本申请项目主要研究随机生物数学模型、传染病模型和生态流行病模型的动力学行为。首先通过随机比较定理和随机Lyapunov方法给出系统正解的存在性。在此基础上通过定性分析研究种群的持久性和稳定性，以及疾病的流行和消失。对于具有退化和非退化扩散项的随机模型，通过扩散过程理论、Has'minskii的遍历性理论和Markov半群理论等探讨系统的平稳分布和遍历性。通过上述分析，目的在于揭示随机扰动对模型渐近行为的影响，解释一些实际现象。从而为预测生态社会各物种数量的变化，预测疾病的发展趋势以及分析疾病流行的原因提供理论依据。

This research will mainly study the long time behaviour of mathematical biology models, epidemic models and eco-epidemiology models with stochastic perturbations. First of all, we will give the existence of the positive solution of stochastic systems with stochastic comparison theorem and stochastic Lyapunov method. Then we will show the persistence and stability of populations, and the disappear and outbreak of the disease. Furthermore, for systems with degenerate or nondegenerate diffusion term, we will investigate the stationary distribution and ergodicity of them through diffusion theory, ergodic theory of Has'minskii and Markov semigroup theory etc. Based on these analysis, we want to reveal the effect of stochastic perturbation. And so it can offer the theory support for predicting the change of population, forecasting the development trend of disease and analysing causes of its outbreak.