Dynamics And Optimal Control Problem Of A Delayed SIR Model

The SIR model is a classic model that describes the dynamics of infectious diseases.Traditional SIR models usually consider only time variables and ignore the influence of spatial position.With the development of modern transportation networks,the mobility of the population has accelerated the spread of the disease and has become an important influencing factor of the model.In this paper,we consider time delay based on the SIR model with spatial diffusion.A class of SIR model with time-delay and spatial diffusion is proposed.Firstly,the existence,uniqueness and and positive boundness of its global solution are obtained.Then the local stability of the disease-free equilibrium point and endemic equilibrium point is obtained by the eigenvalue method,and then the global stability is proved by constructing the Lyapunov function.Then the vaccine control is considered in the model,and then an optimal control problem for the model is proposed.The existence of optimal control solutions is proved and Pontryagin maximum principle is used to give the necessary conditions for optimal control.Finally,the model is numerically simulated using the finite difference method.Chapter 1,we introduce the background and significance of epidemiological model research,and the development status of SIR model and its extended model.Chapter 2,we list some related concepts and theorems about infectious disease dynamics,stability theory and optimal control theory.Chapter 3,we combine the research of the existing delayed SIR model and the spatial SIR model,taking into account the two factors of disease latency and population spatial diffusion.We propose a class of SIR model with time-delay and spatial diffusion.Then we prove the existence and uniqueness and positive boundedness of the global solution of the model,and then calculates the model's disease-free equilibrium point,endemic equilibrium point,and the basic regeneration number R₀.Using the eigenvalue method,We prove that the disease-free equilibrium point is locally stable when R₀<1;the endemic disease-balance point is locally stable when R₀>1.Then the global asymptotic stability is deduced by the Lyapunov function method:when R₀<1,the disease-free equilibrium point is globally asymptotically stable;when R₀>1,the endemic equilibrium point is globally asymptotically stable.Chapter 4,we consider vaccine control on the model.Aiming at the specified objective function,we propose the optimal control of the model.Firstly,the existence of the optimal control solution was proved by constructing minimization sequences,and then the necessary conditions for optimal control were derived according to the Pontryagin maximum principle.Chapter 5,we do the numerical simulation of the model.Under different parameters,the equilibrium solution of the model at R₀<1 and R₀>1 is obtained respectively for different delay parameters so that the global stability of the model's disease-free equilibrium and endemic equilibrium are proved.Under a given weighting coefficient,a specific set of parameters is selected and different delay parameters are used.By the Forward-Backward Sweep iterative method,we solve the state equation,the adjoint equation and the corresponding optimal control solution to prove the existence of optimal control.Aiming at the optimal control problem studied in Chapter 4,numerical simulations are used to determine the temporal and spatial distribution of vaccine control adopted in the control of the spread of infectious diseases which has minimized the number of the infected and the cost of the vaccine control.Then the effectiveness of taking vaccine control is verified.Chapter 6,we summarize the paper and look forward to the work done in this paper,and then explain the main innovations and deficiencies.