Numerical Solution of the Spread Model of COVID-19 by Using Muntz Functions
الموضوعات : فصلنامه ریاضیAfkar Kareem Mnahi 1 , Majid Tavassoli Kajani 2 , Mohammed Jasim Mohammed 3 , Masoud Allame 4
1 - .Department of Mathematics, Islamic Azad University, Isfahan (Khorasgan) Branch, Isfahan, Iran.
2 - Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan, Iran.
3 - Department of Mathematics, University of Thi-Qar, Nasiriyah, 64001, Iraq.
4 - Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan, Iran.
الکلمات المفتاحية: Muntz function, Fractional Differential Equations, Collocation Method, Spread model of COVID- 19.,
ملخص المقالة :
The outbreak of COVID-19 has necessitated the development of various mathematical models to understand and predict its spread. Among these, fractional differential equations have gained attention for their ability to capture the complexity and dynamics of infectious disease transmission. However, obtaining analytical solutions for such models is often infeasible. In this paper, we present an approach to approximate solutions of a fractional differential equation that describes the spread of COVID-19. The fractional order in the model reflects the memory and hereditary properties of the disease transmission process, which are not adequately described by traditional integer-order models. To tackle the complexities of this equation, we utilize Muntz functions, which are a class of basis functions used in approximation theory. Muntz functions are particularly useful due to their flexible nature and ability to converge to various types of functions, making them suitable for approximating solutions to differential equations. We perform numerical simulations to evaluate the performance of the Muntz functions in approximating the solution of our model. The findings indicate that the Muntz function approach yields superior accuracy in modeling the spread of COVID-19 compared to these alternative methods.
[1] M. Goyal, H.M. Baskonus, A. Prakash, An efficient technique for a time fractional model of lassa hemorrhagic
fever spreading in pregnant women, Eur. Phys. J. Plus, 134(481) (2019) 1-10.
[2] W. Gao, P. Veeresha, D.G. Prakasha, H.M. Baskonus, G. Yel, New approach for the model describing the
deathly disease in pregnant women using Mittag-Leffler function, Chaos, Solitons Fract, 134 (2020), Article
109696.
[3] D. Kumar, J. Singh, M. Al-Qurashi, D. Baleanu, A new fractional SIRS-SI malaria disease model with application
of vaccines, anti-malarial drugs, and spraying, Adv. Diff. Eq., 278 (2019) 1-10.
[4] K. Shah, M.A. Alqudah, F. Jarad, T. Abdeljawad, Semi-analytical study of pine wilt disease model with convex
rate under Caputo-Febrizio fractional order derivative, Chaos, Solitons Fract, 135 (2020), Article 109754.
[5] M.R. Gandomani, M. Tavassoli Kajani, Numerical solution of a fractional order model of HIV infection of
CD4 + T cells using M¨untz-Legendre polynomials, International Journal Bioautomation, 20(2) (2016) 193-204.
[6] X. Tian, C. Li, A. Huang, S. Xia, S. Lu, Z. Shi, L. Lu, S. Jiang, Z. Yang, Y. Wu, T. Ying Potent binding
of 2019 novel coronavirus spike protein by a SARS coronavirus-specific human monoclonal antibody, Emerg
Microbes Infect, 9(1) (2020) 382-385.
[7] J.F.W. Chan, K.H. Kok, Z. Zhu, H. Chu, K.K.W. To, S. Yuan, K.Y. Yuen, Genomic characterization of
the 2019 novel human-pathogenic coronavirus isolated from a patient with atypical pneumonia after visiting
Wuhan, Emerg Microbes Infect, 9(1) (2020) 221-236.
[8] K. Shah, M. Arfan, I. Mahariq, A. Ahmadian, S. Salahshour, M. Ferrara, Fractal-fractional mathematical
model addressing the situation of corona virus in Pakistan, Result Phys., 19 (2020), Article 103560.
[9] M. Bahmanpour, M. Tavassoli-Kajani, M. Maleki, A M¨untz wavelets collocation method for solving fractional
differential equations, Computational and Applied Mathematics, 37(4) (2018) 5514-5526.
[10] Mohammad Maleki, Majid Tavassoli Kajani, Numerical approximations for Volterra’s population growth
model with fractional order via a multi-domain pseudospectral method, Applied Mathematical Modelling 39
(2015),4300-4308.
[11] Majid Tavassoli Kajani, Numerical solution of fractional pantograph equations via M¨untz-Legendre polynomials,
Mathematical science, (2023) 1-9.
[12] D. Shirani , M. Tavassoli Kajani, S. Salahshour , Numerical Solution of a SIR Fractional Model of the Distribution
of Computer Viruses Using Dickson Polynomials, Int. J. Industrial Mathematics , 13 (3), (2021)
323-331.
[13] M. R. Gandomani, M. T. Kajani, Numerical Solution of a Fractional Order Model of HIV Infection of CD4+T
Cells Using M¨untz-Legendre Polynomials, INT. J . BIO AUTOMATION, 20(2) (2016), 193-204.
[14] M. R. Gandomani , M. Tavassoli Kajani, Application of shifted M¨untz-Legender Polynomials for solving fractional
differential equations, International Journal of Pure and Applied Mathematics, 103 No. 2 (2015), 263-279.
[15] A. Saadatmandi, S. Akhlaghi, Using hybrid of Block-Pulse functions and Bernoulli polynomials to solve fractional
Fredholm-Volterra integro-differential equations, Sains Malaysiana, vol. 49, no. 4 (2020), 953-962.
[16] G. V. Milovanovi´c, M¨untz orthogonal polynomials and their numerical evaluation, Applications and Computation
of Orthogonal Polynomials, Springer, Oberwolfach Germany, 1998, 179-194.
[17] C. Canuto, M. Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods. Fundamentals in Single Domains,
Springer, Berlin Heidelberg New York (2006).
[18] S. Akhlaghi , M. Tavassoli Kajani , M. Allame, Numerical Solution of Fractional Order Integro-Differential
Equations via M¨untz Orthogonal Functions, Journal of Mathematics, 2023, 6647128, 1-13.
[19] Muhammad Arfan , Hussam Alrabaiah, Mati Ur Rahman , Yu-Liang Sun , Ahmad Sobri Hashim , Bruno A.
Pansera , Ali Ahmadian, Soheil Salahshour, Investigation of fractal-fractional order model of COVID-19 in
Pakistan under Atangana-Baleanu Caputo (ABC) derivative, Results in Physics, 24 (2021),104046.
