Qualitative Analysis Of Some Classes Of Epidemic Models And Its Application In COVID-19

Infectious diseases not only endanger personal life and health,but also profoundly influence the historical development of mankind,and even lead to the decline of the country and the disappearance of civilization.Therefore,it is of great significance to study the epidemic model.On the basis of SIR compartment model and SIS compartment model established by Kermack and Mckendrick,the corresponding epidemic models are established,and their dynamic behaviors are analyzed.Then,taking corona virus disease2019(COVID-19)as an example,parameter estimation and numerical simulation are carried out.In the first part,we study a class of SEIRS epidemic model with general nonlinear incidence,distributed time delay and vertical infection.Firstly,the well-posedness of the model solution is given by using the theory of functional differential equations with time delay.Then the basic reproduction number 0 is determined by the next generation matrix method.Finally,by constructing two appropriate Lyapunov functionals and applying LaSalle's invariance principle,the threshold conditions for the global asymptotic stability of the equilibrium point are obtained.The results show that the global dynamics of the model is completely determined by the basic reproduction number under the appropriate assumption of the nonlinear term of the incidence rate,and the distributed delay does not affect the global dynamics of the model.In the second part,taking the epidemic disease in COVID-19 as an example,according to the characteristics of the spread and the isolation of infectious diseases,a class of SEIQRS infectious disease model is established.Firstly,the well-posedness of the solution of the model is analyzed by the method of disproof and comparison theorem.Secondly,the basic reproduction number 0.Then by analyzing the corresponding characteristic equations and using the Routh-Hurwitz criterion,the local asymptotic stability of the disease-free equilibrium point is obtained.Finally,the estimated values of key parameters such as contact rate,isolation rate,recovery rate and basic reproduction number in Hubei,Zhejiang and Hunan provinces are obtained by the least squares method.Numerical simulation shows that this model effectively simulated the spread of disease.In the third part,a parameter estimation method based on Richards model is proposed for the COVID-19 pandemic.First of all,the cumulative number of cases in COVID-19,China is fitted by stages by using the data difference.Then,the estimated values of key parameters such as the inflection point of the COVID-19 epidemic,the final infection number,and the basic reproduction number in China are obtained by least squares fitting,and the analysis is made using these key parameters.Finally,using the number of confirmed COVID-19 cases in the Western Pacific Region,it is proved that the same program can also be used for real-time prediction of multi-wave epidemic.The results show that according to the official epidemic data released by WHO and the Richards model,not only can the estimated values of key parameters and effectively simulate the development of the epidemic situation be obtained,but also the epidemic situation can be predicted and some suggestions on epidemic prevention and control can be put forward.