Dynamic Behavior Of Stochastic Epidemic Models With Markov Switching

Epidemic diseases,including plague,cholera,AIDS and bird flu,have always been a major problem affecting the life and survival of all human beings.By using deterministic differential equations to establish a mathematical model,epidemiology describes and studies the internal transmission laws of diseases,predicts disease trends and gives the optimal control strategy.However,there always exists noise in the environment.Whether it is the change of light intensity in a day or the change of temperature and humidity caused by the alternation of seasons,it will affect the transmission rate of diseases.Random noise is generally divided into two types:one is the sum of many independent tiny random noises,which are usually described in mathematics by white noise;the other is random noise with a small number but a large intensity,such color noise will cause the system to switch between different states and is usually described by continuous-time Markov chains.Based on deterministic epidemic models,scholars established the stochastic epidemic model driven by a mixture of white noise and color noise.They achieved many important results by using the theories of stochastic differential equation and properties of hybrid diffusion system.By mathematical modeling,stochastic analysis and other methods,the SIRS stochastic epidemic models under different noises with nonlinear transmission rate and vaccination are studied in this paper.We further study the SIS epidemic model in complex network under the influence of color noise,and give an improved numerical method.This dissertation is a preliminary study on the dynamic modeling of epidemic with Markov switching.The main idea of this dissertation includes the following aspects:（1）.A stochastic epidemic system with both color noise and white noise is proposed to research the dynamics of the diseases.Nonlinear incidence and vaccination strategies are also considered in the proposed model.By using the method of stochastic analysis,we point out the key parameters that determine the persistence and extinction of the diseases.Specifically,if R₀^sis greater than 0,the stochastic system has a unique ergodic stationary distribution;while if R₀^*is less than 0,the diseases will be extinct at an exponential rate.Numerical simulations of stochastic SIRS epidemic models with different types of noise are given,the results show that the influence of noise on the spread of epidemic is very complex.The disease can resist the interference of small white noise,while large white noise may cause the extinction of the disease.（2）.A stochastic SIRS epidemic model with nonlinear incidence and vaccination was concerned.The disease transmission rate of this model is driven by a semi-Markov process and under the effect of white noise.By studying the threshold dynamics of the proposed model,we define a basic reproduction number R₀^sunder the semi-Markov regime-switching uncertain environment.When R₀^s<1,the disease will be extinct;when R₀^s>1 the disease will be almost surely persistent in the mean.We give the sensitivity analysis of the R₀^swith each semi-Markov switching under different distribution functions.The result proves that the dynamics of the entire system has a positive correlation to its mean sojourn time in each subsystem.The threshold R₀^swe give can be applied to all piecewise-stochastic semi-Markov processes,and the results of the sensitivity analysis can be regarded as a priori work for optimal control.（3）.A complex network epidemic model with demographics for disease transmission is proposed,and its transmission rate is driven by Markov-switching.By studying limit system of subsystem and the Lyapunov exponent of stochastic system,we derive the explicit expression for the basic reproduction number R₀^*of the model.Specifically,if R₀^*<1,the disease will be extinct;if R₀^*>1 the disease persists almost surely.（4）.In order to study the global asymptotic properties of the SIS epidemic model with Markov-switching in complex networks,the dynamics of the cooperative Markov-switching system under sublinearity assumption is devoted.A threshold dynamics determined by the Lyapunov exponent（?）is established.When （?）<0,the solution will always remain in 0;if（?）>0 and the subsystem satisfies some conditions,the positive solution from the forward invariant set will converge to a unique attractor almost surely.The main result shows that the global asymptotic behavior of cooperative sublinear Markov-switching system is completely determined by the local stability of zero solution of the associated cooperative system in each regime.（5）.In order to give the numerical simulation of the SIS epidemic model with Markov switching in a real high-dimensional complex network,a new numerical method is proposed.The improved 1-Lipshcitz neural network model can effectively reduce the influence of random noise on the simulation results.The parameters of the stochastic epidemic model in the real network are fitted by this improved neural network,and the simulation result are exactly the same as the theoretical result when R₀^*>1.