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The paper formulates and solves the problem of identification of unknown parameters of mathematical models of the spread of COVID-19 coronavirus infection, based on SEIR type models, based on additional information about the number of detected cases, mortality, self-isolation coefficient and tests performed for the Moscow city and the Novosibirsk Region from 03.23.2020. Within the framework of the models used, the population is divided into seven (SEIR-HCD) and five (SEIR-D) groups with similar characteristics with transition probabilities between groups depending on a specific region. Identifiability analysis of the SEIR-HCD mathematical model was carried out, which revealed the least sensitive unknown parameters to additional measurements. The tasks of refining the parameters are reduced to minimizing the corresponding target functionals, which were solved using stochastic methods (simulating annealing, differential evolution, genetic algorithm, etc.). For a different amount of tested data, a prognostic scenario for the development of the disease in the city of Moscow and the Novosibirsk region was developed, the peak is predicted the development of the epidemic in Moscow with an error of 2 days and 174 detected cases, and an analysis of the applicability of the developed models was carried out.