Fractional Derivatives Applied to Epidemiology
DOI:
https://doi.org/10.5540/tcam.2021.022.02.00157Keywords:
SIR model, Fractional Derivatives, COVID-19Abstract
We seek investigate the use of fractional derivatives, both analytically and through simulations. We derivate some models and perform investigations about them, discussing difficulties and differences between classic and fractional models. Also, we analyzed the COVID-19 pandemic using a fractional epidemiological SIR model and performed a numerical analysis using finite differences and implementation in MATLAB.References
W. H. Organization et al., “Coronavirus disease 2019 (covid-19): situationreport, 190,” 2020.
K. McIntosh, M. S. Hirsch, and A. Bloom, “Coronavirus disease 2019 (covid-19),”UpToDate Hirsch MS Bloom, vol. 5, 2020.
C. N. Angstmann, B. I. Henry, and A. V. McGann, “A fractional-order in-fectivity and recovery sir model,”Fractal and Fractional, vol. 1, no. 1, p. 11,2017.
W. O. Kermack and A. G. McKendrick, “Contributions to the mathematicaltheory of epidemics–i. 1927.,” 1991.
A. Slavík,Product integration, its history and applications. MatfyzpressPrague, 2007.
R. M. Corless, G. H. Gonnet, D. E. Hare, D. J. Jeffrey, and D. E. Knuth, “Onthe lambertw function,”Advances in Computational mathematics, vol. 5, no. 1,pp. 329–359, 1996.
N. Becker, B. B. GK, P. Caley,et al., “Using mathematical models to assessresponses to an outbreak of an emerged viral respiratory disease. final reportto the australian government department of health and ageing,”National Cen-tre for Epidemiology and Population Health. Canberra: Australian NationalUniversity, 2006.
K. Wang and J. Huang, “High order fast algorithm for the caputo fractionalderivative,”arXiv preprint arXiv:1705.06101, 2017.
M. A. Khan and A. Atangana, “Modeling the dynamics of novel coronavirus(2019-ncov) with fractional derivative,”Alexandria Engineering Journal, 2020.
A. S. Shaikh, I. N. Shaikh, and K. S. Nisar, “A mathematical model of covid-19using fractional derivative: Outbreak in india with dynamics of transmissionand control,” 2020.
H. Sun, W. Chen, H. Wei, and Y. Chen, “A comparative study of constant-order and variable-order fractional models in characterizing memory propertyof systems,”The European Physical Journal Special Topics, vol. 193, no. 1,p. 185, 2011.
E. C. de Oliveira,Solved Exercises in Fractional Calculus. Springer, 2019.
M. Ortigueira and J. Machado, “Fractional definite integral,”Fractal and Frac-tional, vol. 1, no. 1, p. 2, 2017.
A. Dokoumetzidis, R. Magin, and P. Macheras, “A commentary on fraction-alization of multi-compartmental models,”Journal of pharmacokinetics andpharmacodynamics, vol. 37, no. 2, pp. 203–207, 2010.
D. Rostamy and E. Mottaghi, “Numerical solution and stability analysis ofa nonlinear vaccination model with historical effects,”Hacettepe Journal ofMathematics and Statistics, vol. 47, no. 6, pp. 1478–1494, 2017.
I. Ameen,Fractional Calculus: Numerical Methods and SIR Models. PhDthesis, PhD thesis, University of Padova, Padova, Italy, 2017.
R. Almeida, “Analysis of a fractional seir model with treatment,”Applied Math-ematics Letters, vol. 84, pp. 56–62, 2018.
M. R. Islam, A. Peace, D. Medina, and T. Oraby, “Integer versus fractional or-der seir deterministic and stochastic models of measles,”International Journalof Environmental Research and Public Health, vol. 17, no. 6, p. 2014, 2020.
A. C. F. N. Gomes and A. De Cezaro, “Modelo sirc fracionário com múltiplasordens para influenza,”Scientia Plena, vol. 15, no. 4, 2019.
A. Ahmad, M. Farman, M. Ahmad, N. Raza, and M. Abdullah, “Dynamicalbehavior of sir epidemic model with non-integer time fractional derivatives: Amathematical analysis,”Int. J. Adv. Appl. Sci, vol. 5, no. 1, pp. 123–129, 2018.
M. Farman, A. Ahmad, H. Muslim, M. Ahmad,et al., “Dynamical behavior ofhepatitis b fractional-order model with modeling and simulation,”Journal ofBiochemical Technology, vol. 10, no. 3, p. 11, 2019.
S. Hasan, A. Al-Zoubi, A. Freihet, M. Al-Smadi, and S. Momani, “Solutionof fractional sir epidemic model using residual power series method,”AppliedMathematics and Information Sciences, vol. 13, no. 2, pp. 153–161, 2019.
A. Dokoumetzidis, R. Magin, and P. Macheras, “Fractional kinetics in multi-compartmental systems,”Journal of pharmacokinetics and pharmacodynamics,vol. 37, no. 5, pp. 507–524, 2010.
H. Bateman,Higher transcendental functions, vol. 3. McGraw-Hill Book Com-pany, 1953.
C. N. Angstmann, B. I. Henry, and A. V. McGann, “A fractional-order infectiv-ity sir model,”Physica A: Statistical Mechanics and its Applications, vol. 452,pp. 86–93, 2016.
J. Herskovits, “Feasible direction interior-point technique for nonlinear op-timization,”Journal of optimization theory and applications, vol. 99, no. 1,pp. 121–146, 1998.
IBGE, “Taxa bruta de mortalidade por mil habitantes – brasil –2000 a 2015.”https://brasilemsintese.ibge.gov.br/populacao/taxas-brutas-de-mortalidade.html, 2015.
Microsoft/Bing, “Covid-19-data,” 2020. [Online; accessed 29-July-2020].
Wikipedia contributors, “Demographics of italy — Wikipedia, the free ency-clopedia,” 2020. [Online; accessed 29-July-2020].
R. Ranjan, “Estimating the final epidemic size for covid-19 outbreak usingimproved epidemiological models,”medRxiv, 2020.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish in this journal agree to the following terms:
Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.
Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
author. This is in accordance with the BOAI definition of open access
Intellectual Property
All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.