title: ÐÐ¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ ÑÐ¿Ð¸Ð´ÐµÐ¼Ð¸Ð¸ COVID-19
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Modeling of the COVID-19 epidemic
Creators: name: ÐÐ½Ð¾Ð´ÐµÐ»Ñ, Ð®Ð»Ð¸Ñ
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2021
Subjects: covid-19
ÐºÐ¾Ð¼Ð¿Ð°ÑÑÐ¼ÐµÐ½ÑÐ°Ð»ÑÐ½ÑÐµ Ð¼Ð¾Ð´ÐµÐ»Ð¸
Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ ÑÐ¿Ð¸Ð´ÐµÐ¼Ð¸Ð¹
compartmental models
epidemiology

description: Ð Ð°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ ÑÐ¾Ð³Ð¾, ÐºÐ°Ðº ÑÐ¿Ð¸Ð´ÐµÐ¼Ð¸Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð¾Ð¿Ð¸ÑÑÐ²Ð°ÑÑ ÑÐµÑÐµÐ½Ð¸Ðµ ÑÐ¿Ð¸Ð´ÐµÐ¼Ð¸Ð¸ COVID-19 Ð² ÑÐµÐ³Ð¸Ð¾Ð½Ð°Ñ Ð Ð¾ÑÑÐ¸Ð¸, Ð² ÑÐ°ÑÑÐ½Ð¾ÑÑÐ¸ Ð² ÐÐ¾ÑÐºÐ²Ðµ, Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³Ðµ Ð¸ ÐÐ°Ð»Ð¸Ð½Ð¸Ð½Ð³ÑÐ°Ð´ÑÐºÐ¾Ð¹ Ð¾Ð±Ð»Ð°ÑÑÐ¸. ÐÑÐ±Ð¾Ñ Ð»ÑÑÑÐµÐ¹ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð´Ð»Ñ ÐºÐ°Ð¶Ð´Ð¾Ð³Ð¾ Ð¸Ð· ÑÐµÐ³Ð¸Ð¾Ð½Ð¾Ð² Ð¿ÑÐ¾Ð²Ð¾Ð´Ð¸Ð»ÑÑ Ð¿ÑÑÐµÐ¼ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ Ð½Ð° Ð¸Ð¼ÐµÑÑÐ¸ÑÑÑ Ð¾ÑÐ¸ÑÐ¸Ð°Ð»ÑÐ½ÑÑ Ð´Ð°Ð½Ð½ÑÑ Ð¾ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÐºÐµ Ð·Ð°Ð±Ð¾Ð»ÐµÐ²Ð°Ð½Ð¸Ñ Ð¸ Ð´Ð°Ð»ÑÐ½ÐµÐ¹ÑÐµÐ¼ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ð¸ Ð¼ÐµÐ¶Ð´Ñ ÑÐ¾Ð±Ð¾Ð¹. Ð¡ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ Ð±ÑÐ»Ð¸ Ð¿ÑÐ¾ÑÐµÑÑÐ¸ÑÐ¾Ð²Ð°Ð½Ñ Ð³Ð¸Ð¿Ð¾ÑÐµÐ·Ñ Ð¾ ÑÐ°ÑÐ°ÐºÑÐµÑÐµ ÑÐ¿Ð¸Ð´ÐµÐ¼Ð¸Ð¸. Ð¡ ÑÑÐ¾Ð¹ ÑÐµÐ»ÑÑ Ð±ÑÐ»Ð¸ ÑÐ¾Ð·Ð´Ð°Ð½Ñ Ð¸ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ñ Ð¼Ð¾Ð´ÐµÐ»Ð¸, ÑÑÐ¸ÑÑÐ²Ð°ÑÑÐ¸Ðµ Ð¸Ð·Ð¼ÐµÐ½ÑÑÑÑÑÑÑ ÑÐºÐ¾ÑÐ¾ÑÑÑ ÑÐ°ÑÐ¿ÑÐ¾ÑÑÑÐ°Ð½ÐµÐ½Ð¸Ñ Ð¸Ð½ÑÐµÐºÑÐ¸Ð¸, Ð½Ð°Ð»Ð¸ÑÐ¸Ðµ Ð±ÐµÑÑÐ¸Ð¼Ð¿ÑÐ¾Ð¼Ð½Ð¾ Ð±Ð¾Ð»ÐµÑÑÐµÐ¹ ÑÐ°ÑÑÐ¸ Ð½Ð°ÑÐµÐ»ÐµÐ½Ð¸Ñ, ÐºÐ¾Ð½ÐµÑÐ½Ð¾ÑÑÑ Ð¸Ð¼Ð¼ÑÐ½Ð¸ÑÐµÑÐ° Ð¸ Ð½Ð°Ð»Ð¸ÑÐ¸Ðµ Â«ÑÑÐ¿ÐµÑÑÐ°ÑÐ¿ÑÐ¾ÑÑÑÐ°Ð½Ð¸ÑÐµÐ»ÐµÐ¹Â», ÑÐ°ÑÐ°ÐºÑÐµÑÐ¸Ð·ÑÑÑÐ¸ÑÑÑ Ð¿Ð¾Ð²ÑÑÐµÐ½Ð½Ð¾Ð¹ ÑÐºÐ¾ÑÐ¾ÑÑÑÑ Ð¿ÐµÑÐµÐ´Ð°ÑÐ¸ Ð¸Ð½ÑÐµÐºÑÐ¸Ð¸. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ Ð²ÑÑÐ²Ð»ÐµÐ½Ð¾ ÑÑÐ¸ ÑÑÐ¾Ð²Ð½Ñ Ð² Ð´Ð¸Ð½Ð°Ð¼Ð¸ÐºÐµ ÑÐºÐ¾ÑÐ¾ÑÑÐ¸ ÑÐ°ÑÐ¿ÑÐ¾ÑÑÑÐ°Ð½ÐµÐ½Ð¸Ñ Ð¸Ð½ÑÐµÐºÑÐ¸Ð¸ (R_t = 2.5, 0.91 Ð¸ 1.84 Ð´Ð»Ñ Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³Ð°), ÑÐ¾Ð¾ÑÐ²ÐµÑÑÑÐ²ÑÑÑÐ¸Ðµ ÑÑÐ°Ð¿Ð°Ð¼ Ð²Ð²ÐµÐ´ÐµÐ½Ð¸Ñ Ð¸ ÑÐ½ÑÑÐ¸Ñ ÐºÐ°ÑÐ°Ð½ÑÐ¸Ð½Ð½ÑÑ Ð¾Ð³ÑÐ°Ð½Ð¸ÑÐµÐ½Ð¸Ð¹. ÐÑÑÑÐ¸Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð´Ð»Ñ Ð²ÑÐµÑ ÑÐµÐ³Ð¸Ð¾Ð½Ð¾Ð² Ð²ÐºÐ»ÑÑÐ°ÑÑ Ð² ÑÐµÐ±Ñ Ð½ÐµÑÑÑÑÐ½Ð½ÑÑ Ð² Ð¾ÑÐ¸ÑÐ¸Ð°Ð»ÑÐ½Ð¾Ð¹ ÑÑÐ°ÑÐ¸ÑÑÐ¸ÐºÐµ Ð¿ÐµÑÐµÐ±Ð¾Ð»ÐµÐ²ÑÐ¸Ñ (92% Ð¾Ñ Ð²ÑÐµÑ Ð¿ÐµÑÐµÐ±Ð¾Ð»ÐµÐ²ÑÐ¸Ñ Ð² Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³Ðµ). ÐÐ¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ Ð¿ÑÐµÐ´ÑÐºÐ°Ð·ÑÐ²Ð°ÐµÑ Ð´Ð»Ð¸ÑÐµÐ»ÑÐ½Ð¾ÑÑÑ Ð¸Ð¼Ð¼ÑÐ½Ð¸ÑÐµÑÐ° Ð¾Ñ 4 Ð´Ð¾ 8 Ð¼ÐµÑÑÑÐµÐ². ÐÐ¸Ð¿Ð¾ÑÐµÐ·Ð° Ð¾ ÑÑÑÐµÑÑÐ²ÐµÐ½Ð½Ð¾Ð¼ Ð²Ð»Ð¸ÑÐ½Ð¸Ð¸ ÑÑÐ¿ÐµÑÑÐ°ÑÐ¿ÑÐ¾ÑÑÑÐ°Ð½Ð¸ÑÐµÐ»ÐµÐ¹ Ð½Ð° ÑÐ¾Ð´ ÑÐ¿Ð¸Ð´ÐµÐ¼Ð¸Ð¸ Ð½Ðµ Ð½Ð°ÑÐ»Ð° Ð¿Ð¾Ð´ÑÐ²ÐµÑÐ¶Ð´ÐµÐ½Ð¸Ð¹. ÐÐ¾Ð´ÐµÐ»Ð¸ Ð²Ð¾ Ð²ÑÐµÑ ÑÐµÐ³Ð¸Ð¾Ð½Ð°Ñ Ð¿ÑÐµÐ´ÑÐºÐ°Ð·ÑÐ²Ð°ÑÑ Ð½Ð°ÑÑÑÐ¿Ð°ÑÑÑÑ ÑÑÐµÑÑÑ Ð²Ð¾Ð»Ð½Ñ ÑÐ¿Ð¸Ð´ÐµÐ¼Ð¸Ð¸. ÐÐ° Ð¿ÑÐ¸Ð¼ÐµÑÐµ Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³Ð° Ð¸Ð·ÑÑÐµÐ½Ð¾ Ð²Ð»Ð¸ÑÐ½Ð¸Ðµ Ð²Ð°ÐºÑÐ¸Ð½Ð°ÑÐ¸Ð¸ Ð¸ ÑÑÑÐ°Ð½Ð¾Ð²Ð»ÐµÐ½Ð¾, ÑÑÐ¾ Ð¿Ð¾Ð»Ð½Ð¾Ðµ Ð¿Ð¾Ð´Ð°Ð²Ð»ÐµÐ½Ð¸Ðµ ÑÐ¿Ð¸Ð´ÐµÐ¼Ð¸Ð¸ ÑÑÐµÐ±ÑÐµÑ ÑÐ½Ð¸Ð¶ÐµÐ½Ð¸Ñ R_t Ð½Ð° 40%. ÐÐ¾Ð»ÑÑÐµÐ½Ð½ÑÐµ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÑ Ð´Ð°ÑÑ Ð¸Ð½ÑÐ¾ÑÐ¼Ð°ÑÐ¸Ñ Ð¾ ÑÐ°ÑÐ°ÐºÑÐµÑÐ½ÑÑ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ°Ñ ÑÐ¿Ð¸Ð´ÐµÐ¼Ð¸Ð¸ Ð¸ Ð¸Ñ Ð·Ð°Ð²Ð¸ÑÐ¸Ð¼Ð¾ÑÑÐ¸ Ð¾Ñ ÑÐµÐ³Ð¸Ð¾Ð½Ð¾Ð². Ð Ð°Ð·ÑÐ°Ð±Ð¾ÑÐ°Ð½Ð½ÑÐµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð¼Ð¾Ð³ÑÑ Ð±ÑÑÑ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ñ Ð´Ð»Ñ ÑÐµÑÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ Ð½Ð¾Ð²ÑÑ Ð³Ð¸Ð¿Ð¾ÑÐµÐ· Ð¸ Ð¼Ð¾Ð´Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¹, ÑÑÐ¸ÑÑÐ²Ð°ÑÑÐ¸Ñ Ð½Ð¾Ð²ÑÐµ Ð´Ð°Ð½Ð½ÑÐµ Ð¾Ð± ÑÐ¿Ð¸Ð´ÐµÐ¼Ð¸Ð¸.
description: The work is devoted to the study of how epidemiological models describe the course of the COVID-19 epidemic in the regions of Russia, in particular in Moscow, St. Petersburg and the Kaliningrad region. The selection of the best model for each of the regions was carried out by training on the available official data on the dynamics of the disease and further comparison with each other. Several hypotheses about the nature of the epidemic were tested using simulations. To this end, models were created and studied that take into account the changing rate of infection spread, the presence of an asymptomatically ill part of the population, the finiteness of immunity, and the presence of "super-spreaders" characterized by an increased rate of infection transmission. As a result, three levels in the dynamics of the rate of infection spread (R_t = 2.5, 0.91 and 1.84 for St.Petersburg) were identified, corresponding to the stages of introduction and removal of quarantine restrictions. The best models for all regions include those who were not included in the official statistics (92% of all those who were ill in St.Petersburg). The simulation predicts the duration of immunity from 4 to 8 months. The hypothesis of a significant influence of super-spreaders on the course of the epidemic has not been confirmed. Models in all regions predict the coming third wave of the epidemic. On the example of St. Petersburg, the impact of vaccination was studied and it was found that the complete suppression of the epidemic requires a 40% reduction in R_t. The results obtained provide information about the characteristic parameters of the epidemic and their dependence on the regions. The developed models can be used to test new hypotheses and modifications that take into account new data about the epidemic.