We investigate inverse problems of finding unknown parameters of mathematical models SEIR-HCD and SEIR-D of COVID-19 spread with additional information about the number of detected cases, mortality, self-isolation coefficient, and tests performed for the city of Moscow and Novosibirsk region since 23.03.2020. In SEIR-HCD the population is divided into seven groups, and in SEIR-D into five groups with similar characteristics and transition probabilities depending on the specific region of interest. An identifiability analysis of SEIR-HCD is made to reveal the least sensitive unknown parameters as related to the additional information. The parameters are corrected by minimizing some objective functionals which is made by stochastic methods (simulated annealing, differential evolution, and genetic algorithm). Prognostic scenarios for COVID-19 spread in Moscow and in Novosibirsk region are developed, and the applicability of the models is analyzed.