Stochastic SEIR model with jumps

The paper establishes stochastic SEIR models with jumps; obtains system (2) and system (3) by using two different disturbance manners, respectively, which are used to describe the wide spread of the infectious diseases due to the medical negligence, etc. By adding the first random perturbations, we obtain Lyapunov function and examine that the solutions of model (2) are stochastically stable with the average time; summarize the conditions for persistence and extinction of the infectious diseases. By adding another random disturbance, we get the stochastic system (3). We prove that the positive solution of the system is stochastic asymptotically stable by using Lyapunov method. The conditions that the infectious diseases spread continually are obtained. These studies are good for the prevention and control of the infectious diseases which are with the incubation period especially with long ones.