This paper presents a novel Bayesian inference algorithm for estimating unknown parameters in a stochastic susceptible–exposed–infected–recovered (SEIR) model, aiming to predict the extinction and persistence of infectious diseases. The posterior distribution is constructed using Gaussian processes, and sampling is generated via particle Markov Chain Monte Carlo. A key feature of our method is its gradient-based proposal mechanism, which enhances efficiency compared to traditional random-walk proposals. The algorithm can converge to the stationary distribution within a reasonable time frame, even when handling multiple parameters. Numerical simulations illustrate the effectiveness of our algorithm in parameter estimation. Additionally, several useful theoretical properties of the stochastic SEIR model are discussed. As an application example, the algorithm has been applied to estimate parameters from COVID-19 data in Iceland.