This article examines the use of mathematical modeling for analyzing and predicting the spread of infectious diseases, focusing on the SIR (Susceptible–Infectious–Recovered) model, which is widely used in epidemiology. A general overview of the model is presented, followed by the derivation of its three fundamental differential equations that describe the dynamics of changes in each population group. Numerical solutions are obtained using Euler’s method for two sample cases, and the results are then analyzed to determine the epidemic peak and the point at which it begins to decline. Additionally, the relationship between the height of the peak and the initial number of susceptible individuals is investigated through graphical analysis. The paper discusses the limitations of both the SIR model and Euler’s method, emphasizing that the choice of parameters—such as infection and recovery rates—significantly affects the modeling results. The purpose of this work is to deepen the understanding of disease transmission mechanisms and to demonstrate how mathematical methods can support the development of effective epidemic control and management strategies. Furthermore, the study highlights the practical importance of mathematical modeling in epidemiology: such models make it possible to predict potential epidemic scenarios, assess the effectiveness of preventive measures (vaccination, isolation, contact restrictions), and optimize the allocation of healthcare resources. The SIR model also serves as a foundation for constructing more complex models—such as SEIR, SIRS, SEIRD, and others—making it a fundamental tool for further research into the dynamics of infectious processes.
numerical analysis, epidemic modeling, SIR model, disease dynamics, peak incidence, infectious diseases, Euler’s method, susceptible and infected, epidemic management
1. Anderson, R. M., & May, R. M. (1991), Infectious Diseases of Humans: Dynamics and Control, Oxford: Oxford University Press.
2. Brauer, F., Castillo-Chavez, C., & Feng, Z. (2019), Mathematical Models in Epidemiology, New York: Springer.
3. Colizza, V., Barrat, A., Barthélemy, M., & Vespignani, A. (2007), The Role of the Airline Transportation Network in the Global Spread of Pandemic Influenza, Proceedings of the National Academy of Sciences, 104(6), 1915-1920.
4. Diekmann, O., Jansen, V. A. A., & Heesterbeek, H. (2013), Mathematical Epidemiology: Past, Present, and Future, Berlin: Springer.
5. Ferguson, N. M., et al. (2005), Strategies for Mitigating an Influenza Pandemic, Nature, 437(7056), 209-214.
6. Hethcote, H. W. (2000), The Mathematics of Infectious Diseases, SIAM Review, Vol. 42, No. 4, pp. 599–653.
7. Kermack, W. O., & McKendrick, A. G. (1927), A Contribution to the Mathematical Theory of Epidemics, London: Proceedings of the Royal Society A.
8. Keeling, M. J., & Rohani, P. (2008), Modeling Infectious Diseases in Humans and Animals, Princeton: Princeton University Press.
9. Murray, J. D. (2002), Mathematical Biology: I. An Introduction, New York: Springer.
10. Weiss, H. (2013), The SIR Model and the Foundations of Public Health, Barcelona: Universitat Autònoma de Barcelona.
11. T.A. Morozova, V.N.Gel'miyarova, T.A. Gorshunova, Manaenkova T.A., Korneev A.D. Primenenie instrumentov matematicheskoy statistiki v izuchenii urovnya i dinamiki proizvoditel'nosti truda. . // Moskovskiy ekonomicheskiy zhurnal (VAK) – 2024. – T. 9, № 9. – S. 340-355.
12. Gorshunova T. A., Morozova T. A., Pihtil'kova O. A., Pronina E. V. Matematicheskoe modelirovanie optimal'nyh cen tovarov na osnove elastichnosti sprosa // Moskovskiy ekonomicheskiy zhurnal. 2025. №. 10. S. 221-247. DOI: https://doi.org/10.55186/2413046X_2025_10_10_234 (data obrascheniya: 27.10.2025).
13. Sidorov Andrei (2024). The impact of announcements on cryptocurrency prices. Revista Economică, Lucian Blaga University of Sibiu, Faculty of Economic Sciences, 76(4), 69–94, December. https://doi.org/10.56043/reveco-2024-0035
14. Astaf'ev, R. U. Rol' imitacionnyh modeley v sistemah podderzhki prinyatiya resheniy v oblasti razrabotki programmnyh produktov / R. U. Astaf'ev // Opticheskie tehnologii, materialy i sistemy (Optoteh - 2024) : Mezhdunarodnaya nauchno-tehnicheskaya konferenciya, Moskva, 02–08 dekabrya 2024 goda. – Moskva: MIREA - Rossiyskiy tehnologicheskiy universitet, 2024. – S. 789-790. – EDN JTFOGS.
