Study Of Influenza Models With Cross-immune And Optimal Control

In this paper,a class of infectious disease model is 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 SIRC model for Influenza A with time delay is established and discussed.The basic reproduction number R0 is calculated by using the method of the spectral radius of regenerated matrix.For the model without delay,we demonstrate the conditions for global stability of equilibria.And we show that the delay can only change the stability of the enclemic equilibrium and lead to the existence of Hopf bifurcation.We also derive some explicit formu-lae determining the bifurcation direction and the stability of the bifurcated periodic solutions.Finally,numerical simulation is given to support our resultsIn Chapter 3,On the basis of the previous chapter,the saturated incidence was selected to replace the bilinear incidence to discuss influenza A system.The SIRC model are established and studied.Firstly,First,the basic regeneration number R0 is obtained,then we study the existence and stability of the disease free equilibrium point.Secondly,when time delay ?=0,endemic equilibrium stability is discussed.When R0>1,?=0,E*is locally asymptotically stable.Further,the critical value of delay ?0 is obtained,and when the delay increases through ?0,the stability of the endemic equilibrium point of the model is changed and the Hopf bifurcation is generated.By applying the normal form method and center manifold theorem,we also derive some explicit formulae determining the bifurcation direction and the stability of the bifurcated periodic solutions.Finally,numerical simulations are provided to support our main conclusion.In Chapter 4,further,to minimize the number of susceptible and infected persons as well as cost associated to the implement of the control measures,based on the SIRC model with saturated incidence and time delay,The optimal control problem of the model is presented.The two control functions u1(t)and u2(t)represent the prevention of susceptible persons and the treatment of infected persons respectively.We discuss the existence of the optimal controls and find the optimal control solution.Finally,numerical simulation and conclusion are given.