Zhou Baoquan, Jiang Daqing, Dai Yucong, Hayat Tasawar, Alsaedi Ahmed
College of Science, China University of Petroleum (East China), Qingdao 266580, P.R. China.
Nonlinear Analysis and Applied Mathematics(NAAM)-Research Group, Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia.
Chaos Solitons Fractals. 2021 Feb;143:110601. doi: 10.1016/j.chaos.2020.110601. Epub 2020 Dec 24.
Considering the great effect of vaccination and the unpredictability of environmental variations in nature, a stochastic Susceptible-Vaccinated-Infected-Susceptible (SVIS) epidemic model with standard incidence and vaccination strategies is the focus of the present study. By constructing a series of appropriate Lyapunov functions, the sufficient criterion is obtained for the existence and uniqueness of the ergodic stationary distribution of the model. In epidemiology, the existence of a stationary distribution indicates that the disease will be persistent in a long term. By taking the stochasticity into account, a quasi-endemic equilibrium related to the endemic equilibrium of the deterministic system is defined. By means of the method developed in solving the general three-dimensional Fokker-Planck equation, the exact expression of the probability density function of the stochastic model around the quasi-endemic equilibrium is derived, which is the key aim of the present paper. In statistical significance, the explicit density function can reflect all dynamical properties of an epidemic system. Next, a simple result of disease extinction is obtained. In addition, several numerical simulations and parameter analyses are performed to illustrate the theoretical results. Finally, the corresponding results and conclusions are discussed at the end of the paper.
考虑到疫苗接种的巨大作用以及自然界环境变化的不可预测性，本文聚焦于一个具有标准发病率和疫苗接种策略的随机易感-接种-感染-易感（SVIS）流行病模型。通过构造一系列合适的李雅普诺夫函数，得到了该模型遍历平稳分布存在唯一性的充分判据。在流行病学中，平稳分布的存在表明疾病将长期持续存在。考虑到随机性，定义了一个与确定性系统的地方病平衡点相关的准地方病平衡点。借助求解一般三维福克-普朗克方程所发展的方法，推导了随机模型在准地方病平衡点附近概率密度函数的精确表达式，这是本文的关键目标。在统计学意义上，显式密度函数可以反映流行病系统的所有动力学性质。接下来，得到了疾病灭绝的一个简单结果。此外，进行了一些数值模拟和参数分析以说明理论结果。最后，在本文结尾讨论了相应的结果和结论。
