We consider disruption detection problems for statistical models with dependent observations given by Markov chains over a finite time interval in the Bayesian setting for the uniform prior distribution. To do this, using the methods of optimal stopping, a new optimal sequential detection procedure is constructed that minimizes the average delay time in the class of sequential procedures with the probabilities of false alarms bounded by some fixed value. The main difference between the proposed detection procedure and the usual ones is that it is based not on the posterior probabilities, but on Roberts statistics. This makes it possible to provide optimal detection in a non-asymptotic sense over a finite time interval without using additional unknown parameters, in contrast to the well-known Bayesian procedures based on a priori geometric distribution containing an unknown parameter. Then we apply the constructed procedures to the problem of early detection of the beginning of the spread of the epidemic. To this end we use two epidemic models: the binomial models proposed in [2] and the models based on the Gaussian approximation introduced in [9].