Analysis On Dynamics Of Several Stochastic Epidemic Model With Lévy Jumps

People are always paying attention to prevention and control of infectious disease all the time, since it will endanger people’s life and even make some critical effects on the survival of the race and the country once the disease outbreaks and is out of control.So prevention and control of the infectious disease is not only a public health problem but also a public safety issue. Therefore, studying it from the medical standpoint is not enough, it will be of greater importance to prevent and control it by studying its spreading law. Currently, people know little about some large-scale infectious diseases as SARS and avian influenza which are highly pathogenic and fatal, cannot find safe and effective drugs to fight these diseases and there is not enough time to research and develop the preventive drugs when they break out. In these circumstances studying the transmission rules and cutting the transmission ways of the infectious disease through human intervention appear particularly important. Studying the transmission rules by the way of establishing the mathematical model is a proven and effective method.In order to precisely describe the transmission rules of the infectious diseases whose characteristics are great harmful, fast transmitted and seriously destroyed to the ecosystem of the population, we establish some stochastic models with L′evy jumps in this dissertation. The research contents are as following:1. Study on the SIR model with L′evy jumps which is used in describing the transmission rules of highly pathogenic infectious diseases as SARS and avian influenza. In this thesis we get two kinds of different stochastic models by introducing two kinds of disturbance manners, and then we study these two models by using the stochastic analysis and Lyapunov method, respectively. First, we prove that the solutions of the first model are globally positive, and we obtain the relationship between the solutions of our model and the equilibrium of the corresponding deterministic model at relatively low level of random disturbance. Second, the stochastically asymptotical stability of the equilibrium of the second model is proved. Finally, we give the numerical simulations which fit the reality well.2. Study on the stochastic SEIR model with L′evy jumps. In order to precisely describe the widespread of the diseases with incubation period which are caused by the medical negligence, two stochastic models are deduced by using two different disturbancemanners. By adding the first random perturbations, we get that the solutions of the first model are globally positive and stable in the average time by using Lyapunov Function and It ?o Formula with jumps. Then we obtain the relationship between stochastic SEIR model with L′evy jumps and its corresponding deterministic model, and we summarize su?cient conditions for the extinction and continually spreading of the infectious diseases. By adding another random disturbance, we prove that the positive equilibrium of the system is stochastically asymptotically stable by using the stochastic analysis and Lyapunov method, and su?cient conditions for the infectious diseases spread continually are obtained. Then we further testify the accuracy of the theorem through numerical simulations.3. Study on the AIDS transmission model with L′evy jumps which is used in describing the widespread of AIDS which is caused by drug taking and medical negligence.Infected people are classified by the individual differences of patients in this model. Lyapunov Function and It ?o Formula are adopted to analyze the stochastic model with L′evy noise proportionally. We prove the globality of the positive solutions and obtain su?cient conditions for stability in the sense of time average. Su?cient conditions for the AIDS extinction or continually spreading are summarized at relatively low level of random disturbance. Moreover we establish another stochastic model by adding L′evy jumps to the equilibrium of the former system, and summarize su?cient conditions that make the epidemic equilibrium stochastically asymptotically stable and the numerical simulation is presented as well. Our conclusions show the dynamic properties of the AIDS system, and have great instructive significance in preventing and controlling the transmission of AIDS caused by drugs and medical negligence.The stochastic model with L′evy jumps can clearly reflect the transmission rules of the infectious diseases which are highly pathogenic, fast transmitted, extensively prevalent and seriously destroyed to the ecosystem of the population. The study of my dissertation shows a great instructive significance to prevent and control the transmission of the large-scale infectious diseases for the relevant departments.