Research On Dynamics Modeling And Optimal Control For Infectious Disease Based On Fractional Differential Equations

From time immemorial,infectious diseases have been evolving with the development of human society and threatening the safety of human life and property.In the research of infectious diseases,mathematical models are effective tools to analyze the relevant characteristics of infectious diseases and to prevent and control the spread of infectious diseases.According to the characteristics of the spread of infectious diseases,many researchers establish appropriate models of infectious diseases and analyze the dynamic behavior of infectious diseases in the process of transmission through qualitative theory,so as to effectively predict and control the spread of infectious diseases.The modeling,dynamic analysis and control of infectious disease model are important research topics,which has great research value in theoretical analysis and practical application.In the natural environment,the dynamic behavior of infectious diseases is affected by various factors due to the differences of infectious diseases types,transmission areas and infected individuals.The fractional characteristics in the environment,random noise and time delay all affect the dynamic behavior of infectious diseases more or less.In order to reflect the transmission rules of infectious diseases under different interference factors,this paper used Caputo fractional differential equation and stochastic differential equation to establish corresponding models,analyze the relevant dynamic behavior of infectious diseases,and provide theoretical guidance for the prevention and control of infectious diseases.The specific work is as follows:1.Considering the heredity,memory and infection time delay of infectious diseases,a time delay SIRS model with generalized incidence is established by using Caputo fractional differential equation.Some existing epidemic models are special forms of this model.Firstly,a series of qualitative properties of the model are analyzed,including the nonnegativity of the solution,the existence and uniqueness of disease-free equilibrium point and endemic equilibrium point.According to the relevant fractional order theory,the local asymptotic stability of disease-free equilibrium point and endemic equilibrium point are discussed in detail for the cases with and without time delay respectively.The Hopf bifurcation of the model is studied by selecting the time delay as the bifurcation parameter.A class of time-delay self-feedback controller is designed and the Hopf bifurcation of the controlled model is analyzed.The theoretical results are verified by numerical simulation,and it is found that better dynamic characteristics of the infectious disease model can be obtained by adjusting the controller reasonably.2.Under the interference of environmental noise,the transmission of infectious diseases between predator and prey populations is researched.For the two cases that infectious diseases only spread among predators and infectious diseases spread intraspecific between predators and prey,the corresponding models are established by using stochastic differential equations.A series of qualitative properties of these two models are analyzed,including the existence and uniqueness of global positive solutions,the stochastic ultimate boundedness of the solution and the extinction of the population.For the model in which infectious diseases only spread among predators,the long-term dynamic behavior of the population and the global attractiveness of the solution are further analyzed.In addition,for the model in which infectious diseases spread between predators and prey,the sufficient conditions for the persistent coexistence of infectious diseases and populations are established.3.In order to research the variation trend of isolated individuals during the spread of infectious diseases,two kinds of generalized fractional order SEIR models are proposed.A series of qualitative properties of the two kinds of fractional epidemic models are analyzed,including the nonnegativity of the solution,the existence and local asymptotic stability of the disease-free equilibrium point and endemic equilibrium point.Taking into account the practical policies adopted during the coronavirus disease in2019(COVID-19)propagation period,a fractional COVID-19 model with time-varying coefficients is proposed on the basis of these two types of generalized fractional SEIR models.The parameters of the fractional order COVID-19 model are identified through the real case data in different regions.Based on the results of parameter identification,the transmission trend of COVID-19 is effectively fitted and predicted.In order to further predict the trend of the epidemic in the second growth,an effective processing method is given.4.For the widespread vaccination of COVID-19,a fractional COVID-19 model that can characterize immune individuals is proposed.A series of qualitative properties of the model are analyzed,including the nonnegativity of the solution,the existence and local asymptotic stability of disease-free equilibrium point and endemic equilibrium point.Two kinds of control strategies are designed: vaccination and detection isolation.Under the condition of medical resource shortage and cost limitation,the corresponding fractional optimal control problem is discussed.Real case data are used to identify the model parameters,and the validity of the fractional COVID-19 model in describing the development trend of the epidemic is verified.The fractional optimal control problem is numerically solved,which provides a theoretical reference for epidemic control.