Dynamic Behavior Study Based On SIVS Epidemic Model

With the deterioration of the ecological environment,all kinds of infectious diseases are breaking out all over the world,which poses a great threat to the world economy and human health.Mathematical model is one of the important tools to study the essential law of things.In recent years,it has been widely used to reveal the occurrence and development law of all kinds of infectious diseases.The transmission process of infectious diseases is affected by the season,climate,politics,humanity and other random factors.The application of stochastic differential equation to describe the transmission process of infectious diseases is more consistent with the objective facts of the transmission of infectious diseases.Firstly,considering the influence of seasonal incidence,a random white noise term,which is proportional to the state of the system,is introduced,and a random SIVS infectious disease model with nonlinear incidence and nonlinear disturbance under the Markov chain is established.Then the existence conditions of ergodic stationary distribution were discussed by using stability theory and stochastic differential equation theory respectively,and the persistence threshold and the extinction threshold in the average sense of the infected persons were obtained.Then,considering the periodic variation of the environment and ignoring the influence of colored noise,the random boundedness and random persistence of the solutions are discussed by using Has’ Minskii periodic solution theory and inequality estimation method,and the existence of nontrivial positive periodic solutions is proved.Finally,the theoretical results are supported by numerical simulation.Secondly,because the traditional models of uniformly mixed infectious diseases ignore the limitations of the local contact pattern of the population,and more and more scholars are using scale-free networks to describe the non-uniformly mixed characteristics of the population.In the second work of this paper,we establish a SIVS time-delay infectious disease model with latency effect in a scale-free network.The global stability of the disease-free equilibrium and the persistence of the endemic equilibrium are discussed by constructing the appropriate Lyapunov function.In this paper,two types of SIVS infectious disease models were established respectively considering the uniform and non-uniform mixture of the population,and abundant theoretical and numerical results were obtained,which has important theoretical significance for the prevention and control strategies of the disease.