Non-smooth Dynamical Models Analysis Driven By Media Influence

With the advent of the new media era,the rising of the Internet industry,various social platforms bring people insight and also promote the spread of false information.Rumors are typical representatives of false information,it not only misleads the public,but also may cause group panic and cause social unrest.Therefore,according to the law of rumor communication,the establishment and analysis of the corresponding rumor spreading model is helpful to understand the internal mechanism of rumor communication and provide corresponding theoretical reference for the formulation of reasonable prevention and control measures for rumor communication.A SIRI rumor spreading model with re-propagation is presented in this thesis,and a state-dependent pulse control model under bilinear incidence and general incidence is further presented.With the help of the definition and nature of Poincare mapping,the main contents are as follows:First of all,considering the impact of the continuous fermentation and spread of rumors on its transmission process,a SIRI rumor spreading model with re-propagation is established.The basic reproductive number R0 of the model is defined,and the existence and stability of the equilibria of the model under different conditions are analyzed theoretically and numerically.The results reveal that there exists a backward bifurcation at R0=1 when the rate of re-propagation is large enough,implying that the rumor cannot die out when R0<1 and it spreads persistently due to the existence of the stable propagation equilibrium.Then the sensitivity and the influence of model parameters are conducted by numerical simulations.Furthermore,the proposed model is applied to the empirical data and the effect of the related parameters on curbing rumor spreading is compared.Secondly,on the basis of the SIRI rumor spreading model with re-propagation,consider the intervention influence of media,and further introduce the state-dependent pulse control model induced by the unknown.With the help of the definition and properties of the Poincare map,the existence of positive fixed points of the model in different dynamical states is analyzed theoretically,and the relevant conclusions are verified by numerical simulations.In addition,the stability of model rumor-free periodic solution(RFPS)is analyzed and the branching of several key parameters is discussed.The study shows that RFPS can still exist stably when certain conditions are met,and then the rumor spread can be effectively controlled.The numerical simulation results show that the timely blocking of rumor information and reducing the repropagation of rumor information can effectively control the further spread of rumor information.In order to verify the effectiveness of the model,the model was also applied to the same actual data set for example analysis,and the results show that the pulse intervention to some extent can quickly and effectively suppress the spread of rumor information in a short time.Finally,considering the complexity of rumor spread,on the basis of the bilinear incidence model,the state-dependent pulse control model with general incidence rate is further considered.Analyzing the existence and stability conditions of equilibria,the Bendixsin-Dulac discriminant method is used to prove that there is no limit cycle in the system.With the help of the definition and properties of the Poincare map,analyzing the existence of fixed point of the system under different dynamic states.Then discussing the existence and stability of the RFPS,and analyzing the bifurcation circumstances of several key parameters nearby.Research shows that when R0>1 and meets certain conditions,the RFPS can still exist stably.At this time,the spreading of rumors can be effectively controlled.Besides,taking the saturation incidence as an example,performing numerical simulation of the existence and stability conditions of equilibria,verifying the existence of branch parameters and order-1 periodic solutions near RFPS.Results show that reducing the re-propagation rate can effectively control the spreading of rumors.