The Dynamics Of Two Kinds Of Epidemic Model With Time Delay On Complex Networks

The mathematical modeling of infectious disease propagation has been extensively studied by both medical practice and the academia for a long time,it is of great significant to study the mechanisms of infectious diseases and take effective measures to control the spread of diseases.These existing epidemic models,based on the homogeneous mixing hypothesis,ignore the differences of the individuals,therefore,they are not suitable to describe epidemic propagation in large-scale systems which have complex topology structures.Furthermore,delay is unavoidable in social networks and biological networks,many traditional researches on epidemic are built by using ordinary differential equations,and discount the influences of temporal delay.The investigation of time delayed epidemic model on networks is insufficient,therefore,it is necessary to study the propagation process with time delay on complex networks in detail.Considering the important roles of time delay in propagation process,this paper constructs two types of epidemically models on complex networks: the SIR model with discrete delay and the SIR model with distributed delay.With the help of delay different equations and Lyapunov functional,we study the long-time global dynamics detailedly.The paper is organized as following.To begin with,we provide sufficient analysis of the studies progressing of epidemic modeling,and exhibit the background of the epidemic models that consider the effects of time delay exhaustively.In addition,the methods that used in this paper are introduced in detail.Next,by considering the birth and death of individuals,the topology structures of the networks and time delay,a novel class of SIR model with distributed time delay is constructed.We calculate the basic reproduction number of the system in the first place,and the existence of the equilibriums of the system is analyzed.Next,by employing the Lyapunov functional and the Kirchhoff's matrix tree theorem,we study the stability of the equilibriums.With the help of theoretical proof and numerical simulation,the following results are obtained.At first,when the basic reproduction number is not more than one,the disease-free equilibrium is globally asymptotically stable.Secondly,the system converge to the endemic point,i.e.,the endemic equilibrium is globally asymptotically stable;Then the densities of the infected individuals are higher as the degrees rise.At last,moreover,the system with distributed delay converge faster than the system with discrete delay.In the third place,a new SIR model with discrete delay is developed on heterogeneous networks.We calculate the basic reproduction number of the system at first and discuss the existence of the equilibrium.With the help of the method of Lyapunov functional and the LaSalle invariance principle,we study the permanence of the disease.From theoretical analysis,the novel results are obtain.First of all,when the basic reproduction number is not more than one,the disease disappear gradually,otherwise,the disease will be permanent in the population.Furthermore,when the degree are higher,the densities of the infected individuals are higher.Last but not least,by comparison,we find that the density is higher when the delay is larger.Numerical examples demonstrate the validness and effectiveness of the obtained results.