Novel numerical methods for solving the SEIRD model

The purpose of the research. The COVID-19 pandemic has shown that mathematical modeling has become important in the management of infectious diseases. The relevance of the study lies in understanding the dynamics of the spread of COVID-19 using mathematical modeling methods that play a key role in developing control strategies. Infectionspecific models make it possible to analyze patterns, predict trajectories, and evaluate the effect of measures, including quarantine, social distancing, and vaccination.

The purpose of the research is to develop and analyze an improved SEIRD model using a hybrid numerical method designed to improve the accuracy of forecasting the occurrence and development of pandemic waves and assessing the impact of sanitary measures.

Methods. The research objectives include building a new SEIRD model as an extension of the classic SIR model with the addition of additional categories – "Exposed", "Recovered" and "Dead". To implement the proposed categories, the following methods were applied: explicit Euler method, fourth – order Runge – Kutta and adaptive Runge-Kutta schemes to increase reliability. Methodologically, the SEIRD system is solved using a hybrid numerical scheme combining the advantages of classical and adaptive methods, which made it possible to obtain accurate simulations and assess the impact of interventions.

Results. The results showed that the proposed refined SEIRD model provides reliable forecasts of the occurrence and development of pandemic waves.

Conclusion. An analysis of the results shows that a 10% increase in the number of infections signals the beginning of a new wave that requires adjustments to the parameters and rapid response of public health services, as well as the implementation of rapid sanitary and epidemiological measures. The SEIRD model with hybrid methods reflects the dynamics of COVID-19, and can also be adapted to model future epidemics.

1. Zhang Y., Xiong J., Mao N. Epidemic model of Covid-19 with public health interventions consideration: a review. Authorea. Available at: https://authorea.com/users/623464/articles/646186-epidemicmodel-of-covid-19-with-public-health-interventions-consideration-areview2023 (accessed 11.06.2025). https://doi.оrg/10.22541/au.168539048.84429551/v1

2. Kermack W., McKendrick A. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 1927;115(772):700–721. https://doi.org/10.1098/rspa.1927.0118(1927)

3. Ahmad H.G., Eguda F.Y., Lawan B.M., Andrawus J., Babura B.I. Basic reproduction number and sensitivity analysis of Legionnaires’ disease model. Gadau Journal of Pure and Allied Sciences. 2023;2(1):1–8.

4. Harrington W.N., Kackos C.M., Webby R.J. The evolution and future of influenza pandemic preparedness. Experimental & Molecular Medicine. 2021;53(5):737–749. https://doi.оrg/10.1038/s12276-021-00603-0

5. Al-Saedi H.M., Hameed H.H. Mathematical modeling for COVID-19 pandemic in Iraq. Journal of Interdisciplinary Mathematics. 2021;24(5):1407–27.

6. Deeb M. Role of Mathematical Models in Epidemiology. Tishreen University JournalBasic Sciences Series. 2018;40(1):1–13

7. Fraser C., Riley S., Anderson R.M., Ferguson N.M. Factors that make an infectious disease outbreak controllable. Proceedings of the National Academy of Sciences. 2014;101(16):6146–51.

8. Konstantinov I.S., Taha A.T.T. Mathematical analysis of SIR model with incubation period. Research Result. Information Technologies. 2024;9(3):3–9. https://doi.оrg/10.18413/2518-1092-2024-9-3-0-1.

9. Brauer F., Castillo-Chavez C., Feng Z. Mathematical Models in Epidemiology. In: Texts in Applied Mathematics. Vol. 69. Springer; 2019. 619 p.

10. Keeling M.J., Rohani P. Modeling Infectious Diseases in Humans and Animals: Past, Present, and Future. Journal of Mathematical Biology. 2018;77(1):1–15. https://doi.оrg/ 10.1007/s00285-018-1235-6

11. Youssef H.M., Alghamdi N.A., Ezzat M.A., El-Bary A.A., Shawky A.M. A new dynamical modeling SEIR with global analysis applied to the real data of spreading COVID-19 in Saudi Arabia. Mathematical Biosciences and Engineering. 2020;17(6):7018–7044. https://doi.оrg/10.3934/mbe.2020362

12. Chapra S.C. Applied Numerical Methods with MATLAB for Engineers and Scientists. 4th ed. New York: McGraw-Hill; 2018. 697 p.

13. Schäfer M. Epidemiological Modelling of the Spread and Transmission of Infectious Diseases. Universität Koblenz; 2023. 256 p. (In German)

14. Zeb A., Khan M., Zaman G., Momani S., Ertürk V.S. Comparison of numerical methods of the SEIR epidemic model of fractional order. Journal of Natural History A. 2014;69(1–2):81–9. (In German)

15. Nitzsche C., Simm S. Agent-based modeling to estimate the impact of lockdown scenarios and events on a pandemic exemplified on SARS-CoV-2. Scientific Reports. 2024;14:13391. https://doi.оrg/10.1038/s41598-024-63795-1

16. Agha H.R., Taha A.T. Methods of the epidemics spread mathematical modeling. Research Result. Information Technologies. 2022;7(4):25–33. https://doi.org/10.18413/2518-1092-2022-7-4-0-3

17. Taha A.T.T. Modeling COVID-19 spreading protocol. Information Systems and Technologies. 2025;2(148):48–59.

18. Taha A.T.T., Konstantinov I.S. Analyzing pandemic dynamics through traveling waves: A mathematical model. Izvestiya Yugo-Zapadnogo gosudarstvennogo universiteta. Serija: Upravlenie, vychislitelnaja tekhnika, informatika. Meditsinskoe priborostroenie = Proceedings of the Southwest State University. Series: Control, Computer Engineering, Information Science. Medical Instruments Engineering. 2025;15(1):170–179. (In Russ.). https://doi.org/10.21869/2223-1536-2025-15-1-170-179

19. Karimi R., Farrokhi M., Izadi N., et al. Basic Reproduction Number (R0), Doubling Time, and Daily Growth Rate of the COVID-19 Epidemic: An Ecological Study. Archives of Academic Emergency Medicine. 2024;12(1):e66. https://doi.org/10.22037/aaem.v12i1.2376

20. Yousif R., Jarad F., Abdeljawad T., Awrejcewicz J. Fractional-Order SEIRD Model for Global COVID-19. Mathematics. 2023;11(4):1036. https://doi.org/10.3390/math11041036