Global Dynamics Of Two Filippov Epidemic Models

Based on the classical epidemic model,by considering the threshold strategy,the global dynamics of two kinds of Filippov epidemic models are studied in this paper.The dynamic behaviors of the epidemic model under the corresponding threshold strategy are qualitatively analyzed by using the theory of differential equation with a discontinuous righthand side in Filippov sense.In addition,the global asymptotic stability of the disease free equilibrium,endemic equilibrium or pseudo-equilibrium is investigated by full Poincaré maps as well.The main contents of this paper are as follows:In the opening chapter,the research background,historical significance and research status at home and abroad of epidemic are introduced.Moreover,the preparatory knowledge is summarized and the main work of this paper is briefly illustrated.In the second chapter,the global dynamics of a class of Filippov epidemic model based on media coverage are discussed.Firstly,the model studied in this chapter and the threshold strategy considered are introduced,the existence of equilibia in subsystems and their local and global stability are analyzed briefly.The sliding region,pseudo-equilibrium and sufficient and necessary condition for existence of the pseudo-equilibrium are obtained.Then,according to the real equilibrium,virtual equilibrium and the direction of vector fields,the appropriate semi-Poincaré map is constructed.Next,the full-Poincaré map is defined by applying the properties of composite mapping,and the global asymptotic stability of various equilibriua is studied by full-Poincaré map and its properties.Finaly,the dynamic properties of the model with the incidence of non-linearity are further analyzed.In the third chapter,the global dynamics of a Filippov epidemic model with constant immigration is discussed,the epidemic model with constant immigration in both susceptible and infected persons is considered.Assuming that immigrants carry pathogens,the threshold strategy is as follows: when the number of infected persons is below a positive number,no treatment is given,and once the number of infected persons exceeds the threshold,treatment measures will be taken.The situation that input ratio of immigration in infected population is positive is considered and the dynamic properties of the model with the incidence of non-linearity are further analyzed.The research method and chapter structure are similar to the second chapter.Finally,the diagram of trajectory is clearly depicted by using the numerical simulation of Matlab,which vividly reflects the global asymptotic stability of various equilibria,the results obtained in this paper will be proved further.