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 . 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.
Keywords: Markov chain; invariant measure; logarithmic truncated EM method; positivity; stochastic SIV epidemic model.