Stability Analysis On High Dimensional Coupled Epidemic Networked Systems

It is acknowledged that infectious diseases can not only resulting in serious health issue to persons,but also enormous loss to human society,so that it has historically been an important subject to investigate spreading mechanism and prevention of epidemic diseases.During the spreading process,behavior pattern of person plays critical role,especially a kind of behavior that are order and specification to prevent further contagion.Therefore,it become one of research aspects to make the crowd form a unified and standardized behavior pattern in this work,called controlled behavior synchronization aroused by epidemic spread.Furthermore,it is so top priority to prevent and control the infectious disease that there are also discussion about the control strategy on epidemic systems.And the main content are listed as following.Firstly,based on the quenched mean-field method,we construct the feedback control epidemic-synchronization coupled model,and use the information about the infected and susceptible to design two distinct feedback controllers to achieve two kinds behavior synchronous patterns.Differ from the other feedback strategies,the controllers designed here will be activated to work on behaviors only when epidemic spreading.In addition,the inhibition of contact behavior effect is also considered in our systems,which means that the willing to contact with each other is influenced by the disease's information and individual differences in behavior.By theoretical analysis,the transmission threshold is derived and proving the stability of disease-free equilibrium.And under the external controlling,we obtain the sufficient conditions to achieve behavior synchronization as well as guarantee stability of positive equilibrium.From which we can find when spread law is large than the threshold,there will be endemic,and if giving appropriate control strength,the desired synchronous behavior will come.Which implies that people's behavior will quickly reach a consensus to combat the disease.Secondly,we concern the cluster collective behavior aroused by epidemic spread with patch dispersal effect.We combine patchy dispersal epidemic spreading systems based on constant population and cluster behavior synchronization network systems to construct a class of compound models,in which the number of cluster is same as patch's,while the basis of clustering dose not depend on the spatial position of nodes and it isto divide the relationship network among them.We design feedback controllers based on each patch disease information and the whole difference of behavior,and prove the stability of patch systems theoretically.Furthermore,under the premise of spreading,we derive the sufficient conditions to guarantee global stability of endemic equilibrium as well as achieve cluster synchronous behaviors.We find that there are relevance between behavior synchronization in each subgroup and patch dispersal as well as the relationship network structure of each subgroup.By enhancing the relationship in each subgroup,there will get desired target.Finally,we investigate the optimal control of nonlinear incidence epidemic systems in directed networks.Depend on in-degree and out-degree,we build a kind of epidemic systems with birth and death,in which the empty sites in networks are seen as new input of birth,and in order to reflect the saturation contact rate,there introduces nonlinear incidence affected by psychological effect.We obtain local and global stability results by theoretically analyzing the equilibrium of systems.And then we construct control systems by means of optimal control theorem,and analyze to obtain the corresponding dynamical optimal control strategies and optimal control systems.Several examples are provided to demonstrate the theoretical results,and the optimal control systems are also solved in numeric and reach the aim of controlling.