In this article, we consider the development and adaptation of a mathematical SIR model to the initial spread of COVID-19 infection in the Stavropol region of the Russian Federation. The model is developed according to regional peculiarities, such as daily data on the number of diseased, recovered, and dead individuals in the Stavropol region. The initial conditions of the model were obtained using statistical data. The model parameters (infection and recovery rates) were calculated using the least-squares method. The numerical solution of the system of differential equations of the SIR model was found by the Runge-Kutta method of the 4th order and presented in the form of graphs. Experiments for different values of the infection and recovery coefficients have been carried out, and the corresponding numerical solutions of the system of equations of the SIR model have been found, showing the existence of different scenarios of epidemic development depending on the values of the coefficients. For each pair of infection and recovery coefficients, there was calculated the value of the basic reproductive number (reproduction index). The dependence of infection spread dynamics on the value of the basic reproductive number is clearly illustrated.