Dynamical Analysis Of Two Types Of Disease Models

Two types of infectious disease models are studied in this paper.One is the--₍（6）infectious disease model in which the asymptomatic patients are considered,and the other is a Barbour model with random terms.Some related system stability theorems are used to systematically analyze these two types of infectious disease models.The existence and stability conditions of equilibrium are obtained and proofs are given.In order to study the influence of asymptomatic patients on the spread of infectious diseases,a model of asymptomatic patients with infectious diseases was established.The basic reproduction number of the model is obtained according to the second generation matrix method and then the model has two boundary equilibrium and one positive equi-librium under certain conditions.By choosing appropriate Lyapunov function and Dulac function,the global asymptotic stability of the equilibrium is proved by using Lasalle invariance principle and Bendixson-Delac principle.It is also found that the model will show the bifurcation branch and Bogdanov-Takens branch phenomenon at the bound-ary equilibrium.It follows that the impact of asymptomatic patients on the spread of infectious diseases can lead to complex dynamics.Considering that there are many random factors affecting the transmission process of schistosomiasis,this paper introduces random terms based on the Barbour model to establish a new stochastic dynamic model of schistosomiasis transmission.The mathe-matically analysis of this model is developed by Lyapunov function and It（?）integral.and the uniqueness of the positive solution existence and its ultimate boundedness for the random model are obtained then.