Mixed Single Population And SIQRS Infectious Disease Model Under Markovian Switching

In recent years,the research on stochastic biomathematics model with stochastic perturbation has been widely concerned by scholars at home and abroad.In this paper,we have launched a detailed research about the SIQRS model with constant input and Markov Switching and a class of stochastic switching population model between Gompertz and Logistic growth with Markovian Chain.Then we discussed dynamics of different population with stochastic perturbation.The main research contents can be arranged as follows:1.In the first part,we first introduce the ecological background and current re-searches.Secondly,we give some results about population models with environmental disturbance.Finally,we give the major work about our paper.2.In the second part,we propose a class of SIQRS epidemic model with constant input and Markovian switching.By using Lyapunov method,theorem of stochastic dif-ferential equations,differential and integral inequality and together with some properties of the markov chain and random mathematical biology investigate the long time behavior of this stochastic model.Firstly,we analyse the global positivity and boundedness of the model.Then we established the threshold value:the disease can be eradicated almost surely if the basic reproductive number R₀<1,whilethe disease will spread to continue and become”endemic”in our region if the basic reproductive number R₀>1.Finally,the accuracy of the theoretical results are validated by numerical simulation,the sufficient condition of persistent and eliminate for disease are analysed.3.In the third part,we propose a class of stochastic switching population model between Gompertz and Logistic growth with Markovian Chain,analyzes the existence of global positive solutions,obtained the sufficient condition of random persistent,proved that the system is asymptotic stable in distribution.The numerical simulation is taken by using mathematical software-Matlab,it clearly shows the dynamics behavior of population system,discussed the influence of random disturbance on the population system,and analyzes the distribution of the solutions on white noise.