The Research Of Two Types Of Complex Networks Dynamics Models And Its’ Properties

In this thesis,taking into account the incubation period of the disease and immunity is not persistent,we propose two novel models on complex networks: SLIS model and SIRS model on the basis of the existed models.By employing stability theory of differential equations,LaSalle invariance principle and other methods,the long time behavior of two models are deeply studied via detailed analytical methods.This thesis is divided into three parts.In charter one,firstly,the background of two models is briefly presented.Then we give an outline of the propagation of epidemic on complex networks and the main research contents.In charter two,a novel epidemic model(SLIS model)of diseases with the incubation period is investigated,based on heterogeneous network,and the dynamic behaviors of model are detailed researched.To begin with,the basic reproduction number is presented by detailed analytical methods and discuss the existence of equilibrium point of model.Secondly,the stability of the virus-free equilibrium,the stability of and the permanence of the positive equilibrium is investigated by Lyapunov method and the LaSalle invariance principle.Then analysing the sensitivity of the model parameters,which implies: the basic reproduction number is closely related to the infection of the disease,and when the degree distribution exponent remained unchanged,the eventually infection density is proportional to the degree of the node.We further discuss the impact of different patterns of infection on the magnitude of infection,research shows:on the one hand,an scale-free network with smaller power-law exponent is more beneficial to prevent the disease from spreading as long as the infectivity of nodes φ(k)take the constant form;on the other hand,when the infectivity of nodes φ(k)= k,an scale-free network with smaller power-law exponent is more beneficial to the spread of epidemics.Finally,through numerical simulation,we compare the convergence time of equilibrium point in different patterns of infection.In chapter three,a novel epidemic model(SIRS model)of diseases with nonlinear infectious is detailed investigated,based on scale-free networks.By using stability theory,LaSalle invariance principle and Lyapunov function,the basic reproduction number of model is presented,the dynamical properties of model are investigated.For example,the existence and stability of equilibrium point of model,the permanence of epidemics.We further discuss the sensitivity of the model parameters,it shows: the basic reproduction number is closely related to the infection of the disease and as long as the degree distribution exponent remained unchanged,the eventually endemic level is proportional to the degree of the node.Finally,numerical results demonstrate the correctness of the conclusion by Matlab.