Caputo Derivative as Weighted Average of Historical Values: some consequences illustrated via COVID-19 data
DOI:
https://doi.org/10.5540/03.2023.010.01.0030Palavras-chave:
Caputo Derivative, Weighted Average, Dimensional Analysis, critical state, COVIDResumo
This paper uses the formula previously proposed by the authors themselves, in which the Caputo fractional derivative is written proportionally to the weighted average of historical values of the classical derivative (Equation (2)). Three consequences of this formula are treated in this work. The first explicitly shows the dimension of the Caputo derivative, the second indicates which historical values of the classical derivative have greater/lower weight for the Caputo operator at the current instant, and finally, the third shows that the Caputo derivative is zero at instants after the critical point occurred (allowing interpretations for the order of the derivative, for example in the dynamics of some disease). To illustrate these three results, we used examples previously obtained by the authors themselves, modeling the curve of active COVID-19 cases with the SIR model. This approach captures the memory effect well in epidemiological models.
Downloads
Referências
L. C. de Barros et al. “The memory effect on fractional calculus: an application in the spread of COVID-19”. In: Computational and Applied Mathematics 40 (2021), pp. 1–21. doi:10.1007/s40314-021-01456-z.
K. Diethelm. “A fractional calculus based model for the simulation of an outbreak of dengue fever”. In: Nonlinear Dynamics (2013), pp. 613–619. doi: 10.1007/s11071-012-0475-2.
L. Edelstein-Keshet. Mathematical models in biology. SIAM, 2005. isbn: 9780394355078.
A. Kilicman et al. “A fractional order SIR epidemic model for dengue transmission”. In: Chaos, Solitons & Fractals 114 (2018), pp. 55–62. doi: 10.1016/j.chaos.2018.06.031.
M. M. Lopes. “Fractional calculus and fuzzy sets theory: a study with epidemiological models for COVID-19”. PhD thesis. IMECC/Unicamp, 2023.
M. Mazandarani and A. V. Kamyad. “Modified fractional Euler method for solving fuzzy fractional initial value problem”. In: communications in Nonlinear Science and Numerical Simulation 18.1 (2013), pp. 12–21. doi: 10.1016/j.cnsns.2012.06.008.
I. Podlubny. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Vol. 198. Elsevier, 1998. isbn: 9780080531984.
M. Saeedian et al. “Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model”. In: Physical Review E 95.2 (2017), p. 022409. doi: 10.1103/PhysRevE.95.022409.
G S. Teodoro, J. A. T. Machado, and E. C. De Oliveira. “A review of definitions of fractional derivatives and other operators”. In: Journal of Computational Physics 388 (2019), pp. 195–208. doi: 10.1016/j.jcp.2019.03.008.
WORLDOMETERS. Covid-19 coronavirus pandemic. Online. Available at: https://www.worldometers.info/coronavirus/. Accessed on March 2021. 2021.
