The COVID-19 pandemics remains one of the largest global challenges. Necessity of effective systemic aids for the minimization of losses leads to the requirement of adequate models allowing to predict the impact of different factors on the spread of the disease. Agent-based simulation models provide a suitable solution with the possibility to accurately account for such factors as age structure of a population, characteristics of isolation, self-isolation strategies and testing strategies, presence of super-spreaders etc. In this paper we report on the results of simulating the spread of COVID-19 in several representative regions of Russia using an agent-based model with a general pool combined with the simulation of population testing strategy. The model accounts for the following key epidemiologic characteristics: population age distribution, reproducibility rate, distributions of infectivity period, a period of clinical manifestation, and age-dependent probability of critical disease. It is demonstrated that the daily epidemiologic curves can be predicted well for different territories with the same model parameters, except for the initial number of infected agents and region-dependent testing as well as isolation strategies, which are considered to be tuning parameters of the model. The developed approach can be further expanded to other regions of different countries, while the determined model parameters could be used as starting values for such simulations.