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Математическое моделирование эпидемии COVID-19: методы, сценарии и практическая применимость

https://dx.doi.org/10.51350/zdravkg2026.1.3.18.148.153
Математическое моделирование эпидемии COVID-19: методы, сценарии и практическая применимость
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Аннотация Об авторах Список литературы

Аннотация

Введение. Пандемия COVID-19 актуализировала необходимость оперативного принятия управленческих решений в условиях высокой неопределенности. Одним из ключевых инструментов стало математи- ческое моделирование, позволяющее интерпретировать текущую эпидемиологическую ситуацию и прогнозировать возможные сценарии ее развития. Цель исследования. Систематизировать существующие направления математического моделирования COVID-19 и оценить их релевантность для регионов с ограниченными ресурсами. Материалы и методы. В статье представлен систематизированный литературный обзор современных подходов к математическому моделированию эпидемии COVID-19. Рассматриваются классические компарт ментные модели (SIR, SEIR и модификации), временные модели (ARIMA, SSA), гибридные модели с элементами искусственного интеллекта, а также агент-ориентированные и стохастические модели. Результаты и обсуждение. Проведен анализ их применимости в различных эпидемиологических контекстах, включая страны с ограниченными ресурсами, такие как Кыргызская Республика. Отдельное вни- мание уделено использованию модели SEIR в рамках платформы CoMo Consortium. Описаны основные преимущества и ограничения различных моделей, приведены рекомендации по адаптации под локальные условия. Заключение. В контексте пандемии COVID-19 особенно важным стало сочетание моделирования с эпиднадзором, оперативной статистикой и оценкой риска, что позволило оперативно адаптировать стратегии вмешательств к быстро меняющейся ситуации.

Об авторах

Дооронбекова Aйжан Жакыпбековна, соискатель, научный сотрудник, Центра анализа, управления рисками общественному здравоохранению и профилактики заболеваний Национального института общественного здоровья МЗ, Бишкек, Кыргызская Республика

Dooronbekova Aizhan Zhakypbekovna, applicant, research fellow of the Center for Analysis and Management of Public Health Risks National Institute of Public Health of the Ministry of Health, Bishkek, Kyrgyz Republic

Дооронбекова Айжан Жакыпбековна, талапкер, Саламаттыкты сактоо министрлигинин Коомдук саламаттыкты сактоо боюнча улуттук институтунун Коомдук саламаттыкты сактоо тобокелдиктерин талдоо жана башкаруу борборунун илимий кызматкери, Бишкек, Кыргыз Республикасы

Список литературы

1. Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc A. 1927;115(772):700-721. https://doi.org/10.1098/rspa.1927.0118

2. Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press; 1991.

3. Hethcote HW. The mathematics of infectious diseases. SIAM Rev. 2000;42(4):599-653. https://doi.org/10.1137/S0036144500371907

4. Diekmann O, Heesterbeek JAP, Roberts MG. The construction of next-generation matrices for compartmental epidemic models. J R Soc Interface. 2010;7(47):873-885. https://doi.org/10.1098/rsif.2009.0386

5. Moore S, Hill EM, Tildesley MJ, Dyson L, Keeling MJ. Vaccination and non-pharmaceutical interventions for COVID-19: a mathematical modelling study. Lancet Infect Dis. 2021;21(6):793-802. https://doi.org/10.1016/S1473-3099(21)00143-2

6. Prem K, Liu Y, Russell TW, et al. The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China:a modelling study. Lancet Public Health. 2020;5(5):e261-e270.https://doi.org/10.1016/S2468-2667(20)30073-6

7. Aguas R, Hupert N, Shretta R, et al. COVID-19 pandemic modelling in context: uniting people and technology across nations. BMJ Preprint. 2020. https://www.researchgate.net/publication/342747645

8. CoMo Consortium. COVID-19 Modelling for Resource-Constrained Settings. Oxford: University of Oxford; 2021. https://www.co momodel.net/

9. World Health Organization. Considerations for implementing and adjusting public health and social measures. Geneva: WHO; 2021. https://apps.who.int/iris/handle/10665/337575

10. World Health Organization. COVID-19 Strategic Preparedness and Response Plan. Geneva: WHO; 2021. https://www.who.int/pub lications/i/item/WHO-WHE-2021.02 11. UNICEF. COVID-19 Vaccine Delivery Partnership: Central Asia Case Studies. Final Report. New York: UNICEF; May 2022. https://www.unicef.org/kyrgyzstan/media/8711/file

12. Hunter E, Mac Namee B, Kelleher JD. A comparison of agent-based models and equation based models for infectious disease epi demiology. Complex Syst. 2018;27(6):519-559.

13. Wangping J, Ke H, Yang S, et al. Extended SIR prediction of the epidemics trend of COVID-19 in Italy and compared with Hunan, China. Front Med. 2020;7:169. https://doi.org/10.3389/fmed.2020.00169

14. Cooper I, Mondal A, Antonopoulos CG. A SIR model assumption for the spread of COVID-19 in different communities. Chaos Solitons Fractals. 2020;139:110057. https://doi.org/10.1016/j.chaos.2020.110057

15. Ndaïrou F, Area I, Nieto JJ, Torres DFM. Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos Solitons Fractals. 2020;135:109846. https://doi.org/10.1016/j.chaos.2020.109846

16. Kissler SM, Tedijanto C, Goldstein E, Grad YH, Lipsitch M. Projecting the transmission dynamics of SARS-CoV-2 through the postpandemic period. Science. 2020;368(6493):860-868. https://doi.org/10.1126/science.abb5793

17. Giordano G, Blanchini F, Bruno R, et al. Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nat Med. 2020;26(6):855-860. https://doi.org/10.1038/s41591-020-0883-7

18. Radev ST, Graw F, West R, Rieger MO, Lasser J. OutbreakFlow: model-based Bayesian inference of disease outbreaks. PLoS Comput Biol. 2021;17(5):e1008854. https://doi.org/10.1371/journal.pcbi.1008854

19. Ceylan Z. Estimation of COVID-19 prevalence in Italy, Spain, and France. Sci Total Environ. 2020;729:138817. https://doi.org/10.1016/j.scitotenv.2020.138817

20. Ajelli M, Gonçalves B, Balcan D, et al. Comparing large-scale computational approaches to epidemic modeling: agent-based versus structured metapopulation models. BMC Infect Dis. 2010;10:190. https://doi.org/10.1186/1471-2334-10-190

21. Zhang X, Ma R, Wang L. Predicting turning point, duration and attack rate of COVID-19 outbreaks in major Western countries. Chaos Solitons Fractals. 2020;135:109829. https://doi.org/10.1016/j.chaos.2020.109829

22. Alzahrani SI, Aljamaan IA, Al-Fakih EA. Forecasting COVID-19 pandemic in Saudi Arabia using ARIMA model. J Infect Public Health. 2020;13(7):920-925. https://doi.org/10.1016/j.jiph.2020.06.001

23. Arifi F, Moldokmatova A, Aitmatov K, et al. Application of the CoMo model in the Kyrgyz Republic: scenario-based planning for COVID-19 response. CoMo Consortium Report. Oxford: University of Oxford; 2021.

24. Elsawah S, Spencer T, Ryan MJ, et al. A review of computational model robustness evaluation strategies. Int J Environ Res Public Health. 2020;17(18):6280. https://doi.org/10.3390/ijerph17186280

25. Cramer EY, Huang Y, Wang Y, et al. The United States COVID-19 Forecast Hub dataset. J Am Med Inform Assoc. 2022;29(8): 1490-1499. https://doi.org/10.1093/jamia/ocac066

26. Taylor SJ, Letham B. Forecasting at scale. Am Stat. 2018;72(1):37-45. https://doi.org/10.1080/00031305.2017.1380080

27. Petropoulos F, Makridakis S. Forecasting the novel coronavirus COVID-19. PLoS One. 2020;15(3):e0231236. https://doi.org/10.1371/journal.pone.0231236

28. Hassani H, Silva ES, Gupta R. Forecasting the COVID-19 pandemic with singular spectrum analysis. J Data Sci. 2020;18(3):531- 544. https://doi.org/10.6339/JDS.202007_18(3).00003

29. Rustam F, Reshi AA, Mehmood A, et al. COVID-19 future forecasting using supervised machine learning models. IEEE Access. 2020;8:101489-101499. https://doi.org/10.1109/ACCESS.2020.2997311

30. Aboluwarin OA, Falade OO. COVID-19 forecasting using deep learning models. Sci Afr. 2022;16:e01163. https://doi.org/10.101 6/j.sciaf.2022.e01163

31. Singh RK, Rani M, Bhagavathula AS, et al. Prediction of the COVID-19 pandemic for the top 15 affected countries: advanced autoregressive integrated moving average (ARIMA) model. JMIR Public Health Surveill. 2020;6(2):e19115. https://doi.org/ 10.2196/19115

32. Hellewell J, Abbott S, Gimma A, et al. Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts. Lancet Glob Health. 2020;8(4):e488-e496. https://doi.org/10.1016/S2214-109X(20)30074-7

33. Ferguson NM, Laydon D, Nedjati-Gilani G, et al. Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mor tality and healthcare demand. Imperial College COVID-19 Response Team. 2020. https://doi.org/10.25561/77482

34. Dooronbekova AZh. Epidemiological analysis of COVID-19 spread in the Kyrgyz Republic using mathematical modeling and forecasting methods [dissertation]. Bishkek; 2025. 120 p.

1. Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc A. 1927;115(772):700-721. https://doi.org/10.1098/rspa.1927.0118

2. Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press; 1991.

3. Hethcote HW. The mathematics of infectious diseases. SIAM Rev. 2000;42(4):599-653. https://doi.org/10.1137/S0036144500371907

4. Diekmann O, Heesterbeek JAP, Roberts MG. The construction of next-generation matrices for compartmental epidemic models. J R Soc Interface. 2010;7(47):873-885. https://doi.org/10.1098/rsif.2009.0386

5. Moore S, Hill EM, Tildesley MJ, Dyson L, Keeling MJ. Vaccination and non-pharmaceutical interventions for COVID-19: a mathematical modelling study. Lancet Infect Dis. 2021;21(6):793-802. https://doi.org/10.1016/S1473-3099(21)00143-2

6. Prem K, Liu Y, Russell TW, et al. The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China:a modelling study. Lancet Public Health. 2020;5(5):e261-e270.https://doi.org/10.1016/S2468-2667(20)30073-6

7. Aguas R, Hupert N, Shretta R, et al. COVID-19 pandemic modelling in context: uniting people and technology across nations. BMJ Preprint. 2020. https://www.researchgate.net/publication/342747645

8. CoMo Consortium. COVID-19 Modelling for Resource-Constrained Settings. Oxford: University of Oxford; 2021. https://www.co momodel.net/

9. World Health Organization. Considerations for implementing and adjusting public health and social measures. Geneva: WHO; 2021. https://apps.who.int/iris/handle/10665/337575

10. World Health Organization. COVID-19 Strategic Preparedness and Response Plan. Geneva: WHO; 2021. https://www.who.int/pub lications/i/item/WHO-WHE-2021.02 11. UNICEF. COVID-19 Vaccine Delivery Partnership: Central Asia Case Studies. Final Report. New York: UNICEF; May 2022. https://www.unicef.org/kyrgyzstan/media/8711/file

12. Hunter E, Mac Namee B, Kelleher JD. A comparison of agent-based models and equation based models for infectious disease epi demiology. Complex Syst. 2018;27(6):519-559.

13. Wangping J, Ke H, Yang S, et al. Extended SIR prediction of the epidemics trend of COVID-19 in Italy and compared with Hunan, China. Front Med. 2020;7:169. https://doi.org/10.3389/fmed.2020.00169

14. Cooper I, Mondal A, Antonopoulos CG. A SIR model assumption for the spread of COVID-19 in different communities. Chaos Solitons Fractals. 2020;139:110057. https://doi.org/10.1016/j.chaos.2020.110057

15. Ndaïrou F, Area I, Nieto JJ, Torres DFM. Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos Solitons Fractals. 2020;135:109846. https://doi.org/10.1016/j.chaos.2020.109846

16. Kissler SM, Tedijanto C, Goldstein E, Grad YH, Lipsitch M. Projecting the transmission dynamics of SARS-CoV-2 through the postpandemic period. Science. 2020;368(6493):860-868. https://doi.org/10.1126/science.abb5793

17. Giordano G, Blanchini F, Bruno R, et al. Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nat Med. 2020;26(6):855-860. https://doi.org/10.1038/s41591-020-0883-7

18. Radev ST, Graw F, West R, Rieger MO, Lasser J. OutbreakFlow: model-based Bayesian inference of disease outbreaks. PLoS Comput Biol. 2021;17(5):e1008854. https://doi.org/10.1371/journal.pcbi.1008854

19. Ceylan Z. Estimation of COVID-19 prevalence in Italy, Spain, and France. Sci Total Environ. 2020;729:138817. https://doi.org/10.1016/j.scitotenv.2020.138817

20. Ajelli M, Gonçalves B, Balcan D, et al. Comparing large-scale computational approaches to epidemic modeling: agent-based versus structured metapopulation models. BMC Infect Dis. 2010;10:190. https://doi.org/10.1186/1471-2334-10-190

21. Zhang X, Ma R, Wang L. Predicting turning point, duration and attack rate of COVID-19 outbreaks in major Western countries. Chaos Solitons Fractals. 2020;135:109829. https://doi.org/10.1016/j.chaos.2020.109829

22. Alzahrani SI, Aljamaan IA, Al-Fakih EA. Forecasting COVID-19 pandemic in Saudi Arabia using ARIMA model. J Infect Public Health. 2020;13(7):920-925. https://doi.org/10.1016/j.jiph.2020.06.001

23. Arifi F, Moldokmatova A, Aitmatov K, et al. Application of the CoMo model in the Kyrgyz Republic: scenario-based planning for COVID-19 response. CoMo Consortium Report. Oxford: University of Oxford; 2021.

24. Elsawah S, Spencer T, Ryan MJ, et al. A review of computational model robustness evaluation strategies. Int J Environ Res Public Health. 2020;17(18):6280. https://doi.org/10.3390/ijerph17186280

25. Cramer EY, Huang Y, Wang Y, et al. The United States COVID-19 Forecast Hub dataset. J Am Med Inform Assoc. 2022;29(8): 1490-1499. https://doi.org/10.1093/jamia/ocac066

26. Taylor SJ, Letham B. Forecasting at scale. Am Stat. 2018;72(1):37-45. https://doi.org/10.1080/00031305.2017.1380080

27. Petropoulos F, Makridakis S. Forecasting the novel coronavirus COVID-19. PLoS One. 2020;15(3):e0231236. https://doi.org/10.1371/journal.pone.0231236

28. Hassani H, Silva ES, Gupta R. Forecasting the COVID-19 pandemic with singular spectrum analysis. J Data Sci. 2020;18(3):531- 544. https://doi.org/10.6339/JDS.202007_18(3).00003

29. Rustam F, Reshi AA, Mehmood A, et al. COVID-19 future forecasting using supervised machine learning models. IEEE Access. 2020;8:101489-101499. https://doi.org/10.1109/ACCESS.2020.2997311

30. Aboluwarin OA, Falade OO. COVID-19 forecasting using deep learning models. Sci Afr. 2022;16:e01163. https://doi.org/10.101 6/j.sciaf.2022.e01163

31. Singh RK, Rani M, Bhagavathula AS, et al. Prediction of the COVID-19 pandemic for the top 15 affected countries: advanced autoregressive integrated moving average (ARIMA) model. JMIR Public Health Surveill. 2020;6(2):e19115. https://doi.org/ 10.2196/19115

32. Hellewell J, Abbott S, Gimma A, et al. Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts. Lancet Glob Health. 2020;8(4):e488-e496. https://doi.org/10.1016/S2214-109X(20)30074-7

33. Ferguson NM, Laydon D, Nedjati-Gilani G, et al. Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mor tality and healthcare demand. Imperial College COVID-19 Response Team. 2020. https://doi.org/10.25561/77482

34. Dooronbekova AZh. Epidemiological analysis of COVID-19 spread in the Kyrgyz Republic using mathematical modeling and forecasting methods [dissertation]. Bishkek; 2025. 120 p.

1. Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc A. 1927;115(772):700-721. https://doi.org/10.1098/rspa.1927.0118

2. Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press; 1991.

3. Hethcote HW. The mathematics of infectious diseases. SIAM Rev. 2000;42(4):599-653. https://doi.org/10.1137/S0036144500371907

4. Diekmann O, Heesterbeek JAP, Roberts MG. The construction of next-generation matrices for compartmental epidemic models. J R Soc Interface. 2010;7(47):873-885. https://doi.org/10.1098/rsif.2009.0386

5. Moore S, Hill EM, Tildesley MJ, Dyson L, Keeling MJ. Vaccination and non-pharmaceutical interventions for COVID-19: a mathematical modelling study. Lancet Infect Dis. 2021;21(6):793-802. https://doi.org/10.1016/S1473-3099(21)00143-2

6. Prem K, Liu Y, Russell TW, et al. The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China:a modelling study. Lancet Public Health. 2020;5(5):e261-e270.https://doi.org/10.1016/S2468-2667(20)30073-6

7. Aguas R, Hupert N, Shretta R, et al. COVID-19 pandemic modelling in context: uniting people and technology across nations. BMJ Preprint. 2020. https://www.researchgate.net/publication/342747645

8. CoMo Consortium. COVID-19 Modelling for Resource-Constrained Settings. Oxford: University of Oxford; 2021. https://www.co momodel.net/

9. World Health Organization. Considerations for implementing and adjusting public health and social measures. Geneva: WHO; 2021. https://apps.who.int/iris/handle/10665/337575

10. World Health Organization. COVID-19 Strategic Preparedness and Response Plan. Geneva: WHO; 2021. https://www.who.int/pub lications/i/item/WHO-WHE-2021.02 11. UNICEF. COVID-19 Vaccine Delivery Partnership: Central Asia Case Studies. Final Report. New York: UNICEF; May 2022. https://www.unicef.org/kyrgyzstan/media/8711/file

12. Hunter E, Mac Namee B, Kelleher JD. A comparison of agent-based models and equation based models for infectious disease epi demiology. Complex Syst. 2018;27(6):519-559.

13. Wangping J, Ke H, Yang S, et al. Extended SIR prediction of the epidemics trend of COVID-19 in Italy and compared with Hunan, China. Front Med. 2020;7:169. https://doi.org/10.3389/fmed.2020.00169

14. Cooper I, Mondal A, Antonopoulos CG. A SIR model assumption for the spread of COVID-19 in different communities. Chaos Solitons Fractals. 2020;139:110057. https://doi.org/10.1016/j.chaos.2020.110057

15. Ndaïrou F, Area I, Nieto JJ, Torres DFM. Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos Solitons Fractals. 2020;135:109846. https://doi.org/10.1016/j.chaos.2020.109846

16. Kissler SM, Tedijanto C, Goldstein E, Grad YH, Lipsitch M. Projecting the transmission dynamics of SARS-CoV-2 through the postpandemic period. Science. 2020;368(6493):860-868. https://doi.org/10.1126/science.abb5793

17. Giordano G, Blanchini F, Bruno R, et al. Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nat Med. 2020;26(6):855-860. https://doi.org/10.1038/s41591-020-0883-7

18. Radev ST, Graw F, West R, Rieger MO, Lasser J. OutbreakFlow: model-based Bayesian inference of disease outbreaks. PLoS Comput Biol. 2021;17(5):e1008854. https://doi.org/10.1371/journal.pcbi.1008854

19. Ceylan Z. Estimation of COVID-19 prevalence in Italy, Spain, and France. Sci Total Environ. 2020;729:138817. https://doi.org/10.1016/j.scitotenv.2020.138817

20. Ajelli M, Gonçalves B, Balcan D, et al. Comparing large-scale computational approaches to epidemic modeling: agent-based versus structured metapopulation models. BMC Infect Dis. 2010;10:190. https://doi.org/10.1186/1471-2334-10-190

21. Zhang X, Ma R, Wang L. Predicting turning point, duration and attack rate of COVID-19 outbreaks in major Western countries. Chaos Solitons Fractals. 2020;135:109829. https://doi.org/10.1016/j.chaos.2020.109829

22. Alzahrani SI, Aljamaan IA, Al-Fakih EA. Forecasting COVID-19 pandemic in Saudi Arabia using ARIMA model. J Infect Public Health. 2020;13(7):920-925. https://doi.org/10.1016/j.jiph.2020.06.001

23. Arifi F, Moldokmatova A, Aitmatov K, et al. Application of the CoMo model in the Kyrgyz Republic: scenario-based planning for COVID-19 response. CoMo Consortium Report. Oxford: University of Oxford; 2021.

24. Elsawah S, Spencer T, Ryan MJ, et al. A review of computational model robustness evaluation strategies. Int J Environ Res Public Health. 2020;17(18):6280. https://doi.org/10.3390/ijerph17186280

25. Cramer EY, Huang Y, Wang Y, et al. The United States COVID-19 Forecast Hub dataset. J Am Med Inform Assoc. 2022;29(8): 1490-1499. https://doi.org/10.1093/jamia/ocac066

26. Taylor SJ, Letham B. Forecasting at scale. Am Stat. 2018;72(1):37-45. https://doi.org/10.1080/00031305.2017.1380080

27. Petropoulos F, Makridakis S. Forecasting the novel coronavirus COVID-19. PLoS One. 2020;15(3):e0231236. https://doi.org/10.1371/journal.pone.0231236

28. Hassani H, Silva ES, Gupta R. Forecasting the COVID-19 pandemic with singular spectrum analysis. J Data Sci. 2020;18(3):531- 544. https://doi.org/10.6339/JDS.202007_18(3).00003

29. Rustam F, Reshi AA, Mehmood A, et al. COVID-19 future forecasting using supervised machine learning models. IEEE Access. 2020;8:101489-101499. https://doi.org/10.1109/ACCESS.2020.2997311

30. Aboluwarin OA, Falade OO. COVID-19 forecasting using deep learning models. Sci Afr. 2022;16:e01163. https://doi.org/10.101 6/j.sciaf.2022.e01163

31. Singh RK, Rani M, Bhagavathula AS, et al. Prediction of the COVID-19 pandemic for the top 15 affected countries: advanced autoregressive integrated moving average (ARIMA) model. JMIR Public Health Surveill. 2020;6(2):e19115. https://doi.org/ 10.2196/19115

32. Hellewell J, Abbott S, Gimma A, et al. Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts. Lancet Glob Health. 2020;8(4):e488-e496. https://doi.org/10.1016/S2214-109X(20)30074-7

33. Ferguson NM, Laydon D, Nedjati-Gilani G, et al. Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mor tality and healthcare demand. Imperial College COVID-19 Response Team. 2020. https://doi.org/10.25561/77482

34. Dooronbekova AZh. Epidemiological analysis of COVID-19 spread in the Kyrgyz Republic using mathematical modeling and forecasting methods [dissertation]. Bishkek; 2025. 120 p.

Ключевые слова
Сценарий , COVID-19 , Кыргызстан , ARIMA , Математическое моделирование , SEIR , Прогноз
Для цитирования

Дооронбекова А.Ж. Математическое моделирование эпидемии COVID-19: методы, сценарии и практическая применимость. Научнопрактический журнал «Здравоохранение Кыргызстана» 2026, № 1, с. 148-153.  https://dx.doi.org/10.51350/zdravkg2026.1.3.18.148.153
For citation

Dooronbekova A. Zh. Mathematical Modeling of the COVID -19 Epidemic: Methods, Scenarios, and Practical Applica bility. Scientific practical journal “Health care of Kyrgyzstan” 2026, No.1, p. 148-153.  https://dx.doi.org/10.51350/zdravkg2026.1.3.18.148.153
Цитата үчүн

Дооронбекова А.Ж. COVID-19 эпидемиясынын математикалык моделдөө ыкмалары: методдор, сценарийлер жана практикалык колдонулушу. Кыргызстандын саламаттык сактоо илимий-практикалык журналы 2026, № 1, б. 148-153. https://dx.doi.org/10.51350/zdravkg2026.1.3.18.148.153
Авторы Дооронбекова А.Ж.
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Ключевые слова Сценарий, COVID-19, Кыргызстан, ARIMA, Математическое моделирование, SEIR, Прогноз
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Об авторах

Дооронбекова Aйжан Жакыпбековна, соискатель, научный сотрудник, Центра анализа, управления рисками общественному здравоохранению и профилактики заболеваний Национального института общественного здоровья МЗ, Бишкек, Кыргызская Республика

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Список литературы

1. Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc A. 1927;115(772):700-721. https://doi.org/10.1098/rspa.1927.0118

2. Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press; 1991.

3. Hethcote HW. The mathematics of infectious diseases. SIAM Rev. 2000;42(4):599-653. https://doi.org/10.1137/S0036144500371907

4. Diekmann O, Heesterbeek JAP, Roberts MG. The construction of next-generation matrices for compartmental epidemic models. J R Soc Interface. 2010;7(47):873-885. https://doi.org/10.1098/rsif.2009.0386

5. Moore S, Hill EM, Tildesley MJ, Dyson L, Keeling MJ. Vaccination and non-pharmaceutical interventions for COVID-19: a mathematical modelling study. Lancet Infect Dis. 2021;21(6):793-802. https://doi.org/10.1016/S1473-3099(21)00143-2

6. Prem K, Liu Y, Russell TW, et al. The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China:a modelling study. Lancet Public Health. 2020;5(5):e261-e270.https://doi.org/10.1016/S2468-2667(20)30073-6

7. Aguas R, Hupert N, Shretta R, et al. COVID-19 pandemic modelling in context: uniting people and technology across nations. BMJ Preprint. 2020. https://www.researchgate.net/publication/342747645

8. CoMo Consortium. COVID-19 Modelling for Resource-Constrained Settings. Oxford: University of Oxford; 2021. https://www.co momodel.net/

9. World Health Organization. Considerations for implementing and adjusting public health and social measures. Geneva: WHO; 2021. https://apps.who.int/iris/handle/10665/337575

10. World Health Organization. COVID-19 Strategic Preparedness and Response Plan. Geneva: WHO; 2021. https://www.who.int/pub lications/i/item/WHO-WHE-2021.02 11. UNICEF. COVID-19 Vaccine Delivery Partnership: Central Asia Case Studies. Final Report. New York: UNICEF; May 2022. https://www.unicef.org/kyrgyzstan/media/8711/file

12. Hunter E, Mac Namee B, Kelleher JD. A comparison of agent-based models and equation based models for infectious disease epi demiology. Complex Syst. 2018;27(6):519-559.

13. Wangping J, Ke H, Yang S, et al. Extended SIR prediction of the epidemics trend of COVID-19 in Italy and compared with Hunan, China. Front Med. 2020;7:169. https://doi.org/10.3389/fmed.2020.00169

14. Cooper I, Mondal A, Antonopoulos CG. A SIR model assumption for the spread of COVID-19 in different communities. Chaos Solitons Fractals. 2020;139:110057. https://doi.org/10.1016/j.chaos.2020.110057

15. Ndaïrou F, Area I, Nieto JJ, Torres DFM. Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos Solitons Fractals. 2020;135:109846. https://doi.org/10.1016/j.chaos.2020.109846

16. Kissler SM, Tedijanto C, Goldstein E, Grad YH, Lipsitch M. Projecting the transmission dynamics of SARS-CoV-2 through the postpandemic period. Science. 2020;368(6493):860-868. https://doi.org/10.1126/science.abb5793

17. Giordano G, Blanchini F, Bruno R, et al. Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nat Med. 2020;26(6):855-860. https://doi.org/10.1038/s41591-020-0883-7

18. Radev ST, Graw F, West R, Rieger MO, Lasser J. OutbreakFlow: model-based Bayesian inference of disease outbreaks. PLoS Comput Biol. 2021;17(5):e1008854. https://doi.org/10.1371/journal.pcbi.1008854

19. Ceylan Z. Estimation of COVID-19 prevalence in Italy, Spain, and France. Sci Total Environ. 2020;729:138817. https://doi.org/10.1016/j.scitotenv.2020.138817

20. Ajelli M, Gonçalves B, Balcan D, et al. Comparing large-scale computational approaches to epidemic modeling: agent-based versus structured metapopulation models. BMC Infect Dis. 2010;10:190. https://doi.org/10.1186/1471-2334-10-190

21. Zhang X, Ma R, Wang L. Predicting turning point, duration and attack rate of COVID-19 outbreaks in major Western countries. Chaos Solitons Fractals. 2020;135:109829. https://doi.org/10.1016/j.chaos.2020.109829

22. Alzahrani SI, Aljamaan IA, Al-Fakih EA. Forecasting COVID-19 pandemic in Saudi Arabia using ARIMA model. J Infect Public Health. 2020;13(7):920-925. https://doi.org/10.1016/j.jiph.2020.06.001

23. Arifi F, Moldokmatova A, Aitmatov K, et al. Application of the CoMo model in the Kyrgyz Republic: scenario-based planning for COVID-19 response. CoMo Consortium Report. Oxford: University of Oxford; 2021.

24. Elsawah S, Spencer T, Ryan MJ, et al. A review of computational model robustness evaluation strategies. Int J Environ Res Public Health. 2020;17(18):6280. https://doi.org/10.3390/ijerph17186280

25. Cramer EY, Huang Y, Wang Y, et al. The United States COVID-19 Forecast Hub dataset. J Am Med Inform Assoc. 2022;29(8): 1490-1499. https://doi.org/10.1093/jamia/ocac066

26. Taylor SJ, Letham B. Forecasting at scale. Am Stat. 2018;72(1):37-45. https://doi.org/10.1080/00031305.2017.1380080

27. Petropoulos F, Makridakis S. Forecasting the novel coronavirus COVID-19. PLoS One. 2020;15(3):e0231236. https://doi.org/10.1371/journal.pone.0231236

28. Hassani H, Silva ES, Gupta R. Forecasting the COVID-19 pandemic with singular spectrum analysis. J Data Sci. 2020;18(3):531- 544. https://doi.org/10.6339/JDS.202007_18(3).00003

29. Rustam F, Reshi AA, Mehmood A, et al. COVID-19 future forecasting using supervised machine learning models. IEEE Access. 2020;8:101489-101499. https://doi.org/10.1109/ACCESS.2020.2997311

30. Aboluwarin OA, Falade OO. COVID-19 forecasting using deep learning models. Sci Afr. 2022;16:e01163. https://doi.org/10.101 6/j.sciaf.2022.e01163

31. Singh RK, Rani M, Bhagavathula AS, et al. Prediction of the COVID-19 pandemic for the top 15 affected countries: advanced autoregressive integrated moving average (ARIMA) model. JMIR Public Health Surveill. 2020;6(2):e19115. https://doi.org/ 10.2196/19115

32. Hellewell J, Abbott S, Gimma A, et al. Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts. Lancet Glob Health. 2020;8(4):e488-e496. https://doi.org/10.1016/S2214-109X(20)30074-7

33. Ferguson NM, Laydon D, Nedjati-Gilani G, et al. Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mor tality and healthcare demand. Imperial College COVID-19 Response Team. 2020. https://doi.org/10.25561/77482

34. Dooronbekova AZh. Epidemiological analysis of COVID-19 spread in the Kyrgyz Republic using mathematical modeling and forecasting methods [dissertation]. Bishkek; 2025. 120 p.
Для цитирования

Дооронбекова А.Ж. Математическое моделирование эпидемии COVID-19: методы, сценарии и практическая применимость. Научнопрактический журнал «Здравоохранение Кыргызстана» 2026, № 1, с. 148-153.  https://dx.doi.org/10.51350/zdravkg2026.1.3.18.148.153
Английский
About authors

Dooronbekova Aizhan Zhakypbekovna, applicant, research fellow of the Center for Analysis and Management of Public Health Risks National Institute of Public Health of the Ministry of Health, Bishkek, Kyrgyz Republic

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References

1. Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc A. 1927;115(772):700-721. https://doi.org/10.1098/rspa.1927.0118

2. Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press; 1991.

3. Hethcote HW. The mathematics of infectious diseases. SIAM Rev. 2000;42(4):599-653. https://doi.org/10.1137/S0036144500371907

4. Diekmann O, Heesterbeek JAP, Roberts MG. The construction of next-generation matrices for compartmental epidemic models. J R Soc Interface. 2010;7(47):873-885. https://doi.org/10.1098/rsif.2009.0386

5. Moore S, Hill EM, Tildesley MJ, Dyson L, Keeling MJ. Vaccination and non-pharmaceutical interventions for COVID-19: a mathematical modelling study. Lancet Infect Dis. 2021;21(6):793-802. https://doi.org/10.1016/S1473-3099(21)00143-2

6. Prem K, Liu Y, Russell TW, et al. The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China:a modelling study. Lancet Public Health. 2020;5(5):e261-e270.https://doi.org/10.1016/S2468-2667(20)30073-6

7. Aguas R, Hupert N, Shretta R, et al. COVID-19 pandemic modelling in context: uniting people and technology across nations. BMJ Preprint. 2020. https://www.researchgate.net/publication/342747645

8. CoMo Consortium. COVID-19 Modelling for Resource-Constrained Settings. Oxford: University of Oxford; 2021. https://www.co momodel.net/

9. World Health Organization. Considerations for implementing and adjusting public health and social measures. Geneva: WHO; 2021. https://apps.who.int/iris/handle/10665/337575

10. World Health Organization. COVID-19 Strategic Preparedness and Response Plan. Geneva: WHO; 2021. https://www.who.int/pub lications/i/item/WHO-WHE-2021.02 11. UNICEF. COVID-19 Vaccine Delivery Partnership: Central Asia Case Studies. Final Report. New York: UNICEF; May 2022. https://www.unicef.org/kyrgyzstan/media/8711/file

12. Hunter E, Mac Namee B, Kelleher JD. A comparison of agent-based models and equation based models for infectious disease epi demiology. Complex Syst. 2018;27(6):519-559.

13. Wangping J, Ke H, Yang S, et al. Extended SIR prediction of the epidemics trend of COVID-19 in Italy and compared with Hunan, China. Front Med. 2020;7:169. https://doi.org/10.3389/fmed.2020.00169

14. Cooper I, Mondal A, Antonopoulos CG. A SIR model assumption for the spread of COVID-19 in different communities. Chaos Solitons Fractals. 2020;139:110057. https://doi.org/10.1016/j.chaos.2020.110057

15. Ndaïrou F, Area I, Nieto JJ, Torres DFM. Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos Solitons Fractals. 2020;135:109846. https://doi.org/10.1016/j.chaos.2020.109846

16. Kissler SM, Tedijanto C, Goldstein E, Grad YH, Lipsitch M. Projecting the transmission dynamics of SARS-CoV-2 through the postpandemic period. Science. 2020;368(6493):860-868. https://doi.org/10.1126/science.abb5793

17. Giordano G, Blanchini F, Bruno R, et al. Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nat Med. 2020;26(6):855-860. https://doi.org/10.1038/s41591-020-0883-7

18. Radev ST, Graw F, West R, Rieger MO, Lasser J. OutbreakFlow: model-based Bayesian inference of disease outbreaks. PLoS Comput Biol. 2021;17(5):e1008854. https://doi.org/10.1371/journal.pcbi.1008854

19. Ceylan Z. Estimation of COVID-19 prevalence in Italy, Spain, and France. Sci Total Environ. 2020;729:138817. https://doi.org/10.1016/j.scitotenv.2020.138817

20. Ajelli M, Gonçalves B, Balcan D, et al. Comparing large-scale computational approaches to epidemic modeling: agent-based versus structured metapopulation models. BMC Infect Dis. 2010;10:190. https://doi.org/10.1186/1471-2334-10-190

21. Zhang X, Ma R, Wang L. Predicting turning point, duration and attack rate of COVID-19 outbreaks in major Western countries. Chaos Solitons Fractals. 2020;135:109829. https://doi.org/10.1016/j.chaos.2020.109829

22. Alzahrani SI, Aljamaan IA, Al-Fakih EA. Forecasting COVID-19 pandemic in Saudi Arabia using ARIMA model. J Infect Public Health. 2020;13(7):920-925. https://doi.org/10.1016/j.jiph.2020.06.001

23. Arifi F, Moldokmatova A, Aitmatov K, et al. Application of the CoMo model in the Kyrgyz Republic: scenario-based planning for COVID-19 response. CoMo Consortium Report. Oxford: University of Oxford; 2021.

24. Elsawah S, Spencer T, Ryan MJ, et al. A review of computational model robustness evaluation strategies. Int J Environ Res Public Health. 2020;17(18):6280. https://doi.org/10.3390/ijerph17186280

25. Cramer EY, Huang Y, Wang Y, et al. The United States COVID-19 Forecast Hub dataset. J Am Med Inform Assoc. 2022;29(8): 1490-1499. https://doi.org/10.1093/jamia/ocac066

26. Taylor SJ, Letham B. Forecasting at scale. Am Stat. 2018;72(1):37-45. https://doi.org/10.1080/00031305.2017.1380080

27. Petropoulos F, Makridakis S. Forecasting the novel coronavirus COVID-19. PLoS One. 2020;15(3):e0231236. https://doi.org/10.1371/journal.pone.0231236

28. Hassani H, Silva ES, Gupta R. Forecasting the COVID-19 pandemic with singular spectrum analysis. J Data Sci. 2020;18(3):531- 544. https://doi.org/10.6339/JDS.202007_18(3).00003

29. Rustam F, Reshi AA, Mehmood A, et al. COVID-19 future forecasting using supervised machine learning models. IEEE Access. 2020;8:101489-101499. https://doi.org/10.1109/ACCESS.2020.2997311

30. Aboluwarin OA, Falade OO. COVID-19 forecasting using deep learning models. Sci Afr. 2022;16:e01163. https://doi.org/10.101 6/j.sciaf.2022.e01163

31. Singh RK, Rani M, Bhagavathula AS, et al. Prediction of the COVID-19 pandemic for the top 15 affected countries: advanced autoregressive integrated moving average (ARIMA) model. JMIR Public Health Surveill. 2020;6(2):e19115. https://doi.org/ 10.2196/19115

32. Hellewell J, Abbott S, Gimma A, et al. Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts. Lancet Glob Health. 2020;8(4):e488-e496. https://doi.org/10.1016/S2214-109X(20)30074-7

33. Ferguson NM, Laydon D, Nedjati-Gilani G, et al. Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mor tality and healthcare demand. Imperial College COVID-19 Response Team. 2020. https://doi.org/10.25561/77482

34. Dooronbekova AZh. Epidemiological analysis of COVID-19 spread in the Kyrgyz Republic using mathematical modeling and forecasting methods [dissertation]. Bishkek; 2025. 120 p.
For citation

Dooronbekova A. Zh. Mathematical Modeling of the COVID -19 Epidemic: Methods, Scenarios, and Practical Applica bility. Scientific practical journal “Health care of Kyrgyzstan” 2026, No.1, p. 148-153.  https://dx.doi.org/10.51350/zdravkg2026.1.3.18.148.153
Кыргызский
Авторлор жөнүндө

Дооронбекова Айжан Жакыпбековна, талапкер, Саламаттыкты сактоо министрлигинин Коомдук саламаттыкты сактоо боюнча улуттук институтунун Коомдук саламаттыкты сактоо тобокелдиктерин талдоо жана башкаруу борборунун илимий кызматкери, Бишкек, Кыргыз Республикасы
Шилтемелер

1. Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc A. 1927;115(772):700-721. https://doi.org/10.1098/rspa.1927.0118

2. Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press; 1991.

3. Hethcote HW. The mathematics of infectious diseases. SIAM Rev. 2000;42(4):599-653. https://doi.org/10.1137/S0036144500371907

4. Diekmann O, Heesterbeek JAP, Roberts MG. The construction of next-generation matrices for compartmental epidemic models. J R Soc Interface. 2010;7(47):873-885. https://doi.org/10.1098/rsif.2009.0386

5. Moore S, Hill EM, Tildesley MJ, Dyson L, Keeling MJ. Vaccination and non-pharmaceutical interventions for COVID-19: a mathematical modelling study. Lancet Infect Dis. 2021;21(6):793-802. https://doi.org/10.1016/S1473-3099(21)00143-2

6. Prem K, Liu Y, Russell TW, et al. The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China:a modelling study. Lancet Public Health. 2020;5(5):e261-e270.https://doi.org/10.1016/S2468-2667(20)30073-6

7. Aguas R, Hupert N, Shretta R, et al. COVID-19 pandemic modelling in context: uniting people and technology across nations. BMJ Preprint. 2020. https://www.researchgate.net/publication/342747645

8. CoMo Consortium. COVID-19 Modelling for Resource-Constrained Settings. Oxford: University of Oxford; 2021. https://www.co momodel.net/

9. World Health Organization. Considerations for implementing and adjusting public health and social measures. Geneva: WHO; 2021. https://apps.who.int/iris/handle/10665/337575

10. World Health Organization. COVID-19 Strategic Preparedness and Response Plan. Geneva: WHO; 2021. https://www.who.int/pub lications/i/item/WHO-WHE-2021.02 11. UNICEF. COVID-19 Vaccine Delivery Partnership: Central Asia Case Studies. Final Report. New York: UNICEF; May 2022. https://www.unicef.org/kyrgyzstan/media/8711/file

12. Hunter E, Mac Namee B, Kelleher JD. A comparison of agent-based models and equation based models for infectious disease epi demiology. Complex Syst. 2018;27(6):519-559.

13. Wangping J, Ke H, Yang S, et al. Extended SIR prediction of the epidemics trend of COVID-19 in Italy and compared with Hunan, China. Front Med. 2020;7:169. https://doi.org/10.3389/fmed.2020.00169

14. Cooper I, Mondal A, Antonopoulos CG. A SIR model assumption for the spread of COVID-19 in different communities. Chaos Solitons Fractals. 2020;139:110057. https://doi.org/10.1016/j.chaos.2020.110057

15. Ndaïrou F, Area I, Nieto JJ, Torres DFM. Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos Solitons Fractals. 2020;135:109846. https://doi.org/10.1016/j.chaos.2020.109846

16. Kissler SM, Tedijanto C, Goldstein E, Grad YH, Lipsitch M. Projecting the transmission dynamics of SARS-CoV-2 through the postpandemic period. Science. 2020;368(6493):860-868. https://doi.org/10.1126/science.abb5793

17. Giordano G, Blanchini F, Bruno R, et al. Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nat Med. 2020;26(6):855-860. https://doi.org/10.1038/s41591-020-0883-7

18. Radev ST, Graw F, West R, Rieger MO, Lasser J. OutbreakFlow: model-based Bayesian inference of disease outbreaks. PLoS Comput Biol. 2021;17(5):e1008854. https://doi.org/10.1371/journal.pcbi.1008854

19. Ceylan Z. Estimation of COVID-19 prevalence in Italy, Spain, and France. Sci Total Environ. 2020;729:138817. https://doi.org/10.1016/j.scitotenv.2020.138817

20. Ajelli M, Gonçalves B, Balcan D, et al. Comparing large-scale computational approaches to epidemic modeling: agent-based versus structured metapopulation models. BMC Infect Dis. 2010;10:190. https://doi.org/10.1186/1471-2334-10-190

21. Zhang X, Ma R, Wang L. Predicting turning point, duration and attack rate of COVID-19 outbreaks in major Western countries. Chaos Solitons Fractals. 2020;135:109829. https://doi.org/10.1016/j.chaos.2020.109829

22. Alzahrani SI, Aljamaan IA, Al-Fakih EA. Forecasting COVID-19 pandemic in Saudi Arabia using ARIMA model. J Infect Public Health. 2020;13(7):920-925. https://doi.org/10.1016/j.jiph.2020.06.001

23. Arifi F, Moldokmatova A, Aitmatov K, et al. Application of the CoMo model in the Kyrgyz Republic: scenario-based planning for COVID-19 response. CoMo Consortium Report. Oxford: University of Oxford; 2021.

24. Elsawah S, Spencer T, Ryan MJ, et al. A review of computational model robustness evaluation strategies. Int J Environ Res Public Health. 2020;17(18):6280. https://doi.org/10.3390/ijerph17186280

25. Cramer EY, Huang Y, Wang Y, et al. The United States COVID-19 Forecast Hub dataset. J Am Med Inform Assoc. 2022;29(8): 1490-1499. https://doi.org/10.1093/jamia/ocac066

26. Taylor SJ, Letham B. Forecasting at scale. Am Stat. 2018;72(1):37-45. https://doi.org/10.1080/00031305.2017.1380080

27. Petropoulos F, Makridakis S. Forecasting the novel coronavirus COVID-19. PLoS One. 2020;15(3):e0231236. https://doi.org/10.1371/journal.pone.0231236

28. Hassani H, Silva ES, Gupta R. Forecasting the COVID-19 pandemic with singular spectrum analysis. J Data Sci. 2020;18(3):531- 544. https://doi.org/10.6339/JDS.202007_18(3).00003

29. Rustam F, Reshi AA, Mehmood A, et al. COVID-19 future forecasting using supervised machine learning models. IEEE Access. 2020;8:101489-101499. https://doi.org/10.1109/ACCESS.2020.2997311

30. Aboluwarin OA, Falade OO. COVID-19 forecasting using deep learning models. Sci Afr. 2022;16:e01163. https://doi.org/10.101 6/j.sciaf.2022.e01163

31. Singh RK, Rani M, Bhagavathula AS, et al. Prediction of the COVID-19 pandemic for the top 15 affected countries: advanced autoregressive integrated moving average (ARIMA) model. JMIR Public Health Surveill. 2020;6(2):e19115. https://doi.org/ 10.2196/19115

32. Hellewell J, Abbott S, Gimma A, et al. Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts. Lancet Glob Health. 2020;8(4):e488-e496. https://doi.org/10.1016/S2214-109X(20)30074-7

33. Ferguson NM, Laydon D, Nedjati-Gilani G, et al. Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mor tality and healthcare demand. Imperial College COVID-19 Response Team. 2020. https://doi.org/10.25561/77482

34. Dooronbekova AZh. Epidemiological analysis of COVID-19 spread in the Kyrgyz Republic using mathematical modeling and forecasting methods [dissertation]. Bishkek; 2025. 120 p.
Цитата үчүн

Дооронбекова А.Ж. COVID-19 эпидемиясынын математикалык моделдөө ыкмалары: методдор, сценарийлер жана практикалык колдонулушу. Кыргызстандын саламаттык сактоо илимий-практикалык журналы 2026, № 1, б. 148-153. https://dx.doi.org/10.51350/zdravkg2026.1.3.18.148.153
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