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A Comparative Study between Discrete Stochastic Arithmetic and Floating-Point Arithmetic to Validate the Results of Fractional Order Model of Malaria Infection

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  • Samad Noeiaghdam

    (Department of Applied Mathematics and Programming, South Ural State University, Lenin Prospect 76, 454080 Chelyabinsk, Russia
    Industrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, Russia)

  • Aliona Dreglea

    (Industrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, Russia)

  • Hüseyin Işık

    (Department of Engineering Science, Bandırma Onyedi Eylül University, Bandırma 10200, Balıkesir, Turkey)

  • Muhammad Suleman

    (Department of Mathematics, Comsats University Islamabad, Islamabad 45550, Pakistan)

Abstract

The researchers aimed to study the nonlinear fractional order model of malaria infection based on the Caputo-Fabrizio fractional derivative. The homotopy analysis transform method (HATM) is applied based on the floating-point arithmetic (FPA) and the discrete stochastic arithmetic (DSA). In the FPA, to show the accuracy of the method we use the absolute error which depends on the exact solution and a positive value ε . Because in real life problems we do not have the exact solution and the optimal value of ε , we need to introduce a new condition and arithmetic to show the efficiency of the method. Thus the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are applied. The CESTAC method is based on the DSA. Also, a new termination criterion is used which is based on two successive approximations. Using the CESTAC method we can find the optimal approximation, the optimal error and the optimal iteration of the method. The main theorem of the CESTAC method is proved to show that the number of common significant digits (NCSDs) between two successive approximations are almost equal to the NCSDs of the exact and approximate solutions. Plotting several graphs, the regions of convergence are demonstrated for different number of iterations k = 5, 10. The numerical results based on the simulated data show the advantages of the DSA in comparison with the FPA.

Suggested Citation

  • Samad Noeiaghdam & Aliona Dreglea & Hüseyin Işık & Muhammad Suleman, 2021. "A Comparative Study between Discrete Stochastic Arithmetic and Floating-Point Arithmetic to Validate the Results of Fractional Order Model of Malaria Infection," Mathematics, MDPI, vol. 9(12), pages 1-17, June.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:12:p:1435-:d:578288
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    References listed on IDEAS

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    1. Mittal, R.C. & Goel, Rohit & Ahlawat, Neha, 2021. "An Efficient Numerical Simulation of a Reaction-Diffusion Malaria Infection Model using B-splines Collocation," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
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    3. Naik, Parvaiz Ahmad & Zu, Jian & Owolabi, Kolade M., 2020. "Modeling the mechanics of viral kinetics under immune control during primary infection of HIV-1 with treatment in fractional order," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    4. Vignes, J., 1993. "A stochastic arithmetic for reliable scientific computation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 35(3), pages 233-261.
    5. Kim, Jung Eun & Choi, Yongin & Lee, Chang Hyeong, 2019. "Effects of climate change on Plasmodium vivax malaria transmission dynamics: A mathematical modeling approach," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 616-630.
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