Analysis Of Dynamical Behaviors Of Coupled Epidemic Models

Since ancient times,infectious diseases have always been a threat to the survival and development of human beings,especially the emergence of some new epidemics in recent years,which are more infectious,higher lethal,and more difficult to prevent and control.Therefore,it is very important to establish the mathematical models of epidemics,then study the transmission mechanisms and preventive measures.Since the transmission process of diseases in the real world is not isolated,which is affected by various factors,so the establishment of coupled epidemic models is more in line with the actual transmission case,and the research results have more practical significance and value.With the rapid development of complex network,new research tools and methods for epidemic dynamics are provided,which make up for the fact that the traditional models cannot describe the propagation process in social contact networks with heterogeneous,then the propagation of epidemics and synchronization of behaviors on complex networks are modeled,and the effective prevention and control strategies are designed.Thus,the study of epidemic dynamics and synchronization on complex networks have become hot issues.For better cognitive the coupled epidemic models,in this paper,we study the epidemic dynamics of zoonotic diseases on coupled networks,the epidemic synchronization on complex networks,and the dynamical behaviors of the two-patch coupled epidemic model.Firstly,realistic networks are often composed of several kinds of complex networks interconnected with each other,and the interrelated networks may have different topologies and epidemic dynamics.Moreover,most human infectious diseases are derived from animals,and the propagation process of zoonotic infections is always one-way.Hence,we construct two unidirectional three-layer interactive networks by the heterogeneous mean-field method,one model has direct contact between interactive networks,the other one describes the diseases are transmitted through vectors between interactive networks,then study the epidemic dynamics of zoonotic diseases on the unidirectional circular-coupled networks.Through mathematical analysis,we obtain the basic reproduction numbers of two models,it is found that basic reproduction numbers depend on the infection rates,infection periods,average degrees,and degree ratios,then analysisthe stability of the disease-free and endemic equilibria.Then,some sensitive analyses of the models are performed,the conclusions illustrate and complement the theoretical results very well.Secondly,many real-world networks exhibit community structure: the connections between nodes within each community are dense,while the connections between communities are sparser.Moreover,during the disease outbreak,self-protection behaviors are induced by the emergence of epidemics and in turn influence the propagation process of epidemics.Here,we study the interaction between epidemic spread and collective behavior in scale-free networks with community structure,by constructing a mathematical model that embeds community structure,behavioral evolution,and disease transmission.Due to the differences among individuals' responses in different communities to epidemics,we use nonidentical functions to describe the inherent dynamics of individuals.With the disease transmission,individual behaviors in different communities may tend to cluster synchronization,which is indicated by the analysis of our model.Then,we explore the epidemic threshold of the spreading model,and present the stability analysis of global synchronization and epidemic dynamics by constructing an appropriate Lyapunov function.Numerical simulations are given to illustrate and complement the theoretical results.Finally,in order to study the effect of population mobility between two regions on the transmission dynamics,a two-patch vaccination coupled model with standard incidence for disease transmission is established,in which the three-dimensional system is regarded as the basic structure of each region,and the linear terms are added to the equations to reflect the individual migration due to travel.In addition,vaccination can reduce the risk of infection which is indicated by the parameters in the model.Due to the complexity of the two-patch model,we assume that two regions are symmetric in model parameters,and the model is divided into the connected regions and disconnected regions.Then,we calculate the basic reproduction number of the two-patch system,and obtain the conditions for the existence of multiple endemic equilibria in the system,moreover,the analysis of bifurcations and the stability of equilibria are given.Then,some numerical simulations on the two-patch model are performed.The results show that the model exhibits rich dynamical behaviors,i.e.,the structure of equilibria and bifurcations.Theemergence of multiple stable states and backward bifurcation in the epidemic model,which can increase the difficulty of disease prevention and control.Therefore,the prevention and control strategy should be focus on reducing the basic reproduction number below the critical value to ensure the eradication of infectious diseases.