Ynamical Analysis Of Some Fractional Order Epidemic Models

In this paper,some fractional epidemic models with Caputo derivative are investigated.In the first chapter,the development of fractional-order epidemic.model is reviewed and many important preliminary knowledge is listed.In Chapter 2,the fractional order vector comparison principle and some stability theories are proved;then,some new sufficient conditions are obtained to analyze the stability of fractional order systems.In the second section,we extend the fractional order scalar comparison principle to the fractional order vector comparison principle.On the basis of previous researchers,this method is used to analyze the stability of the fractional order differential systems which have great significance.In the third section,when 0<?<1.sufficient conditions are derived for the global asymptotical stability of nonlinear fractional order differential systems by using the generalized Gronwall inequality and related lemma.Some new sufficient conditions for the global asymptotically stability of nonlinear fractional order differential systems are obtained with order ?in?1,2?.Furthermore,several errors in previous works in the literature are corrected.In Chapter 3,an HIV-1 infection model with CTL immune response is investigated.By using the suitable Lyapunov functions and stability theories,we can analyze the stability of the equilibrium and obtain the threshold parameters.When o<1,the infection-free equilibrium is globally asymptotically stable.When R₁<1<R₀,the CTI-inactivated infection equilibrium is globally asymptotically stable.The CTL-ractivated equilibrium is globally asymptotically stable if R₁>1.Furthermore,the fractional optimal control problem?FOCP?of this model is also investigated.Chapter 4 investigates the qualitative dynamics of a fractional SVIRS vaccination model with waning natural and vaccine-induced immunity.Firstly,we discuss the reasons of backward bifurca-tion and give the associated reproduction number Rcv.Then,we analyze the stability of the original system equilibrium,the corresponding standard system and limit system are built.Some related conclusions about the stability of equilibrium points are obtained by using the Lyapunov functions methods.Finally,numerical simulations are present to illustrate our mathematical findings.In the last chapter,we make a summary and have a outlook of the relevant research works in the future.