Study On Population And Epidemic Model With Markov State Switching

The population model analyzes and predicts the changes of population quantity in space and time by constructing mathematical model,so as to regulate and control the development of population in nature.By describing and studying the inherent law of disease transmission,the infectious disease model can predict its development trend and provide a theoretical basis for people to prevent and control diseases.However,in the ecosystem,all kinds of random disturbances are everywhere.The growth process of population and the spread of disease are not only disturbed by white noise,but also by other forms of random disturbances,such as the disturbance of color noise,leading to the random transformation of the system in two or more environments.In this paper,we establish three kinds of population and epidemic models with Markov state switching,and discuss the influence of white noise and color noise on population and epidemic.The content of this paper is as follows:1.We develop a predator-prey model with diseased prey under Markov switching.We firstly obtain that the system admits unique positive global solution starting from the positive initial value.Secondly,by constructing some suitable Lyapunov function,we prove that there are unique stationary distribution and they are ergodic.Finally,The sufficient conditions for the weak mean survival and extinction of the system are given.2.We investigate a stochastic predator-prey system with markov switching and partial benefit relationship.By constructing Lyapunov functions,we show that our model has a global positive solution.Sufficient conditions of the system to be extinct and persistent in mean are also established.Then we find that the system has a unique stationary distribution and ergodicity.Finally,Some numerical simulation is provided to demonstrate our main result.3.We investigate a stochastic SIRS epidemic model with double diseases and Standard incidence rate under regime switching.By constructing an appropriate Lyapunov functions and using the It(?)formula,we show that our model has a global positive solution.By using the theory of stochastic differential equations,When R*>1,we prove that system is ergodic.And establish the threshold condition between the population's weakly persistence in mean and extinction.Finally,numerical simulations are carried out to support our theoretical results.