Dynamic Analysis Of Stochastic SIQRS Epidemic Models

The outbreak of infectious diseases has brought tremendous harm and loss to human beings.From the perspective of the dynamics of infectious diseases,the study of the factors affecting the transmission of infectious diseases is beneficial to the prevention and control of infectious diseases.But in real life,there are many random factors,such as temperature and humidity.We found that many random factors in the natural environment have a certain degree of inhibition on the population.In addition,as the outbreak of disease will bring warning warnings to human beings,the government and other relevant departments will carry out certain preventive and intervention measures,such as media coverage and publicity to strengthen the awareness of prevention and protection,and the isolation of infected and suspected cases.Based on the above concerns,the main contents of this thesis is as follows:Firstly,considering the influence of media publicity and isolation measures on disease transmission,a stochastic SIQRS epidemic model with saturation incidence and media coverage is considered.By constructing the random Lyapunov function and using Ito’s formula,the definition of bivariate variation,the strong large number theorem of Martingale,and so on,we prove that the infectious disease model has a unique global positive solution.We then show the sufficient conditions of existence and uniqueness of the stochastic epidemic model.Through analysis,we conclude that the number of infected persons can be effectively reduced by means of media propaganda.That is,media propaganda and reports have a significant effect on disease control.We also show that disease outbreaks can be effectively suppressed by increasing the intensity of white noise.In the end,the correctness of the theoretical results is verified by numerical simulation.Finally,the results are fitted by numerical simulation.Secondly,when the mortality rate is high and there is environmental resistance,the number of inputs to the susceptible is usually not a constant,but rather a function that corresponds to a logistic increase,as the outbreak of SARS in 2003,some scholars used logistic growth model to predict,and the data obtained were very accurate.Therefore,a stochastic SIQRS epidemic model with logistic growth and Beddington-De Angelis incidence is established.By constructing stochastic Lyapunov function and using Ito’s formula,stopping time,and strong large number theorem of Martingale,we prove that the global positive solution of the epidemic model exists and is unique.On this basis,sufficient conditions for the eventual extinction and persistence of the disease are also given.We conclude that as long as the white noise is large enough,no matter what the threshold value,the disease will eventually become extinct.Finally,the theoretical results obtained in this paper are visualized by numerical simulation.At last,considering that in real life,it is possible for the same population to have double diseases at the same time,such as AIDS and Hepatitis B,although the two infectious diseases are unrelated,may be infected at the same time.So we establish a stochastic SIQRS epidemic model with double diseases.The existence and uniqueness of global positive solutions for the stochastic system are proved.Furthermore,we discuss sufficient conditions for the eventual extinction and persistence of the disease.Finally,numerical simulation is carried out for the results obtained.