Dynamics Of Epidemic Models On Complex Networks

Infectious diseases are always harmful to human beings’ health.Building mathe-matical models to study the transmission of infectious diseases is crucial.The tradi-tional models,based on uniformly mixing populations.However,in reality,different individuals may have different number of acquaintances.It is obvious that a given in-fective individual does not have equal probability of infecting others.A large number of complex system can describe by social network.Therefore,it is meaningful to study epidemic transmission on complex network.In this thesis,there are two models are proposed based on complex networks.By employing the theory of complex networks,stability theory of differential equations,and so on.The dynamic behaviors of two models are studied via detailed analytical methods and simulations.Meanwhile,Some effective measures in preventing and controlling epidemic spreading are presented.This thesis is divided into three parts.In chapter one,at first,we briefly introduce the background of two models that are investigated in this paper.Then we give an outline of the propagation of epidemic on complex networks and the main research contents.In chapter two,by taking full consideration of exposed period and quarantined period,a novel epidemic model(SEIQRS model)of diseases is investigated based on SIRS model.To begin with,we obtain the basic reproduction number of model by detailed analytical methods.Once more,we discuss the existence of equilibrium point of model.The stability of the virus-free equilibrium is discussed via analyzing the distribution of roots of the characteristic equation to the linearized system,and The stability of and the permanence of the endemic equilibrium is investigated.We further discuss the magnitude of infection,and a surprising result is presented.Finally,numerical simulations show the correctness of our conclusion.In chapter three,we consider the global stabiliy of SIQRS model on heterogeneous networks.Firstly,we obtain the basic reproduction number of model by detailed ana-lytical methods.Secondly,we discuss the existence of equilibrium point of model.Once more,by constructing suitable Lyapunov functions,then global stability of free-disease equilibrium and endemic equilibrium of the model can be investigated under a certain condition.Finally,numerical simulations show the correctness of our conclusion.