Dynamic Analysis Of Two Fractional-Order Mathematical Models For HBV

In this paper,two kinds of fractional-order HBV models are established and studied.The whole paper is divided into four chapters:The preface is in chapter 1,we introduce the research background of this article,the main task and some important preliminaries.In chapter 2,a fractional-order model for HBV with Holling ? functional response is es-tablished and discussed.Firstly,the existence and uniqueness of positive solutions are proved.Secondly,the basic reproduction number R0 is calculated by using the method of the spectral radius of next generation matrix.Thirdly,the sufficient conditions for the existence of two equilibriums are obtained.The local stability of equilibriums are established by analyzing the characteristic equations.The global stability of the disease-free equilibrium is obtained by using Lyapunov functional and LaSalle's invariance principle.Finally,some numerical simulations are performed to verify the theoretical prediction.In chapter 3,on the basis of the previous chapter,two new compartments and time delay ?are taken into the model.Therefore,a fractional-order delayed model with Holling ? functional response and CTL immune response is established and studied.Firstly,the existence and uniqueness of positive solutions are proved.Secondly,the basic reproduction number R0 is calculated by using the method of the spectral radius of next generation matrix.Then,the sufficient conditions for the existence of three equilibriums are obtained.Thirdly,calculation indicate that:the disease-free equilibrium is globally asymptotically stable when R0<1,??0;the immune-free equilibrium is locally asymptotically stable when R1<1<R0,?>0;the endemic equilibrium is locally asymptotically stable when R0>R1>1,??[0,?0);while the endemic equilibrium is unstable when ?=?0 and the Hopf bifurcation occur near the endemic equilibrium.Finally,some numerical simulations are performed to verify the theoretical prediction.In chapter 4,the main work of this paper is summarized and the future research work is prospected.