Epidemiyolojideki Kompartman Modellerinin Eşlenmiş Hamilton Analizi

Year 2021, Volume: 33 Issue: 2, 265 - 276, 31.03.2021

Begüm Ateşli Oğul Esen Serkan Sütlü

https://doi.org/10.7240/jeps.796442

Abstract

Epidemiyolojideki SIR, SEIR, 2-SIR ve 2-SEIR kompartman modellerinin Hamilton formülasyonuna yer verildi. Eşlenmiş Lie-Poisson sistemleri hatırlatıldı. SIR ve SEIR modellerinin eşlenmiş Lie-Poisson sistemi oldukları gösterildi. Bükülmüş eşçevrim genişlemesi kullanılarak eşlenmiş Lie-Poisson denklemlerinin bir genelleştirilmesi elde edildi. SIR ve SEIR kompartman modellerinin iki popülasyon karşılığı olan 2-SIR ve 2-SEIR modellerinin bükülmüş eşçevrim genişlemesiyle elde edilmiş Lie-Poisson sistemi olarak ifade edilebilecekleri gösterildi.

Keywords

Eşlenmiş Lie cebirleri, Eşlenmiş Lie-Poisson sistemleri, Bükülmüş eşçevrim genişlemesi, Epidemiyolojik Kompartman Modelleri

Supporting Institution

Tübitak

Project Number

117F426

Thanks

Bu çalışma, TÜBITAK’ın 117F426 numaralı "Eşlenmis Lagrange ve Hamilton Sistemleri" isimli projesinin bir parçasıdır. Desteği için TÜBITAK’a teşekkür ederiz.

References

• [1] Abbey, H. (1952). An examination of the Reed-Frost theory of epidemics. Human biology, 24(3), 201.
• [2] Abou-Ismail, A. (2020). Compartmental Models of the COVID-19 Pandemic for Physicians and Physician-Scientists. Sn Comprehensive Clinical Medicine, 1.
• [3] Abraham, R., Marsden, J. E., & Marsden, J. E. (1978). Foundations of mechanics (Vol. 36). Reading, Massachusetts: Benjamin/Cummings Publishing Company.
• [4] Agore, A. L., & Militaru, G. (2014). Extending structures for Lie algebras. Monatshefte für Mathematik, 174(2), 169-193.
• [5] Agore, A., & Militaru, G. (2019). Extending Structures: Fundamentals and Applications. CRC Press.
• [6] Anderson, R. M. (2013). The population dynamics of infectious diseases: theory and applications. Springer.
• [7] Arnol'd, V. I. (2013). Mathematical methods of classical mechanics (Vol. 60). Springer Science & Business Media.
• [8] Ay, A., Gürses, M., & Zheltukhin, K. (2003). Hamiltonian equations in R 3. Journal of mathematical physics, 44(12), 5688-5705.
• [9] Ballesteros, A., Blasco, A., & Gutierrez-Sagredo, I. (2020). Hamiltonian structure of compartmental epidemiological models. arXiv preprint arXiv:2006.00564.
• [10] Brauer, F., Castillo-Chavez, C., & Castillo-Chavez, C. (2012). Mathematical models in population biology and epidemiology (Vol. 2, p. 508). New York: Springer.
• [11] Esen, O., Grmela, M., Gümral, H., & Pavelka, M. (2019). Lifts of symmetric tensors: fluids, plasma, and grad hierarchy. Entropy, 21(9), 907.
• [12] Esen, O., Pavelka, M., & Grmela, M. (2017). Hamiltonian coupling of electromagnetic field and matter. International Journal of Advances in Engineering Sciences and Applied Mathematics, 9(1), 3-20.
• [13] Esen, O., & Sütlü, S. (2016). Hamiltonian dynamics on matched pairs. International Journal of Geometric Methods in Modern Physics, 13(10), 1650128.
• [14] Esen, O., & Sütlü, S. (2020). Matched pair analysis of the Vlasov plasma. arXiv preprint arXiv:2004.12595.
• [15] Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM review, 42(4), 599-653.
• [16] Bäuerle, G. G., Kerf, E. A., & ten Kroode, A. P. E. (1997). Finite and infinite dimensional Lie algebras and applications in physics (Vol. 2). Elsevier.
• [17] Kermack, W. O., & McKendrick, A. G. (1927). 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, 115(772), 700-721.
• [18] Majid, S. (1990). Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations. Pacific Journal of Mathematics, 141(2), 311-332.
• [19] Marsden, J. E., Misiołek, G., Perlmutter, M., & Ratiu, T. S. (1998). Symplectic reduction for semidirect products and central extensions. Differential Geometry and its Applications, 9(1-2), 173-212.
• [20] Marsden, J. E., & Ratiu, T. S. (2013). Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems (Vol. 17). Springer Science & Business Media.
• [21] Marsden, J. E., Ratiu, T. S., & Weinstein, A. (1984). Reduction and Hamiltonian structures on duals of semidirect product Lie algebras. Cont. Math. AMS, 28, 55-100.
• [22] Murray, J. D. (2007). Mathematical biology: I. An introduction (Vol. 17). Springer Science & Business Media.
• [23] Nakamura, G. M., & Martinez, A. S. (2019). Hamiltonian dynamics of the SIS epidemic model with stochastic fluctuations. Scientific reports, 9(1), 1-9.
• [24] Nutku, Y. (1990). Bi-Hamiltonian structure of the Kermack-McKendrick model for epidemics. Journal of Physics A: Mathematical and General, 23(21), L1145.
• [25] Oliveira, G. (2020). Refined compartmental models, asymptomatic carriers and COVID-19. arXiv preprint arXiv:2004.14780.
• [26] Olver, P. J. (2000). Applications of Lie groups to differential equations (Vol. 107). Springer Science & Business Media.
• [27] Schottenloher, M. (2008). A mathematical introduction to conformal field theory (Vol. 759). Springer.
• [28] Șuhubi, E. S. (2008). Dış form analizi. Türkiye Bilimler Akademisi.
• [29] Vaisman, I. (2012). Lectures on the geometry of Poisson manifolds (Vol. 118). Birkhäuser.
• [30] Weinstein, A. (1983). The local structure of Poisson manifolds. Journal of differential geometry, 18(3), 523-557.