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2024 | OriginalPaper | Chapter

Simple Equations Method (SEsM): Exact Solutions for Description of COVID-19 Epidemic Waves

Author : Nikolay K. Vitanov

Published in: New Trends in the Applications of Differential Equations in Sciences

Publisher: Springer Nature Switzerland

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Abstract

We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear nonlinear differential equations connected to the SIR model of epidemic spread. We illustrate the methodology by obtaining a solution on the basis of the use of Bernoulli equation as a simple equation.

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R. Levin R.: Complexity. Life at the Edge of Chaos. The University of Chicago Press, Chicago (1999)

Dimitrova, Z.I.: On the nonlinear dynamics of interacting populations. Effects of delay on populations substitution. Compt. rend. Acad. bulg. Sci 61, 1541–1548 (2008).

Vitanov N.K.: Science Dynamics and Research Production. Indicators, Indexes, Statistical Laws and Mathematical Models. Springer, Cham (2016)

Dimitrova, Z. I.: Fluctuations and dynamics of the chaotic attractor connected to an instability in a heated from below rotating fluid layer. Compt. rend. Acad. bulg. Sci 60, 1065–1070 (2007).

May, R. M., Levin, S. A., Sugihara, G.: Ecology for bankers. Nature 451, 893–894, (2008). https://doi.org/10.1038/451893a

Vitanov, N. K., Vitanov, K. N.: Discrete-time model for a motion of substance in a channel of a network with application to channels of human migration. Physica A: Statistical Mechanics and its Applications 509, 635–650 (2018). https://doi.org/10.1016/j.physa.2018.06.076

Nikolova, E. V., Serbezov, D. Z., Jordanov, I.: Nonlinear spread waves in population dynamics including a human-induced Allee effect. AIP Conference Proceedings vol. 2075, 150004 (2019). https://doi.org/10.1063/1.5091327

Z. I. Dimitrova, Z. I., Hoffmann N. P.: On the probability for extreme water levels of the river Elba in Germany. Compt. rend. Acad. bulg. Sci 65, 153–160 (2012).

Kutner, R., Ausloos, M., Grech, D., Di Matteo, T., Schinckus, C., Stanley, H. E.: Econophysics and sociophysics: Their milestones & challenges. Physica A 516, 240–253 (2019). https://doi.org/10.1016/j.physa.2018.10.019

10.

Vitanov, N. K., Vitanov, K. N.: On the motion of substance in a channel of a network and human migration. Physica A: 490, 1277–1294 (2018). https://doi.org/10.1016/j.physa.2017.08.038

11.

Amaral, L. A. N., Scala, A., Barthelemy, M., Stanley, H. E.: Classes of small-world networks. PNAS USA 97, 11149–11152 (2000). https://doi.org/10.1073/pnas.20032719

12.

Vitanov, N. K., Ausloos, M., Rotundo, G.: Discrete model of ideological struggle accounting for migration. Advances in Complex Systems 15, 1250049 (2012). https://doi.org/10.1142/S021952591250049X

13.

Simon J. H.: The Economic Consequences of Immigration. The University of Michigan Press, Ann Arbor, MI, USA (1999)

14.

Jordanov, I. P., Nikolova, E. V.: On the evolution of nonlinear density population waves in the socio-economic systems. AIP Conference Proceedings vol. 2075, 150002 (2019). https://doi.org/10.1063/1.5091325

15.

Neil Rasband S.: Chaotic Dynamics of Nonlinear Systems. Dover, New York (1990)

16.

Jordanov, I., Nikolova, E.: On nonlinear waves in the spatio-temporal dynamics of interacting populations. Journal of Theoretical and Applied Mechanics 43, 69–76 (2013). https://doi.org/10.2478/jtam-2013-0015. arXiv:1208.5465

17.

Nikolova E., Goranova, E., Dimitrova Z.: Assessment of rupture risk factors of abdominal aortic aneurysms in Bulgarian patients using a finite element based system. Compt. rend. Acad. bulg. Sci 9, 1213–1222 (2016).

18.

Dimitrova, Z. I.: Numerical investigation of nonlinear waves connected to blood flow in an elastic tube with variable radius. Journal of Theoretical and Applied Mechanics 45, 79–92 (2015). https://doi.org/10.1515/jtam-2015-0025

19.

Jordanov, I. P.: On the nonlinear waves in (2+ 1)-dimensional population systems. Compt. rend. Acad. bulg. Sci 61, 307–314 (2008).

20.

Dimitrova, Z.I.: On travelling waves in lattices: the case of Riccati lattices. Journal of Theoretical and Applied Mechanics 42, 3–22 (2012). https://doi.org/10.2478/v10254-012-0011-2). arXiv:1208.2414

21.

Ganji, D. D., Sabzehmeidani, Y., Sedighiamiri A.: Nonlinear Systems in Heat Transfer Elsevier, Amsterdam (2018)

22.

Nikolova, E.V.: Evolution Equation for Propagation of Blood Pressure Waves in an Artery with an Aneurysm. In: Georgiev, K., Todorov, M., Georgiev, I. (eds.) Advanced Computing in Industrial Mathematics. BGSIAM 2017. Studies in Computational Intelligence vol. 793, 327–339 (2019). https://doi.org/10.1007/978-3-319-97277-0_27

23.

Jordanov, I. P., Dimitrova, Z. I.: On Nonlinear Waves of Migration. Journal of Theoretical and Applied Mechanics 40, 89–96 (2010).

24.

Dimitrova, Z. I., Ausloos, M.: Primacy analysis in the system of Bulgarian cities. Open Physics 13, 218–225 (2015). https://doi.org/10.1515/phys-2015-0029

25.

Nikolova, E. V.: On nonlinear waves in a blood-filled artery with an aneurysm. In AIP Conference Proceedings vol. 1978, 470050 (2018). https://doi.org/10.1063/1.5044120

26.

Kantz, H., Schreiber T.: Nonlinear Time Series Analysis. Cambridge University Press, Cambridge, UK (2004)

27.

Struble R.: Nonlinear Differential Equations. Dover, New York (2018)

28.

Vitanov, N. K.: Upper bounds on the heat transport in a porous layer. Physica D 136, 322–339 (2000). https://doi.org/10.1016/S0167-2789(99)00165-7

29.

Dimitrova, Z. I., Vitanov, N. K.: Adaptation and its impact on the dynamics of a system of three competing populations. Physica A: Statistical Mechanics and its Applications 300, 91–115 (2001). https://doi.org/10.1016/S0378-4371(01)00330-2

30.

Vitanov, N. K., Ausloos, M. R.: Knowledge epidemics and population dynamics models for describing idea diffusion. In: Scharnhorst A., Boerner K., van den Besselaar P. (eds.) Models of science dynamics. Understanding Complex Systems. pp. 69–125. Springer, Berlin, Heidelberg (2012). https://doi.org/10.1007/978-3-642-23068-4_3

31.

Dimitrova, Z., Gogova, D.,: Investigation of Differences in Optical Phonons Modes by Principal Component Analysis. Compt. rend. Acad. bulg. Sci 63, 1415–1420 (2010).

32.

Mills T.: Applied Time Series Analysis. Academic Press, London (2019)

33.

Dimitrova, Z. I.: On the Low-Dimensional Dynamics of Blood Flow in Small Peripheral Human Arteries. Compt. rend. Acad. bulg. Sci 63, 55–60 (2010).

34.

Dimitrova, Z. I., Vitanov, N. K.: Chaotic pairwise competition. Theoretical Population Biology 66, 1–12 (2004). https://doi.org/10.1016/j.tpb.2003.10.008

35.

Vitanov, N. K., Vitanov, K. N.: Statistical distributions connected to motion of substance in a channel of a network. Physica A 527, 121174 (2019). https://doi.org/10.1016/j.physa.2019.121174

36.

Borisov, R., Dimitrova, Z. I., Vitanov, N. K.: Statistical characteristics of stationary flow of substance in a network channel containing arbitrary number of arms. Entropy 22, 553 (2020). https://doi.org/10.3390/e22050553

37.

Dimitrova, Z. I., Vitanov, K. N.: Homogeneous balance method and auxiliary equation method as particular cases of simple equations method (SEsM). In AIP Conference Proceedings vol. 2321, 030004 (2021). https://doi.org/10.1063/5.0043070

38.

Vitanov, N. K., Vitanov, K. N., Kantz, H.: On the motion of substance in a channel of a network: Extended model and new classes of probability distributions. Entropy 22, 1240 (2020). https://doi.org/10.3390/e22111240

39.

Vitanov, N. K. Simple Equations Method (SEsM): An effective algorithm for obtaining exact solutions of nonlinear differential equations. Entropy, 24, 1653 (2022). https://doi.org/10.3390/e24111653

40.

Vitanov, N. K., Dimitrova, Z. I., Vitanov, K. N.: Simple Equations Method (SEsM): Algorithm, connection with Hirota method, Inverse Scattering Transform Method, and several other methods. Entropy 23, 10 (2021). https://doi.org/10.3390/e23010010

41.

Vitanov, N. K.: Recent developments of the methodology of the modified method of simplest equation with application. Pliska Studia Mathematica Bulgarica 30, 29–42 (2019).

42.

Vitanov, N.K.: Modified method of simplest equation for obtaining exact solutions of nonlinear partial differential equations: history, recent developments of the methodology and studied of classes of equations. Journal of Theoretical and Applied Mechanics 49, 107–122 (2019).

43.

Vitanov, N. K.: The simple equations method (SEsM) for obtaining exact solutions of nonlinear PDEs: Opportunities connected to the exponential functions. AIP Conference Proceedings vol. 2159, 030038 (2019). https://doi.org/10.1063/1.5127503

44.

Vitanov, N. K., Dimitrova, Z. I.: Simple equations method (SEsM) and other direct methods for obtaining exact solutions of nonlinear PDEs. AIP Conference Proceedings vol. 2159, 030039 (2019). https://doi.org/10.1063/1.5127504

45.

Dimitrova, Z. I., Vitanov, N. K.: Travelling waves connected to blood flow and motion of arterial walls. Gadomski, A. (ed.) In: Water in Biomechanical and Related Systems pp. 243-263. Springer, Cham. (2021)

46.

Martinov, N., Vitanov, N.: On the correspondence between the self-consistent 2D Poisson-Boltzmann structures and the sine-Gordon waves. Journal of Physics A: Mathematical and General 25, L51–L56 (1992). https://doi.org/10.1088/0305-4470/25/2/004

47.

Martinov, N., Vitanov, N.: On some solutions of the two-dimensional sine-Gordon equation. Journal of Physics A: Mathematical and General 25, L419–L426 (1992). https://doi.org/10.1088/0305-4470/25/8/007

48.

Martinov, N. K., Vitanov, N. K.: New class of running-wave solutions of the (2+ 1)-dimensional sine-Gordon equation. Journal of Physics A: Mathematical and General 27, 4611–4618 (1994). https://doi.org/10.1088/0305-4470/27/13/034

49.

Martinov, N. K., Vitanov, N. K.: On self-consistent thermal equilibrium structures in two-dimensional negative-temperature systems. Canadian Journal of Physics 72, 618–624 (1994). https://doi.org/10.1139/p94-079

50.

Vitanov, N. K.: Breather and soliton wave families for the sine-Gordon equation. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 2409–2423 (1998). https://doi.org/10.1098/rspa.1998.0264

51.

Vitanov, N. K., Martinov, N. K.: On the solitary waves in the sine-Gordon model of the two-dimensional Josephson junction. Zeitschrift fur Physik B Condensed Matter 100, 129–135 (1996). https://doi.org/10.1007/s002570050102

52.

Vitanov, N. K.: On travelling waves and double-periodic structures in two-dimensional sine-Gordon systems. Journal of Physics A: Mathematical and General 29, 5195–5207 (1996). https://doi.org/10.1088/0305-4470/29/16/036

53.

Vitanov, N. K., Jordanov, I. P., Dimitrova, Z. I.: On nonlinear dynamics of interacting populations: Coupled kink waves in a system of two populations. Communications in Nonlinear Science and Numerical Simulation 14, 2379–2388 (2009). https://doi.org/10.1016/j.cnsns.2008.07.015

54.

Vitanov, N. K., Jordanov, I. P., Dimitrova, Z. I.: On nonlinear population waves. Applied Mathematics and Computation 215, 2950–2964 (2009). https://doi.org/10.1016/j.amc.2009.09.041

55.

Vitanov, N. K.: Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity. Communications in Nonlinear Science and Numerical Simulation 15, 2050–2060 (2010). https://doi.org/10.1016/j.cnsns.2009.08.011

56.

Vitanov, N. K., Dimitrova, Z. I.: Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics. Communications in Nonlinear Science and Numerical Simulation 15, 2836–2845 (2010). https://doi.org/10.1016/j.cnsns.2009.11.029

57.

Vitanov, N. K., Dimitrova, Z. I., Kantz, H.: Modified method of simplest equation and its application to nonlinear PDEs. Applied Mathematics and Computation 216, 2587–2595 (2010). https://doi.org/10.1016/j.amc.2010.03.102

58.

Vitanov, N. K.: Modified method of simplest equation: powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs. Communications in Nonlinear Science and Numerical Simulation 16, 1176–1185 (2011). https://doi.org/10.1016/j.cnsns.2010.06.011

59.

Vitanov, N. K., Dimitrova, Z. I., Vitanov, K. N.: On the class of nonlinear PDEs that can be treated by the modified method of simplest equation. Application to generalized Degasperis-Processi equation and b-equation. Communications in Nonlinear Science and Numerical Simulation 16, 3033–3044 (2011). https://doi.org/10.1016/j.cnsns.2010.11.013

60.

Vitanov, N. K.: On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: the role of the simplest equation. Communications in Nonlinear Science and Numerical Simulation 16, 4215–4231 (2011). https://doi.org/10.1016/j.cnsns.2011.03.035

61.

Vitanov, N. K.: On modified method of simplest equation for obtaining exact solutions of nonlinear PDEs: case of elliptic simplest equation. Pliska Studia Mathematica Bulgarica 21, 257–266 (2012).

62.

Vitanov, N. K., Dimitrova, Z. I., Kantz, H.: Application of the method of simplest equation for obtaining exact traveling-wave solutions for the extended Korteweg-de Vries equation and generalized Camassa-Holm equation. Applied Mathematics and Computation 219, 7480–7492 (2013). https://doi.org/10.1016/j.amc.2013.01.035

63.

Vitanov, N. K., Dimitrova, Z. I., Vitanov, K. N.: Traveling waves and statistical distributions connected to systems of interacting populations. Computers & Mathematics with Applications 66, 1666–1684 (2013). https://doi.org/10.1016/j.camwa.2013.04.002

64.

Vitanov, N. K., Vitanov, K. N.: Population dynamics in presence of state dependent fluctuations. Computers & Mathematics with Applications 68, 962–971 (2014). https://doi.org/10.1016/j.camwa.2014.03.006

65.

Dimitrova, Z. I.: Relation between G’/G-expansion method and the modified method of simplest equation. Compt. rend. Acad. bulg. Scie 65, 1513–1520 (2012).

66.

Vitanov, N. K., Dimitrova, Z. I., Vitanov, K. N.: Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: further development of the methodology with applications. Applied Mathematics and Computation 269, 363–378 (2015). https://doi.org/10.1016/j.amc.2015.07.060

67.

Vitanov, N. K., Dimitrova, Z. I., Ivanova, T. I.: On solitary wave solutions of a class of nonlinear partial differential equations based on the function 1n\(cosh^n\) (\(\beta \)t). Applied Mathematics and Computation 315, 372–380 (2017). https://doi.org/10.1016/j.amc.2017.07.064

68.

Vitanov, N. K., Dimitrova, Z. I.: On the modified method of simplest equation and the nonlinear Schrödinger equation. Journal of Theoretical and Applied Mechanics 48, 59–68 (2018).

69.

Nikolova, E. V., Jordanov, I. P., Dimitrova, Z. I., Vitanov, N. K.: Evolution of nonlinear waves in a blood-filled artery with an aneurysm. AIP Conference Proceedings vol. 1895, 070002 (2017). https://doi.org/10.1063/1.5007391

70.

Jordanov, I.P., Vitanov, N.K.: On the Exact Traveling Wave Solutions of a Hyperbolic Reaction-Diffusion Equation. In: Georgiev, K., Todorov, M., Georgiev, I. (eds.) Advanced Computing in Industrial Mathematics. BGSIAM 2017. Studies in Computational Intelligence, vol. 793, pp. 199–210. Springer, Cham. (2019). https://doi.org/10.1007/978-3-319-97277-0_16

71.

Nikolova, E. V., Chilikova-Lubomirova, M., Vitanov, N. K.: Exact solutions of a fifth-order Korteweg-de Vries-type equation modeling nonlinear long waves in several natural phenomena. AIP Conference Proceedings vol. 2321, 030026 (2021). https://doi.org/10.1063/5.0040089

72.

Dimitrova, Z. I.: Several examples of application of the simple equations method (SEsM) for obtaining exact solutions of nonlinear PDEs. AIP Conference Proceedings vol. 2459, 030005 (2022). https://doi.org/10.1063/5.0083572

73.

Dimitrova, Z. I.: On several specific cases of the simple equations method (SEsM): Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method, general projective Riccati equations method, and first intergal method. AIP Conference Proceedings, vol. 2459, 030006 (2022). https://doi.org/10.1063/5.0083573

74.

Vitanov, N. K., Dimitrova, Z. I.: Simple Equations Method and non-linear differential equations with non-polynomial non-linearity Entropy 23, 1624 (2021). https://doi.org/10.3390/e23121624

75.

Vitanov, N. K., Vitanov, K. N.: Epidemic waves and exact solutions of a sequence of nonlinear differential equations connected to the SIR model of epidemics. Entropy, 25, 438 (2023). https://doi.org/10.3390/e25030438.

76.

Vitanov, N. K., Dimitrova, Z. I.: Computation of the exact forms of waves for a set of differential equations associated with the SEIR model of epidemics. Computation, 11, 129 (2023). https://doi.org/10.3390/computation11070129

77.

Vitanov, N. K.: Simple equations method (SEsM) and nonlinear PDEs with fractional derivatives. AIP Conference Proceedings 2459, 030040 (2022). https://doi.org/10.1063/5.0083566

78.

Vitanov, N. K.: Simple equations method (SEsM): Review and new results. AIP Conference Proceedings, 2459, 020003 (2022). https://doi.org/10.1063/5.0083565

79.

Vitanov, N. K.: Simple Equations Method (SEsM): Areas of possible applications. Springer Proceedings in Mathematics & Statistics, 412, 15–24 (2023). https://doi.org/10.1007/978-3-031-21484-4_2

80.

Vitanov, N. K.: Special Functions and polynomials connected to the Simple Equations Method (SEsM). Springer Proceedings in Mathematics & Statistics, 412, 73–81 (2023). https://doi.org/10.1007/978-3-031-21484-4_7

81.

Nikolova, E. V., Chilikova-Lubomirova, M.: Exact traveling wave solutions of the generalized Rosenau-Kawahara-RLW equation via Simple Equations Method. Springer Proceedings in Mathematics & Statistics, 412, 141–152 (2023). https://doi.org/10.1007/978-3-031-21484-4_13

82.

Nikolova, E. V.: Exact travelling-wave solutions of the extended fifth- order Korteweg-de Vries equation via Simple Equations Method (SEsM): The case of two simple equations. Entropy, 24, 1288 (2022). https://doi.org/10.3390/e24091288

83.

Nikolova, E. V., Chilikova-Lubomirova, M.: Exact traveling wave solutions of the Nikolaevskiy model for nonlinear seismic waves. AIP Conference Proceedings 2459, 030028 (2022). https://doi.org/10.1063/5.0083633

84.

Jordanov, I. P.: An application of Simplest Equations Method to nonlinear equations of Schrödinger kind. Springer Proceedings in Mathematics & Statistics, vol 412, 187–197 (2023). https://doi.org/10.1007/978-3-031-21484-4_18

85.

Jordanov, I. P.: Simple equations method applied to equations of nonlinear Schrödinger kind. AIP Conference Proceedings, 2459, 030016 (2022). https://doi.org/10.1063/5.0084340

86.

Kermack, W. O., McKendrick, A. G. A.: Contribution to the mathematical theory of epidemics Proc. R. Soc. Lond. Ser. A, 115, 700–721 (1927). https://doi.org/10.1098/rspa.1927.0118

Title: Simple Equations Method (SEsM): Exact Solutions for Description of COVID-19 Epidemic Waves
Author: Nikolay K. Vitanov
Publisher: Springer Nature Switzerland
Book: New Trends in the Applications of Differential Equations in Sciences
Print ISBN: 978-3-031-53211-5

Electronic ISBN: 978-3-031-53212-2

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-53212-2_34

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