Analysis And Research On Stochastic Epidemic Model With Quarantined

Epidemic has always been a major concern,especially the COVID-19 at the late 2019,seriously hindered the people normal life,which shows that the study of infectious diseases has a certain significance.With the development of biomathematics,a large number of experts and scholars make full use of the ideas and methods of mathematical modeling to try to establish compartment models based on the transmission mechanism of epidemic.The established models have SI(Susceptible-Infected),SIR(Susceptible-Infected-Recovered),SIV(Susceptible-Infected-Vaccinated),and so on.While some infectious diseases are highly infectious which need to be quarantined immediately after diagnosis,therefore,SIQ(Susceptible-InfectedQuarantined)model appears.Mathematical tools are used to analyze the epidemic model,understand the dynamic changes of infectious diseases,make reasonable predictions and put forward some suggestions.By the dynamic analysis of the established model,the development trend of epidemic is predicted and reasonable suggestions are put forward.Generally,the established models of epidemic are deterministic systems,which cannot reflect the interference of various random factors in the actual transmission of epidemic.Therefore,considering the random model can be closer to the reality.The SEQIR model with standard incidence is established to analyze the dynamics of deterministic and stochastic models in the second part.For the deterministic model,we obtain the basic reproduction number,and prove the stability of the equilibria.For the stochastic model,the existence and uniqueness of global positive solutions are proved,sufficient condition for the extinction of epidemic and asymptotic property for the existence of epidemic are obtained.Numerical simulations are divided into two parts,one part of the given parameters to verify the conclusions,the other part of fitting the COVID-19 daily cases of France,Italy,the UK from February 24 to May 30,2021 by least-square method,which can be drawn in this period the epidemic situation in these three countries remained a downward trend,but the governments still need to continue to take measures to maintain this trend by enhancing the intensity of quarantined,increasing nucleic acid detection efforts,and control the development of the epidemic.In the third part,the stochastic SIQR epidemic model with Lévy jump is established.Before the analysis,it is first proved that the system has a global positive solution.Further,by constructing the appropriate Lyapunov function and the ?Ito formula with jump,the asymptotic behavior of the stochastic system near the equilibria of the original deterministic system is obtained.Finally,the relationship between the solutions of the deterministic system,the stochastic system with white noise and the stochastic system with white noise and Lévy jump is compared.It is found that the solution of the stochastic system oscillates randomly near the solution of the deterministic system,and the amplitude is related to the noise intensity.This paper mainly considers the stochastic epidemic model with quarantined,and uses different random perturbation methods for different epidemic propagation situations to approach the real propagation of infectious diseases as far as possible.Firstly,the SEQIR model is established combined with the spread of the COVID-19,and the dynamic analysis is carried out.Some data are fitted and reasonable suggestions are put forward.Then,the SIQR epidemic model with Lévy jump is established,the dynamic behavior of the model is analyzed,and the conclusion is obtained which the solution of the stochastic system oscillates randomly near the solution of the corresponding deterministic system.