Dynamic Behavior Analysis Of Stochastic Infectious Disease Model With Multiple Disturbances Of White Noise

In the real world,infectious diseases are inevitably affected by environmental random factors in the process of transmission,which makes the relevant parameters in the infectious disease model(such as contact rate,etc.)to fluctuate around their average value or which causes the whole infectious disease model to be disturbed.In many cases,the effect of random factors(such as white noise)on multiple disturbances in disease transmission cannot be ignored.It is not always ideal to describe and predict the development process and transmission law of disease with deterministic infectious disease model.Therefore,it is of more practical significance to study the dynamic properties of infectious disease model with the effect of random white noise.As is known to all,the harm brought by vertical transmission behavior cannot be ignored in the transmission of infectious diseases.On the other hand,nonlinear incidence is an important index in the study of the spread of infectious diseases.Therefore,in the second part of this paper,the stochastic SIR epidemic model with vertical propagation and nonlinear incidence under multiple interferences of white noise is established and studied,and the dynamic behavior of the stochastic system is discussed and analyzed.First,the existence and uniqueness of the global positive solution of the stochastic system is analyzed,which is the basis of discussing the dynamic behavior of the stochastic system.By constructing an appropriate Lyapunov functional,the threshold of disease extinction and epidemic control was further analyzed and found,and compared with the basic reproduction number of deterministic system without white noise disturbance,the conclusion that disease outbreak can be suppressed with white noise participation was obtained.In addition,the existence of stationary distribution and ergodicity of random system is analyzed.Finally,the asymptotic behavior of solutions of stochastic systems around disease-free equilibrium and endemic equilibrium of deterministic systems is analyzed and discussed.As time delay is also the main factor affecting the spread of infectious diseases,one of the manifestations is temporary immunity.Therefore,the third part of this paper considers stochastic SIR infectious disease model with time delay and nonlinear incidence under multiple interferences of white noise and analyzes its dynamic behavior.It is found that the disease will become extinct in random system under the condition that the global positive solution exists.In addition,by using Kinnally's theory of stationary distribution of stochastic delay differential equations and constructing appropriate Lyapunov functional,the existence and ergodicity of stationary distribution of stochastic systems are obtained,which indicates that diseases will spread.From the biological point of view,the interference of environmental white noise may have certain influence on the stability of biological system: the ability of population to adapt to the environment is limited.If the intensity of white noise in the environment is small enough,the stability of the population will not be damaged.If the intensity of white noise is large in the environment,it may lead to the extinction of species.Therefore,we can make the disease extinct by adjusting the appropriate random disturbance.