Spatial And Temporal Evolution Analysis And Optimal Control Of Several Types Of Infectious Disease Models

So far,the spread of infectious diseases still seriously threatens the safety of human life and the development of world civilization.Since the study of infectious diseases cannot be conducted through large-scale experiments,establishing mathematical models with typical characteristics of infectious disease transmission,conducting kinetic analysis of infectious diseases,exploring the laws of infectious disease transmission,and inferring their development trends are one of the effective methods to prevent and control infectious diseases.Studies have shown that there is an inevitable time delay in the spread of viruses,and the appearance of time delay often destroys the stability of the system and induces more complex dynamic behaviors.At the same time,virus transmission is a dynamic diffusion process,during which susceptible individuals and infectious sources move randomly in the spatial domain,and therefore the reactive diffusion effect of virus transmission needs to be considered when studying the dynamics of infectious diseases.However,there are relatively many studies on neural network dynamics considering diffusion effects,but there are few research efforts on infectious disease dynamics.In this thesis,on the issue of infectious disease dynamics,based on the theoretical results obtained by previous studies,the SIR infectious disease model dynamics under the integer order/fractional order and the application of a two-state feedback controller to consider the effect of spatiotemporal diffusion and individual contact heterogeneity are deeply explored.The influence of the SIR infectious disease model on the dynamic behavior,the main research contents are as follows:(1)Given that fractional order differential equations are more accurate in describing networks,the dynamical behavior of a class of fractional order SIR infectious disease models at different orders was investigated.Considering the population logistic growth and disease saturation transmission rate,the time delay term is selected as the bifurcation parameter,and the local asymptotic stability and Hopf bifurcation of the infectious disease model are analyzed to reveal the influence law of fractional order on the stability domain of the infectious disease model.It is shown that changing the fractional order can advance or delay the Hopf bifurcation phenomenon.By selecting the appropriate fractional order and model parameters,the model can be effectively controlled to be in a stable state at the equilibrium point.(2)The spatio-temporal dynamics of a class of SIR infectious disease models with individual contact heterogeneity was studied in conjunction with the spatial diffusion effect in the transmission of infectious diseases.The diffusion coefficient is chosen as the variational parameter to obtain sufficient conditions for the local asymptotic stability of the equilibrium point of the infectious disease model and the occurrence of Turing destabilization.It is found that diffusion has a great influence on the dynamics of infectious diseases,and the Turing instability of infectious disease models can lead to the formation of patterns.(3)The optimal control of the spatio-temporal dynamics of the reaction-diffusion SIR infectious disease model based on state feedback is investigated.The time delay is chosen as the bifurcation parameter,and sufficient conditions are given for the local asymptotic stability of the equilibrium point of the controlled infectious disease model and the occurrence of Hopf bifurcation.The effect of controller gain on the bifurcation point of the system is further explored by comparing the bifurcation points of the controlled and uncontrolled models.It is shown that adjusting the control gain can effectively advance or lag the bifurcation point,and the controlled model bifurcation point decreases as the controller parameters increase.According to this thesis,it is known that the fractional order of the model,the virus propagation time delay,the spatial diffusion effect and the feedback parameters of the controller all have significant effects on the dynamics of the infectious disease model.Adjusting these parameters can effectively expand or reduce the stability domain of the infectious disease model.The research results obtained in this thesis help to expand the theory of nonlinear dynamics on the one hand,and provide theoretical guidance for the prevention and control of infectious diseases on the other.