The Qualitative Analysis Of Several Types Of Avian Influenza Model With Nonlinear Infection Rate

The spread of avian influenza has a long history,the virus has gradually become one of the diseases that threaten human health,owing to the mutation of avian influenza virus and the changes in the external environment.In this paper,four models of avian influenza are established,which are based on the mechanism of spread of avian influenza.And we demonstrate the global state of each model.The details are as follows:Firstly,an epidemic model of H7N9 avian influenza with the infection rateh ?_hS_h/k+?I_h~m and media coverage is studied,the threshold of model is obtained.The model is analyzed by using Liapunov function,Dulac function and the limit equation theory.It is proved that the disease free equilibrium is globally asymptotically stable if R0?1;the endemic equilibrium is globally asymptotically stable,if R0?1.The results imply that the scale of H7N9 avian influenza can be controlled through media coverage.Then,the numerical simulation of the infection rate which k=1.5,m=2 is carried out.It intuitively reveals the correctness of the previous conclusions and the great impact of media coverage on the prevention of avian influenza.Secondly,in order to investigate the influence of media coverage on the spread of H7N9 avian influenza,a H7N9 avian influenza model with the infection rate (?_h-(?I_h/m+I_h))S_h and impact of media coverage is established.On the basis of obtaining the basic reproduction number of the model,Liapunov function,Lasalle invariance principle,the limit equation theory and so on are used to analyze the model.It is proved that the disease free equilibrium is globally asymptotically stable if R0?1;the endemic equilibrium is globally asymptotically stable if R0?1.The impact of media coverage on the spread of H7N9 avian influenza and the global state of the model with the infection rate are revealed by numerical simulation.Thirdly,according to the reality,an epidemic model of the avian influenza transmission is established in the article to investigate the circumstances of infected humanwith latent period,and the basic reproduction number R0 of model is obtained.Then,the stability of the model has been analyzed by using Liapunov function,and the limit equation theory.It is proved that the disease free equilibrium is globally asymptotically stable if R0 is not larger than 1;the epidemic equilibrium is globally asymptotically stable if R0 is larger than 1.Finally,an avian influenza model with saturated infection rate is established in the article to investigate the circumstances of infected human with latent period.The threshold of avian influenza virus R0 is obtained.Then,the stability of the model has been analyzed by using Liapunov function,the limit equation theory,Bendixsion-Dulac criterion and Hurwitz criterion.It is proved that the disease free equilibrium is globally asymptotically stable if R0 is not larger than 1;the epidemic equilibrium is globally asymptotically stable if R0 is larger than 1.