Infection Clearance Rate in Fractional-Order SEIR Model: Stability Analysis

Muhammad Iqbal, Mohammad Januar Ismail Burhan, Adam Adam

Abstract


Infectious disease has become a serious problem over the past few years. At the same time, research on disease dynamics keeps advancing, especially on diseases caused by viruses. One concept that has been used in epidemic models is fractional calculus, or more specifically, fractional differential equations. This paper discusses the analytical properties of a fractional-order SEIR model, which are then verified by numerical simulations. Fractional-order has the property of memory effect, which represents the past experience effect on the current behavior of people. Mathematically, the present state is affected by previous states. Analytical results have shown that fractional-order value does not change the stability condition for each equilibrium. It is shown that there exists a stronger sufficient condition for disease-free equilibrium to be globally asymptotically stable. For the endemic equilibrium, it is only proven to be locally asymptotically stable when the basic reproduction number is greater than 1. Simulation results from Explicit Fractional Order Runge--Kutta (EFORK) method are confirmed to be in agreement with the basic properties provided by the analysis. The results also illustrate the impact of fractional-order and infection clearance rate, indicating that smaller fractional-orders converge faster compared to larger orders.

Keywords


Fractional-order; Infection Clearance Rate; SEIR Model; Stability Analysis

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References


[1] World Health Organization, The top 10 causes of death: Fact sheet, https://www.who.int/news- room/fact- sheets/detail/the- top- 10- causes- of- death, Accessed:20-06-2025, 2024.

[2] World Health Organization, Mpox, https://www.who.int/news-room/fact-sheets/detail/mpox, Accessed: 20-06-2025, 2024.

[3] I. Korolev, “dentification and estimation of the SEIRD epidemic model for COVID-19,”Journal of Econometrics, vol. 220, pp. 63–85, 2021, https://doi.org/10.1016/j.jeconom.2020.07.038.

[4] S. Mwalili, M. Kimathi, V. Ojiambo, D. Gathungu, and R. Mbogo, “SEIR model forCOVID-19 dynamics incorporating the environment and social distancing,” BMC ResearchNotes, vol. 13, Article number 352, 5 pages, 2020, https://doi.org/10.1186/s13104-020-05192-1.

[5] K. S. Nisar, M. Farman, M. Abdel-Aty, and J. Cao, “A review on epidemic models insight of fractional calculus,” Alexandria Engineering Journal, vol. 75, pp. 81–113, 2023,https://doi.org/10.1016/j.aej.2023.05.071.

[6] C. Yang and J. Wang, “A mathematical model for the novel coronavirus epidemic in Wuhan,China,” Mathematical Biosciences and Engineering, vol. 17, no. 3, pp. 2708–2724, 2020,https://doi.org/10.3934/mbe.2020148.

[7] L. C. de Barros, M. M. Lopes, F. S. Pedro, E. Esmi, J. P. C. dos Santos, and D. E. Sánchez,“The memory effect on fractional calculus: an application in the spread of COVID-19,”Computational and Applied Mathematics, vol. 40, Article number 72, 21 pages, 2021,https://doi.org/10.1007/s40314-021-01456-z.

[8] M. Sadki, S. Harroudi, and K. Allali, “Fractional-order sir epidemic model with treatmentcure rate,” Partial Differential Equations in Applied Mathematics, vol. 8, Article 100593, 9pages, 2023, https://doi.org/10.1016/j.padiff.2023.100593.

[9] S. Paul, A. Mahata, S. Mukherjee, P. C. Mali, and B. Roy, “Dynamical behavior of fractional order SEIR epidemic model with multiple time delays and its stability analysis,”Examples and Counterexamples, vol. 4, Article 100128, 13 pages, 2023, https://doi.org/10.1016/j.exco.2023.100128.

[10] A. Atangana and A. Secer, “A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions,” Abstract and Applied Analysis, vol. 2013, Article ID: 279681, 8 pages, 2013, https://doi.org/10.1155/2013/279681.

[11] A. Boukhouima, K. Hattaf, and N. Yousfi, “Dynamics of a Fractional Order HIV Infection Model with Specific Functional Response and Cure Rate,” International Journal of Differential Equations, vol. 2017, no. 1, Article ID: 8372140, 8 pages, 2017, https://doi.org/10.1155/2017/8372140.

[12] N. Sene, “Analysis of the fractional SEIR epidemic model with Caputo derivative via resolvents operators and numerical scheme,” Discrete and Continuous Dynamical Systems- S, vol. 18, no. 5, pp. 1316–1330, 2025, https://doi.org/10.3934/dcdss.2024149.

[13] E. M. Moumine, S. Khassal, O. Balatif, and M. Rachik, “Modeling and analysis of a fractional order spatio-temporal SEIR model: Stability and prediction,” Results in Control and Optimization, vol. 15, Article 100433, 16 pages, 2024, https://doi.org/10.1016/j.rico.2024.100433.

[14] I. Batiha, S. Alshorman, I. Jebril, and M. A. Hammad, “A Brief Review about Fractional Calculus,” International Journal of Open Problems in Computer Science and Mathematics, vol. 15, no. 4, 2022, https://www.researchgate.net/publication/366839100.

[15] C. Castillo-Garsow and C. Castillo-Chávez, “A Tour of the Basic Reproductive Numberand the Next Generation of Researchers,” in Cham: Springer International Publishing,2020, pp. 87–124, https://doi.org/10.1007/978-3-030-33645-5_2.

[16] J. P. C. dos Santos, E. Monteiro, and G. B. Vieira, “Global stability of fractional SIR epidemic model,” Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, vol. 5, Article 010019, 7 pages, 2017, https://doi.org/10.5540/03.2017.005.01.0019.

[17] I. M. Batiha, A. A. Abubaker, I. H. Jebril, S. B. Al-Shaikh, K. Matarneh, and M. Almuzini,“A Mathematical Study on a Fractional-Order SEIR Mpox Model: Analysis and Vaccination Influence,” Algorithms, vol. 16, Article 418, 18 pages, 9 2023, https://doi.org/10.3390/a16090418.




DOI: https://doi.org/10.18860/cauchy.v11i1.40849

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