Din Anwarud, Li Yongjin
Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275 P. R. China.
Eur Phys J Plus. 2022;137(5):558. doi: 10.1140/epjp/s13360-022-02748-x. Epub 2022 May 6.
This work present a detailed analysis of a stochastic delayed model which governs the transmission mechanism of the Hepatitis B virus (HBV) while considering the white noises and the effect of vaccinations. It is assumed that the perturbations are nonlinear and an individual may lose his/her immunity after the vaccination, that is, the vaccination can produce temporal immunity. Based on the characteristics of the disease and the underlying assumptions, we formulated the associated deterministic model for which the threshold parameter is calculated. The model was further extended to a stochastic model and it is well-justified that the model is both mathematically and biologically feasible by showing that the model solution exists globally, bounded stochastically and is positive. By utilizing the concepts of stochastic theory and by constructing appropriate Lyapunov functions, we developed the theory for the extinction and persistence of the disease. Further, it is shown that the model is ergodic and has a unique stationary distribution. The stochastic bifurcation theory is utilized and a detailed bifurcation analysis of the model is presented. By using the standard curve fitting tools, we fitted the model against the available HBV data in Pakistan from March 2018 to February 2019 and accordingly the parameters of the model were estimated. These estimated values were used in simulating the model, theoretical findings of the study are validated through simulations and predictions were drawn. Simulations suggest that for a complete understanding of HBV dynamics, one must include time delay into such studies, and improvements in every vaccination program are unavoidable.
这项工作对一个随机延迟模型进行了详细分析，该模型在考虑白噪声和疫苗接种效果的情况下，控制乙型肝炎病毒（HBV）的传播机制。假设扰动是非线性的，并且个体在接种疫苗后可能会失去免疫力，即疫苗接种可以产生暂时免疫力。基于该疾病的特征和基本假设，我们制定了相关的确定性模型，并计算了其阈值参数。该模型进一步扩展为随机模型，通过证明模型解全局存在、随机有界且为正，充分证明了该模型在数学和生物学上都是可行的。通过运用随机理论的概念并构建适当的李雅普诺夫函数，我们建立了该疾病灭绝和持续存在的理论。此外，结果表明该模型是遍历的且具有唯一的平稳分布。利用随机分岔理论，对该模型进行了详细的分岔分析。通过使用标准曲线拟合工具，我们将该模型与2018年3月至2019年2月巴基斯坦现有的HBV数据进行拟合，从而估计了模型的参数。这些估计值被用于模拟模型，通过模拟验证了该研究的理论结果并进行了预测。模拟结果表明，为了全面理解HBV动态，必须在这类研究中纳入时间延迟，并且每个疫苗接种计划的改进都是不可避免的。
