| Since the end of 2019,Coronavirus Disease 2019(COVID-19)pandemic has caused hundreds of millions of infections,deaths,social impact,and economic losses.Single discipline research cannot effectively analyze and control the COVID-19 pandemic.Therefore,interdisciplinarity is becoming a research hotspot.As an extension of integral calculus(IC),fractional order(FO)theory has gradually become the study trend in mathematics,physics,economics,biomedicine,and other fields.Especially in control engineering,fractional calculus(FC)can be used to develop and analyze the system more accurately and concisely.FO optimization algorithms can improve the performance of control systems.As we all know,infectious disease systems are nonlinear and time-varying.The modeling systems are easily affected by environmental and policy changes,geographical mobility,and infectious symptom data,which change the accuracy and feasibility of systems.Therefore,based on the FO compartment models,the study of fractional order systems(FOSs)dynamics such as fitting and prediction,numerical analysis,and optimization control for the COVID-19 epidemic is essential and significant.In systems theory,the compartment model is used to describe the time-varying system of reactions between populations in different compartments,which plays an essential role in the modeling of practical phenomena.Such as epidemics,the dynamics of pathogens within the host,and the transport of chemicals(drugs,hormones,nutrients,etc.)between body parts.The governing equations are a set of coupled ordinary differential equations(ODEs).Compared with integer order systems(IOSs),FOSs include the long memory and genetic characteristics of evolutionary systems.Although the FO epidemic model is applied to epidemiological dynamics by introducing FC,the essence of the system changes,impeding the de velopment of FO infectious disease dynamics.For instance,the eigenvalues of the FOS change from a single value to multi values,and the state space changes from the finite dimension to the infinite dimension.Therefore,the theoretical analysis of the system is difficult,for example,the proof of existence and stability of the disease-free equilibrium,the endemic equilibrium,and the physical significance of the left-hand or right-hand side FC of models.The dissertationfocuses on the qualitative theoretical analysis,quantitative experimental simulation,and application of FO epidemiology dynamics for the COVID-19 pandemic.My main contributions are as follows:Firstly,due to the aberration phenomenon caused by the FO non-zero initial values system is inevitable,this dissertation emphasizes the rational approximation and numerical implementation of FOSs with the long memory and the infinite dimension.The limitations of the calculus chain simulation module method based on the RiemannLiouville(RL)FC are analyzed.Considering the pure integral property of 1/sα,a strictly positive approximation model of FC operators in the frequency domain is constructed.Moreover,an integrator chain block diagram for the state-space model is designed to simplify the structure and improve performance.This method is suitable for both RL and Caputo FC.The validity and superiority of the system are verified by the one-term fractional ODE(FODE),the linear multi-term FODE,and the explicit nonlinear FODE(Benchmark equations),respectively.Secondly,based on the modular method of the FO non-zero initial value system,to capture the COVID-19 pandemic with an expanding transmission chain,a class of FO generalized SEIR(FO-GSEIR)models are studied.From the physical perspective,the necessity of developing fractional models is analyzed.It is proved that the global dynamic of the model is determined by the basic reproduction number R0,which is determined by the spectral radius of a linear integral operator.If R0<1,the disease-free equilibrium point is globally asymptotically stable.Otherwise,if R0>1,the disease will become endemic.Based on the real-time statistics reported by the Public Health Department,Simulink Design Optimization(SLDO)blockset is used for the study of COVID-19 dynamic systems for the first time.The least-squares algorithm is used to fit the infection trend and parameters.Moreover,model sensitivity analyses and timeweighted performance indices(for example,ITSE,integral time squared errors)based on optimization are performed using SLDO to adjust parameters and improve the fitting accuracy.Using the Receding Horizon Control(RHC),the persistent short-term forecast and periodic accumulative forecast are obtained.Simulation results demonstrate that the FO-GSEIR model performance is superior to the IO one.For instance,FOSs provide more accurate short-term and long-term pandemic forecasts to guide prevention policies and timely allocate medical equipment and resources.Furthermore,the strong concealment of mutated viruses such as Delta and Omicron has led to a steep increase in the difficulty of epidemic detection,and complete eradication of the COVID-19 pandemic is difficult to achieve.Due to the high infection rate and low fatality rate of viruses,the psychological defense line and social behaviors of susceptible people change.Thus,combined with the real-world Facebook symptom data and state-level Google mobility data,which are the number of COVID-19 consultations and keyword searches,the FO-GSEIR model with nonmonotonic incidence is proposed.The improved model,which reflects psychological effect factors,can better characterize inhomogeneity in human interactions and infection rates,and deliver more accurate predictive performance.Therefore,government policies and infection data of California are taken as an example to study the perturbation control of the protection rate affected by vaccination intentions,and the infection rate affected by social mobility under different intervention policies.The prediction results of the analyzed FO-GESIR models would be able to guide mitigation strategies for various tasks such as social reopening,to minimize casualties and economic losses.Finally,the lack of COVID-19 drug treatment makes it impossible to eliminate the disease pandemic through geographical isolation and medical treatment.Fortunately,periodic vaccination is emerging as an effective response to outbreaks.However,we lack a study of vaccination rate and effective rate,to do so,we propose Recursive Integration Optimal Trajectory Solver(RIOTS)as a vaccination control problem Solver of general fractional order optimal control problems(FOOCPs).Moreover,the effectiveness of RIOTS as a benchmark solution for FOOCPs of vaccination and treatment is verified for the first time.Afterward,a class of FO-GSEIR models with informationinduced functions are proposed to study FOCPs of the COVID-19 vaccination strategies.The proposed optimal vaccine strategy can explicitly consider constraints on the vaccine availability,the speed of replenishing,the vaccine application rate,etc.,and can make the whole management in a closed-loop manner with an optimization performance index.The closed-loop vaccine system can achieve good tracking performance given a future desired and feasible trend.More precisely,forecast trends of infections and death cases in different immunization scenarios can guide individuals and policy decision-making processes in developing mitigation measures and monitoring community risk levels.Based on the COVID-19 pandemic,the dissertation makes up for gaps in the compartment modeling,fitting and prediction,theoretical analysis,and numerical simulation of FOSs for the epidemic dynamics.Moreover,it expands applications of FOOCPs in mitigations,treatments,and vaccination strategies for general infectious diseases. |
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