into account.The model with latent period and saturation incidence is established.The basic reproduction number of the model is obtained by the method of reproducing matrix.The local asymptotic stability of the equilibrium point of the model is proved by Routh-Hurwitz criterion.Then the global asymptotic stability of the model at the equilibrium points are proved by constructing Liapunov function and using LaSalle invariance principle.The conclusion is verified by numerical simulation.Thirdly,the influence of vaccination and treatment on epidemic are taken into account.The model with vaccination and treatment is established.The basic reproduction number of the epidemic is given.Then the global asymptotic stability of the model at the equilibrium points are studied by constructing the Liapunov function and using the method of autonomous convergence theorem.Fourthly,the influence of the media on the epidemic is taken into account.The model affected by the media is established.In this paper,the basic reproduction number of the epidemic is given.The local asymptotic stability of the equilibrium point of the model is proved by Routh-Hurwitz criterion.It is shown that when R0?1 the disease-free equilibrium is globally asymptotically stable.It means that the epidemic is not prevalent.The global asymptotic stability of the model at the equilibrium points are studied by using the method of autonomous convergence theorem.When R0>1,the endemic equilibrium is globally asymptotically stable.It means that the epidemic will become an endemic disease.Finally,based on the analysis of the above models,some measures to prevent and control the epidemic are summarized.The problems encountered in this paper and the direction of future work are explained.