Exact solutions to SIR epidemic models via integrable discretization. (English) Zbl 07891903
Summary: An integrable discretization of the SIR model with vaccination is proposed. Through the discretization, the conserved quantities of the continuous model are inherited to the discrete model, since the discretization is based on the intersection structure of the non-algebraic invariant curve defined by the conserved quantities. Uniqueness of the forward/backward evolution of the discrete model is demonstrated in terms of the single-valuedness of the Lambert W function on the positive real axis. Furthermore, the exact solution to the continuous SIR model with vaccination is constructed via the integrable discretization. When applied to the original SIR model, the discretization procedure leads to two kinds of integrable discretization, and the exact solution to the continuous SIR model is also deduced. It is furthermore shown that the discrete SIR model geometrically linearizes the time evolution by using the non-autonomous parallel translation of the line intersecting the invariant curve.
©2024 American Institute of Physics
©2024 American Institute of Physics
MSC:
| 37N25 | Dynamical systems in biology |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37J70 | Completely integrable discrete dynamical systems |
| 92D30 | Epidemiology |
| 92D25 | Population dynamics (general) |
References:
| [1] | Kermack, W.; McKendrick, A., A contribution to the mathematical theory of epidemics, Proc. R. Soc. A, 115, 700, 1927 · JFM 53.0517.01 · doi:10.1098/rspa.1927.0118 |
| [2] | Hethcote, H. W., Asymptotic behavior and stability in epidemic models, Lecture Notes in Biomathematics, 2, 83, 1974, Springer-Verlag: Springer-Verlag, Berlin, Heidelberg · Zbl 0303.92001 |
| [3] | Dietz, K.; Ludwing, D.; Cook, K. L., Transmission and control of arbovirus diseases, Epidemiology, 104, 1975, Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics, Philadelphia |
| [4] | Schenzle, D., An age-structured model of pre- and post-vaccination measles transmission, Math. Med. Biol., 1, 169, 1984 · Zbl 0611.92021 · doi:10.1093/imammb/1.2.169 |
| [5] | Willox, R.; Grammaticos, B.; Carstea, A. S.; Ramani, A., Epidemic dynamics: Discrete-time and cellular automaton models, Physica A, 328, 13, 2003 · Zbl 1026.92042 · doi:10.1016/s0378-4371(03)00552-1 |
| [6] | Satsuma, J.; Willox, R.; Ramani, A.; Grammaticos, B.; Carstea, A. S., Extending the SIR epidemic model, Physica A, 336, 369, 2004 · doi:10.1016/j.physa.2003.12.035 |
| [7] | Khan, H.; Mohapatora, R. N.; Vajravelu, K.; Liao, S. J., The explicit series solution of SIR and SIS epidemic models, Appl. Math. Comput., 215, 653, 2009 · Zbl 1171.92033 · doi:10.1016/j.amc.2009.05.051 |
| [8] | Shabbir, G.; Khan, H.; Sadiq, M. A., A note on exact solution of SIR and SIS epidemic models, 2010 |
| [9] | Harko, T.; Lobo, F.; Mak, M., Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates, Appl. Math. Comput., 236, 184, 2014 · Zbl 1334.92400 · doi:10.1016/j.amc.2014.03.030 |
| [10] | Adiga, A.; Dubhashi, D.; Lewis, B.; Marathe, M.; Venkatramanan, S.; Vullikanti, A., Mathematical models for COVID-19 pandemic: A comparative analysis, J. Indian Inst. Sci., 100, 793, 2020 · doi:10.1007/s41745-020-00200-6 |
| [11] | Kopfpva, J.; Nábělková, P.; Rachinskii, D.; Rouf, S. C., Dynamics of SIR model with vaccination and heterogeneous behavioral response of individuals modeled by the Preisach operator, J. Math. Biol., 83, 11, 2021 · Zbl 1472.34092 · doi:10.1007/s00285-021-01629-8 |
| [12] | Kubota, S., The macroeconomics of COVID-19 exit strategy: The case of Japan, Jpn. Econ. Rev., 72, 651, 2021 · doi:10.1007/s42973-021-00091-x |
| [13] | Gonzalez-Parra, G.; Martínez-Rodríguez, D.; Villanueva-Micó, R. J., Impact of a new SARS-CoV-2 variant on the population: A mathematical modeling approach, Math. Comput. Appl., 26, 25, 2021 · doi:10.3390/mca26020025 |
| [14] | Kifle, Z. S.; Obsu, L. L., Mathematical modeling for COVID-19 transmission dynamics: A case study in Ethiopia, Results Phys., 34, 105191, 2022 · doi:10.1016/j.rinp.2022.105191 |
| [15] | Tanaka, Y.; Maruno, K., Integrable discretizations of the SIR model, RIMS Kôkyûroku Bessatsu, B94, 085, 2023 · Zbl 1535.39006 |
| [16] | Suris, Y. B., The Problem of Integrable Discretization: Hamiltonian Approach. Progress in Mathematics, 219, 2003, Birkhäuser: Birkhäuser, Basel, Boston, Berlin · Zbl 1033.37030 |
| [17] | Hirota, R., The Direct Method in Soliton Theory. Cambridge Tracts in Mathematics, 155, 2009, Cambridge University Press: Cambridge University Press, Cambridge |
| [18] | Duistermaat, J. J., Discrete Integrable Systems: QRT Maps and Elliptic Surfaces. Springer Monographs in Mathematics, 2010, Springer-Verlag: Springer-Verlag, Berlin, Heidelberg · Zbl 1219.14001 |
| [19] | Tokihiro, T.; Takahashi, D.; Matsukidaira, J.; Satsuma, J., From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett., 76, 3247, 1996 · doi:10.1103/physrevlett.76.3247 |
| [20] | Quispel, G. R. W.; Roberts, J. A. G.; Thompson, C. J., Integrable mappings and soliton equations II, Physica D, 34, 183, 1989 · Zbl 0679.58024 · doi:10.1016/0167-2789(89)90233-9 |
| [21] | Maclagan, D.; Sturmfels, B., Introduction to Tropical Geometry. Graduate Studies in Mathematics, 161, 2015, American Mathematica Society: American Mathematica Society, Providence, RI · Zbl 1321.14048 |
| [22] | Inoue, R.; Takenawa, T., Tropical spectral curves and integrable cellular automata, Int. Math. Res. Not., 2008, rnn019 · Zbl 1155.37012 · doi:10.1093/imrn/rnn019 |
| [23] | Nobe, A., Ultradiscrete QRT maps and tropical elliptic curves, J. Phys. A: Math. Theor., 41, 125205, 2008 · Zbl 1145.35097 · doi:10.1088/1751-8113/41/12/125205 |
| [24] | Nobe, A., An ultradiscrete integrable map arising from a pair of tropical elliptic pencils, Phys. Lett. A, 375, 4178, 2011 · Zbl 1254.37049 · doi:10.1016/j.physleta.2011.10.010 |
| [25] | Nobe, A., A geometric realization of the periodic discrete Toda lattice and its tropicalization, J. Phys. A: Math. Theor., 46, 465203, 2013 · Zbl 1417.37206 · doi:10.1088/1751-8113/46/46/465203 |
| [26] | Hirota, R.; Tsujimoto, S., Conserved quantities of a class of nonlinear difference-difference equations, J. Phys. Soc. Jpn., 64, 3125, 1995 · Zbl 0972.39501 · doi:10.1143/jpsj.64.3125 |
| [27] | Bogoyavlenskij, O., Integrable Lotka-Volterra systems, Regul. Chaotic Dyn., 13, 543, 2008 · Zbl 1229.37097 · doi:10.1134/s1560354708060051 |
| [28] | Nobe, A., The Volterra lattice, Abel’s equation of the first kind, and the SIR epidemic models, 2024 |
| [29] | Lambert, J. H., Observationes variae in mathesim puram, Acta Helvetica, Physico-Mathematico-Anatomico-Botanico-Medica, 3, 1758, 128, The Physical and Medical Society of Basel, Basel |
| [30] | Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffery, D. J.; Knuth, D. E., On the LambertW function, Adv. Comput. Math., 5, 329, 1996 · Zbl 0863.65008 · doi:10.1007/bf02124750 |
| [31] | Mancas, S.; Rosu, H., Integrable Abel equations and Vein’s Abel equation: Integrable Abel equations and Vein’s Abel equation, Math. Methods Appl. Sci., 39, 1376, 2016 · Zbl 1338.34009 · doi:10.1002/mma.3575 |
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