Dynamics Analysis And Optimal Control Of Vector-borne Epidemic Model With Time Delay And Direct Transmission

In this paper,we study the SIRMV model of a class of vector diseases with time delay and direct transmission.Firstly,the dynamic behavior of the model is studied,for any??0,when R₀<1,we obtain the global asymptotic stability of the disease-free equilibrium by using the eigenequation method and constructing Lyapunov functional.It's shown that the existence of time delay does not affect the global behavior of the disease-free equilibrium;When R₀>1,a unique endemic equilibrium exists and time delays destabilized the system.Taking the time delay as the bifurcation parameter,sufficient conditions for local stability and the existence of Hopf bifurcation are obtained.Further,the direction and stability of Hopf bifurcation are studied by using the normal theory and the central manifold theorem.Based on SIRMV model with time delay,three control variables are added to obtain the con-trol system,and the existence of optimal control and the specific expression form of optimal control are proved.Finally,the stability of the equilibrium point and the effectiveness of the optimal control problem are verified by numerical simulation.Chapter 1,We introduce the research background and significance of vector-borne dis-eases,and the development status of SIRMV model.Chapter 2,We introduce relevant preparatory knowledge,including Hopf bifurcation theorem,optimal control theory and related theorems.Chapter 3,In this chapter,we propose a class of time delay SIRMV models with direct transmission.Firstly,the positive of the solution to the infectious disease model is proved,and the basic regeneration number R₀of the model is obtained according to the biological significance.Further,the equilibrium of the model are calculated,by using the characteristic equation method,Routh-Hurwitz theorem and constructing Lyapunov functional,the following results are obtained:when R₀<1,it's shown that the existence of time delay does not affect the global behavior of the disease-free equilibrium;When R₀>1,a unique endemic equilibrium exists,and time delays destabilized the system.Finally,the direction and stability of Hopf bifurcation are studied by using the normal theory and the central popular theorem.Chapter 4,Three control strategies are added into the original model,namely,blood donor screening,vaccination of susceptible persons,and use of insecticides,the first two are to control the susceptible population,the latter is to control the vector.Combined with these three controls,the corresponding optimal control problem is proposed.Firstly,the existence of optimal control for the control system is proved,and then the specific expression of the optimal control is given according to Pontryagin maximum principle.Chapter 5,In this chapter,the model is numerically simulated by Matlab.Firstly,assign a value to the parameter,when?=0,the global stability of the disease-free and endemic equilibria is verified;when??=0,the global stability of the disease-free equi-librium point is verified;when(H₅)is true,the local stability of the endemic equilibria is verified;When(H₆)and(H₇)are true,a Hopf bifurcation appears when?>?₀,and when?<?₀,the endemic equilibrium point is locally asymptotically stable.Finally,the existence of the solution to the optimal control problem and the effectiveness of the selected control is verified.Chapter 6,Summary of the full text,the main innovation points and shortcomings.