Research On A Two-delay SVIR Epidemic Model With Stage Structure And Nonlinear Incidence Rate

Based on the SVIR epidemic model with bilinear incidence rate,this dissertation introduces the stage structure and nonlinear incidence rate,and establishes a two-delay SVIR epidemic model with stage structure and nonlinear incidence.In this dissertation,the dynamic behaviors of the system is studied by using relevant knowledge of the stability theory of delay differential equations and functional differential equations.Based on the theoretical results,numerical simulations are used to verify the rationality of theoretical analysis.Firstly,this dissertation studied the dynamic behavior of a two-delay SVIR epidemic model with the stage structure and nonlinear incidence rate,including the boundedness of the solutions,the existence and local stability of equilibrium points,the existence of the Hopf bifurcation,and the properties of the bifurcating periodic solutions and which indicated the solutions satisfying the initial conditions are bounded.The basic repro-duction number R0 is calculated by the method of regeneration matrix.At the same time,the conditions for the existence of equilibrium are obtained by analysis.Next,the stability of the disease-free equilibrium point M0 is discussed in four cases.It is found that when 0<R0<1 M0is always locally asymptotically stable.When R0>1,M0 is always unstable.To discuss the stability of the positive equilibrium point M*.This dissertation is divided into(?)?1=0??2=0;(?)?1>0??2=0;(?)?1=0??2>0(?)?1>0??2>0??1=?2=?;(?)?1>0??2?[0,?20)five situations to analyze,and the sufficient conditions for the local asymptotic stability of the positive equilibrium point M*and the critical conditions for the Hopf bifurcation are obtained.In addition,using the method of central manifold theorem and normative theory to study the direction and stability of Hopf bifurcation in the fifth case,the calculation formula of bifurcation periodic solution is obtainedSecondly,due to the limited effectiveness of vaccination,in order to reduce the num-ber of infected people as much as possible,this dissertation considers the optimal control model of SVIR with vaccination and uses the principle of Pontryagin maximum principle to obtain the optimal solution u*1,u*2.Finally,in order to verify the rationality of the ob-tained theoretical results,Matlab software is used for numerical simulation,and the results show that the simulation results are consistent with theoretical results.