随机传染病模型的动力学行为

This research will mainly study the dynamic of epidemic models with stochastic perturbations. Based on the existence theory of stochastic differential equations, stochastic comparison theorems and stochastic Lyapunov analysis method, we will give the existence of the positive solution of the model. Then we will investigate their long time behavior. In general, there is neither a disease-free equilibrium nor an endemic equilibrium in stochastic epidemic models which are studied. We will show the extinction and persistence of the disease in the population, and give basic reproduction number. Furthermore, for a system with degenerate or nondegenerate diffusion term, we will investigate the stationary distribution and ergodicity of it through diffusion theory, ergodic theory and Markov semigroup theory etc., which also means the disease will prevail and persist in a population. At last, we will give simulations to support our analysis results, and do comparison with diterministic models to show the effect of white noise. Besides, we will explain some phenomena according to some data of the disease. And so it can offer the theory support for forecasting the development trend of disease, analysing causes of its outbreak, prevention and control of disease.

本项目研究传染病模型中参数在随机扰动下的动力学行为。首先利用随机微分方程解的存在性定理、随机比较定理和Lyapunov分析法给出模型正解的存在性。其次探讨解的渐近行为。我们拟研究的随机模型通常不存在无病平衡点和地方病平衡点，通过解的性质反映疾病的消失和流行，并在此基础上给出系统的基本再生数。另外，通过遍历性理论和Markov半群理论等证明具有非退化和退化扩散项的传染病模型存在平稳分布，这也表明疾病在种群中是持久的。最后，通过数值模拟直观上说明所得的性质，并与确定性模型比较，揭示白噪声对模型渐近行为的影响；并结合一些疾病的实际数据，解释一些现象。总之，通过分析可以为预测疾病的发展趋势、分析疾病流行的原因和关键因素以及指导疾病的预防和控制工作提供理论依据。

本项目按计划进行并达到了预期目标。本项目研究了一些传染病模型在随机扰动下的渐近行为，主要得到下面一些结果：(1)分析了带随机扰动的SIR型，SIRS型传染病模型的渐近行为，给出了随机意义下的基本再生数；(2)通过构造合适的李雅普诺夫函数得到了随机斑块型的SIRS，SEIR传染病模型正解的存在唯一性，以及模型中传染病消失的条件；(3)证明了非退化的随机SIR，SIRS，SEIR等传染病模型存在平稳分布，具有遍历性。本项目共完成5篇论文，其中3篇已经发表，均为SCI检索。