The stochastic susceptible-infected-vaccinated (SIV) epidemic model includes a nonlinear term, making it difficult to obtain analytical solutions. Thus, numerical approximation schemes are an important tool for predicting the dynamics of infectious diseases and establishing optimal control strategies. However, the convergence rate of the existing numerical methods [e.g., Euler-Maruyama (EM) and truncated EM scheme] is only 1/2 order of the time step Δt. This article describes the construction of a logarithmic truncated EM scheme that achieves order-1 convergence and ensures positive numerical solutions of the stochastic SIV epidemic model. The existence of an invariant measure is proved for the stochastic SIV epidemic model with Markov switching. In addition, relaxed controls for the stochastic SIV epidemic model are investigated by using the Markov chain approximation method. It is demonstrated that the approximation schemes converge to the optimal strategy as the mesh size goes to zero. Finally, the results of numerical examples are presented to illustrate the theoretical results derived in this article.

中文翻译：

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具有马尔可夫切换的随机易感感染者疫苗接种流行病模型的保正数值方法和宽松控制。


随机易感者感染者接种疫苗（SIV）流行病模型包含非线性项，因此很难获得解析解。因此，数值逼近方案是预测传染病动态和建立最优控制策略的重要工具。然而，现有数值方法[例如Euler-Maruyama (EM)和截断EM方案]的收敛速度仅为时间步长Δt的1/2量级。本文描述了对数截断 EM 方案的构建，该方案实现了 1 阶收敛并确保随机 SIV 流行病模型的正数值解。证明了马尔可夫切换的随机SIV流行病模型存在不变测度。此外，利用马尔可夫链近似方法研究了随机 SIV 流行病模型的宽松控制。结果表明，当网格尺寸变为零时，近似方案收敛到最优策略。最后，给出了数值算例的结果来说明本文得出的理论结果。