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Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order

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  • Arenas, Abraham J.
  • González-Parra, Gilberto
  • Chen-Charpentier, Benito M.

Abstract

In this paper we construct nonstandard finite difference (NSFD) schemes to obtain numerical solutions of the susceptible–infected (SI) and susceptible–infected–recovered (SIR) fractional-order epidemic models. In order to deal with fractional derivatives we apply the Caputo operator and use the Grünwald–Letnikov method to approximate the fractional derivatives in the numerical simulations. According to the principles of dynamic consistency we construct NSFD schemes to numerically integrate the fractional-order epidemic models. These NSFD schemes preserve the positivity that other classical methods do not guarantee. Additionally, the NSFD schemes hold other conservation properties of the solution corresponding to the continuous epidemic model. We run some numerical comparisons with classical methods to test the behavior of the NSFD schemes using the short memory principle. We conclude that the NSFD schemes, which are explicit and computationally inexpensive, are reliable methods to obtain realistic positive numerical solutions of the SI and SIR fractional-order epidemic models.

Suggested Citation

  • Arenas, Abraham J. & González-Parra, Gilberto & Chen-Charpentier, Benito M., 2016. "Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 121(C), pages 48-63.
    • Handle: RePEc:eee:matcom:v:121:y:2016:i:c:p:48-63
    DOI: 10.1016/j.matcom.2015.09.001
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    1. Michele Caputo, 2014. "The role of memory in modeling social and economic cycles of extreme events," Chapters, in: Francesco Forte & Ram Mudambi & Pietro Maria Navarra (ed.), A Handbook of Alternative Theories of Public Economics, chapter 11, pages 245-259, Edward Elgar Publishing.
    2. Garrappa, Roberto, 2015. "Trapezoidal methods for fractional differential equations: Theoretical and computational aspects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 110(C), pages 96-112.
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    9. Khan, Aziz & Gómez-Aguilar, J.F. & Saeed Khan, Tahir & Khan, Hasib, 2019. "Stability analysis and numerical solutions of fractional order HIV/AIDS model," Chaos, Solitons & Fractals, Elsevier, vol. 122(C), pages 119-128.
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