一类具有疫苗接种的随机SVIR传染病模型的平稳分布

The Stationary Distribution of a Stochastic SVIR Infectious Disease Model with Vaccination

摘要: 在人类与传染病的长期斗争中，疫苗接种已被证实是最成功的防控手段之一。疫苗通过激活机体的特异性免疫反应，能够形成针对特定病原体的防护机制。大量实际案例表明，疫苗不仅能显著降低疾病传播风险，更是最终根除某些传染病的重要途径。当前，这一技术已成为传染病防治研究和实践的核心组成部分。因此，本文研究了一类具有疫苗接种的随机SVIR传染病模型的平稳分布。首先，通过构建合适的Lyapunov函数证明了模型正解的存在唯一性。然后，建立了参数${\Re }_0^S$ ，证明了当${\Re }_0^S>1$ 时，模型的解在${ℝ}_{+}^4$ 上存在唯一的平稳分布。最后，对本文主要研究内容进行了总结，发现随机扰动会影响${\Re }_0^S$ ，并且${\Re }_0^S$ 小于等于确定型SVIR模型的基本再生数${\Re }_0^C$ 。

Abstract: Vaccination has been proven to be one of the most successful prevention and control measures in the long-term struggle between humans and infectious diseases. Vaccines can form protective mechanisms against specific pathogens by activating the body’s specific immune response. Numerous practical cases have shown that vaccines not only significantly reduce the risk of disease transmission, but are also an important way to ultimately eradicate certain infectious diseases. Currently, this technology has become a core component of research and practice in infectious disease prevention and control. Therefore, this paper investigates the stationary distribution of a stochastic SVIR epidemic model incorporating vaccination. Firstly, we prove the existence and uniqueness of the positive solution of the model by constructing Lyapunov function. Then, we establish the parameter ${\Re }_0^S$ and proven that when ${\Re }_0^S>1$ , the solution of the model has a unique stationary distribution in${ℝ}_{+}^4$ . Finally, we summarize the main results of this article and found that ${\Re }_0^S$ is affected by stochastic perturbations. Othermore, ${\Re }_0^S$ is less than or equal to the basic reproduction number ${\Re }_0^C$ of the deterministic SVIR model.