Dynamic Research On Several Classes Of Filippov Models

Due to the influence of the natural law and many other factors,the differential equations with discontinuous right-hand sides are often applied to the mathematical modeling in many practical problems.Based on the classic model of infectious diseases,we put forward a plant disease model with discontinuous control strategies and a smoking cessation model with discontinuous intervention measures,and then discuss the sliding mode dynamics as well as global dynamics of the models by applying the qualitative theory of discontinuous differential equation with discontinuous right-hand sides.The thesis is divided into three chapters.In Chapter one,we first review the history and development of the differential equations with discontinuous right-hand sides and infectious disease model.Afterwards,we introduce the historical background and status of plant disease dynamics and smoking epidemic dy-namics one by one.At last,the motivations,outline and significance of this thesis are briefly presented as well,and the basic mathematical theories which are needed in this thesis are introduced in this chapter.In Chapter two,according to the previous discontinuous plant disease model,we improve its threshold control strategy and present a Filippov plant disease model incorporating an economic threshold of infected-susceptible ratio,above which control strategies of replanting or removing are needed to be carried out.Based on the Filippov approach,we study the slid-ing mode dynamics and further the global dynamics.It is shown that by varying parameters,all the solutions of the Filippov system eventually converge to a unique equilibrium which is a disease-free equilibrium,an endemic equilibrium or a pseudo-equilibrium,or the Filippov system is bistable.Moreover,the equilibrium is proved to be globally asymptotically stable.Through numerical simulation,our results indicate that the control goal can be achieved by taking appropriate replanting and removing rate.In Chapter three,considering that the intervention of government and media in smoking is discontinuous,based on the infectious disease model,a Filippov smoking cessation model with interventions is put forward.Making use of the method of Filippov qualitative analysis,we investigate its sliding mode dynamics and further global dynamics.It is shown that the Filippov system stabilizes at a unique equilibrium which is a smoking-free equilibrium,an endemic equilibrium or a pseudo-equilibrium.The results indicate that appropriate inter-ventions can reduce the number of smokers and keep them within a predetermined threshold level.