Summary: In this study, we consider disorder detection problems for statistical models with dependent observations defined by Markov chains in a Bayesian setting for a uniform prior distribution when the number of observations is limited by some known quantity. To achieve this, a new optimal sequential detection procedure is constructed using optimal stopping methods. This procedure minimizes the average delay time in the class of sequential procedures with false alarm probabilities that do not exceed a fixed value. The main difference between the proposed detection procedure and the conventional ones is that it is based not on the posterior probabilities but on the weighted Shiryaev-Roberts statistic. This allows for optimal detection in a nonasymptotic sense over any bounded time interval. Thereafter, we applied the constructed procedures to the early detection problem for the beginning of the epidemic spread. We used two epidemic models: the binomial model proposed by Baron, Choudhary, and Yu (2013) and the model based on the Gaussian approximation introduced by Pergamenchtchikov, Tartakovsky, and Spivak (2022). The theoretical results were confirmed through numerical simulations using the Monte Carlo method.