Turing Instability And Hopf Bifurcation In Epidemic Models And Its Control

Hopf bifurcation has been widely used in the field of biomathematics as an important indicator to explore the dynamic characteristics of systems in the nonlinear field.In recent years,especially the bifurcation research of epidemic model has been greatly enriched.Early modeling of epidemics often ignored the memory and genetic characteristics that are present in actual transmission,which can be well portrayed by fractional-order derivatives.At the same time,considering that individuals inevitably migrate when epidemics break out,it is necessary to add a diffusion term into the epidemic model,and the existence of diffusion term usually leads to Turing instability.In addition,the incubation period of epidemics is a key factor affecting the dynamic behavior in practical research,which can be simulated by a time delay term.In order to improve the accuracy of the analysis of epidemic dynamic model,a hybrid controller is added to optimize the bifurcation behavior of the model and enhance the steady-state performance of the model,which improves the application prospect to a certain extent.Build on the previous work,this thesis explores the stability and bifurcation phenomena of the delayed epidemic model containing fractional-order derivative,the dynamical behavior of the epidemic model driven by diffusion,and adds a controller to optimize it.Specific work is as follows:(1)A new fractional-order epidemic SIR model with time delay is established,and the distribution of the root of the characteristic equation of the model is calculated.The basic reproductive number is obtained and the steady-state of the model is discussed.Using the fractional-order calculus theory,the local asymptotic stability of the model is investigated with the disease latency time delay as bifurcation parameter,and the conditions for Hopf bifurcation are given.The influence of the fractional-order on the bifurcation point is also investigated.The results show that the bifurcation point of the model decreases with the increase of the order.(2)Turing instability of SIR epidemic model driven by diffusion is analyzed.The position and stability premise of all equilibrium points in the model are determined by using eigenvalue method.The spatio-temporal dynamic characteristics of the model are researched through the partial differential equation theory.The research shows that the addition of diffusion term will induce the occurrence of Turing instability and the corresponding patterns are presented.(3)Bifurcation control of reaction-diffusion epidemic model based on hybrid controller is investigated.The dynamical behavior and control strategy of the model are discussed by stability theory and bifurcation control theory.The controller parameters are effectively adjusted to expand the stability domain of the model.The positions of the Hopf bifurcation periodic solutions are altered,and the dynamic characteristics of the epidemic model are optimized.