We investigate inverse problems of finding unknown parameters ofmathematical models SEIR-HCD and SEIR-D of COVID-19 spread withadditional information about the number of detected cases, mortality,self-isolation coefficient, and tests performed for the city of Moscowand Novosibirsk region since 23.03.2020. In SEIR-HCD the population isdivided into seven groups, and in SEIR-D into five groups with similarcharacteristics and transition probabilities depending on the specificregion of interest. An identifiability analysis of SEIR-HCD is made toreveal the least sensitive unknown parameters as related to theadditional information. The parameters are corrected by minimizing someobjective functionals which is made by stochastic methods (simulatedannealing, differential evolution, and genetic algorithm). Prognosticscenarios for COVID-19 spread in Moscow and in Novosibirsk region aredeveloped, and the applicability of the models is analyzed.