Dynamic Behaviour Of A Stochastic Epidemic Model With Vertical Transmission

The epidemic is always severely influencing the human public health and social harmony.Many scholars use mathematical models to predict the transmission process and reveal their propagation rules.As the maturation of stochastic differential equation,the combination of epidemic and white noise has become one of the main trends.This paper mainly studies the dynamic behavior of a stochastic epidemic model with vertical transmission.In chapter 2,we study the stochastic SIR epidemic model with vertical transmission.Firstly,we show the system has a unique global positive solution.Then we use the methods of stochastic inequality and martingale theory getting the sufficient conditions which can determine the extinction or persistence in mean of the epidemic are presented.Further,in the case of persistence,by Has' minskii ergodic theory,we point out the existence of the stationary distribution and it is ergodicity.Moreover,we give the numberical simulation.In chapter 3,we investigate the deterministic SIS model with vertical transmission.We give the stability of equilibrium and basic reproduction number.Then we explore the dynamic behavior of stochastic SIS epidemic models with vertical transmission.We consider the stochastic models with infectious transmission rate perturbed by white noise,or with multiple white noise.We also establish sufficient conditions for extinction and persistence of the disease.Besides,for the model with system disturbance,by using suitable stochastic Lyapunov function,we establish the sufficient conditions for the existence of a stationary distribution.Moreover,numerical simulations verify the above main conclusions.