Dynamics Of Two Avian Influenza Models

Avian influenza, an acute infectious disease caused by a subtype structure of influenza A virus, firstly occurred in chickens, ducks, geese, pigeons and other birds. Recently, many areas have appeared the cases of human infection with avian in-fluenza virus, and people infected with the bird flu has a high mortality rate, but at present human generally lack of avian influenza virus immune system. The World Health Organization considers the disease to be one of the biggest potential threats to human disease. To better understand the spread and epidemic law of the avian flu in the population, we use the idea of modeling compartment to establish avian influenza mathematical models.Whether the avian influenza has the characteristic of human-to-human trans-mission has not been confirmed, in this paper we establish two avian flu epidemic models by considering the two transmission relationships, susceptible humans and infected birds, susceptible humans and infected humans based on the two ways of human infection with avian influenza, one is by the infected birds, another way is by the infected humans. One is the SI-SIR avian flu epidemic model with no infec-tion between people and another is the SI-SIR model with infection between people. Through the analysis of the SI-SIR avian flu epidemic model with no infection be-tween people, we get the basic reproduction of this model. The system always exists disease-free equilibrium, and there exists an positive equilibrium only an if only Ra>1. If Ra<1, the disease-free equilibrium is globally asymptotically stable, while if Ra>1the disease-free equilibrium is unstable, and the positive equilibrium is globally asymptotically stable. Through the research and analysis we know in this case it need to reduce the contact rate between bird susceptible and bird infection to effectively control the avian influenza outbreaks. Through the analysis of the SI-SIR avian flu epidemic model with infection between people, this model has two thresh-olds. Using the Lyapunov function and LaSalle invariance principle we show the local and global asymptotic stability of the equilibriums in this model. Additionally, we suggest in this case it need to reduce the contact rate between poultry suscep-tible and poultry infection, and between human susceptible an human infection, to effectively control the avian influenza outbreaks.