Qualitative And Quantitative Study Of A Non-smooth Filippov Epidemical Model With Threshold Strategy

Based on the impact of media reports and limited medical resources on the spread of disease,the dynamics of several epidemiological models with discontinuous media publicity strategy and non-smooth treatment strategy are studied.First,By the number of the infected individuals as an index to media publicity and setting a threshold to characterize the infective size which leads to media response to disease spread,a class of SIR epidemic model with discontinuous media influence is formulated.When the number of infected individuals is less than the threshold,the media does not work;otherwise,media-related incidence rate is activated to express this influence.The sliding mode region and sliding mode dynamics properties of the model are studied.The existence of boundary equilibrium bifurcation,and the existence and stability of the real,virtual,pseudo-equilibrium,the tangent point and the boundary equilibrium point are discussed,respectively.Also,nonexistence of limit cycle is proved.The results show that whether the endemic equilibrium is real or virtual and the existence and stability of the pseudo-equilibrium depend on the threshold.In particular,there is a unique asymptotically stable pseudo-equilibrium when the threshold lies between the infected states of two positive pseudo-equilibria.All of the trajectories of the discontinuous model tend to the real or pseudo-equilibrium.Then,based on the limited medical resources,assuming that the hospital's maximum capacity is less than the media response threshold,a class of epidemic model with two switches,i.e.media response switch and treatment one,is established.By employing the Filippov theory and other mathematical methods,one obtains the sliding mode region,sliding dynamics properties,and the existence of the real,virtual,pseudo-equilibrium,boundary nodes(boundary focus)bifurcation and boundary saddle point bifurcation.In the same time,the existence and stability of the equilibrium of the corresponding subsystem are discussed,respectively.Our results show that the discontinuous model affected by media propaganda and treatment strategy has sliding mode region but has not crossing one.However,considering treatment strategy only,it does not have sliding mode area,but have crossing area.In addition,if the number of infected individuals is less than the maximum capacity of the hospital,the discontinuous system has a unique positive equilibrium,and it is asymptotically stable;while if the number of infected individuals is greater than the maximum capacity of the hospital,there are two positive equilibria,and one of which is a saddle point.