Approximate solution for the nonlinear fractional order mathematical model

Kahkashan Mahreen; Qura Tul Ain; Gauhar Rahman; Bahaaeldin Abdalla; Kamal Shah; Thabet Abdeljawad; Kahkashan Mahreen; Qura Tul Ain; Gauhar Rahman; Bahaaeldin Abdalla; Kamal Shah; Thabet Abdeljawad

doi:10.3934/math.20221057

AIMS Mathematics

2022, Volume 7, Issue 10: 19267-19286. doi: 10.3934/math.20221057

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Approximate solution for the nonlinear fractional order mathematical model

1.
Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan
2.
Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, China
3.
Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan
4.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia
5.
Department of Mathematics, University of Malakand, Chakdara Dir(L), 18000, Khyber Pakhtunkhwa, Pakistan
6.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 21 June 2022 Revised: 16 August 2022 Accepted: 22 August 2022 Published: 30 August 2022
MSC : 26A33, 34A08, 92B05

Health organizations are working to reduce the outbreak of infectious diseases with the help of several techniques so that exposure to infectious diseases can be minimized. Mathematics is also an important tool in the study of epidemiology. Mathematical modeling presents mathematical expressions and offers a clear view of how variables and interactions between variables affect the results. The objective of this work is to solve the mathematical model of MERS-CoV with the simplest, easiest and most proficient techniques considering the fractional Caputo derivative. To acquire the approximate solution, we apply the Adomian decomposition technique coupled with the Laplace transformation. Also, a convergence analysis of the method is conducted. For the comparison of the obtained results, we apply another semi-analytic technique called the homotopy perturbation method and compare the results. We also investigate the positivity and boundedness of the selected model. The dynamics and solution of the MERS-CoV compartmental mathematical fractional order model and its transmission between the human populace and the camels are investigated graphically for $ \theta = 0.5, \, 0.7, \, 0.9, \, 1.0 $. It is seen that the recommended schemes are proficient and powerful for the given model considering the fractional Caputo derivative.
- Adomian decomposition method,
- convergent series,
- Laplace transformation,
- Caputo fractional derivative,
- homotopy perturbation method,
- invariant set
Citation: Kahkashan Mahreen, Qura Tul Ain, Gauhar Rahman, Bahaaeldin Abdalla, Kamal Shah, Thabet Abdeljawad. Approximate solution for the nonlinear fractional order mathematical model[J]. AIMS Mathematics, 2022, 7(10): 19267-19286. doi: 10.3934/math.20221057

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Abstract

Health organizations are working to reduce the outbreak of infectious diseases with the help of several techniques so that exposure to infectious diseases can be minimized. Mathematics is also an important tool in the study of epidemiology. Mathematical modeling presents mathematical expressions and offers a clear view of how variables and interactions between variables affect the results. The objective of this work is to solve the mathematical model of MERS-CoV with the simplest, easiest and most proficient techniques considering the fractional Caputo derivative. To acquire the approximate solution, we apply the Adomian decomposition technique coupled with the Laplace transformation. Also, a convergence analysis of the method is conducted. For the comparison of the obtained results, we apply another semi-analytic technique called the homotopy perturbation method and compare the results. We also investigate the positivity and boundedness of the selected model. The dynamics and solution of the MERS-CoV compartmental mathematical fractional order model and its transmission between the human populace and the camels are investigated graphically for $ \theta = 0.5, \, 0.7, \, 0.9, \, 1.0 $. It is seen that the recommended schemes are proficient and powerful for the given model considering the fractional Caputo derivative.

References

[1]	E. I. Azhar, S. A. El-Kafrawy, S. A. Farraj, A. M. Hassan, M. S. Al-Saeed, A. M. Hashem, et al., Evidence for camel-to-human transmission of MERS coronavirus, New Engl. J. Med., 370 (2014), 2499–2505. https://doi.org/10.1056/NEJMoa1401505 doi: 10.1056/NEJMoa1401505
[2]	H. R. Thieme, Mathematics in Population Biology, Princeton University Press, New York, 2018. https://press.princeton.edu/books/paperback/9780691092911/mathematics-in-population-biology
[3]	K. Yunhwan, L. Sunmi, C. Chaeshin, C. Seoyun, H. Saeme, S. Youngseo, The characteristics of Middle Eastern Respiratory Syndrome Coronavirus transmission dynamics in South Korea, Osong Public Health Res. Perspect, 7 (2016), 49–55. https://doi.org/10.1016/j.phrp.2016.01.001 doi: 10.1016/j.phrp.2016.01.001
[4]	J. A. AlTawfiq, K. Hinedi, J. Ghandour, Middle East respiratory syndrome coronavirus: A case-control study of hospitalized patients, Clin. Infect. Dis., 59 (2014), 160–165. https://doi.org/10.1093/cid/ciu226 doi: 10.1093/cid/ciu226
[5]	Y. M. Arabi, A. A. Arifi, H. H. Balkhy, Clinical course and outcomes of critically ill patients with Middle East respiratory syndrome corona virus infection, Ann. Intern. Med., 160 (2014), 389–97. https://doi.org/10.7326/M13-2486 doi: 10.7326/M13-2486
[6]	A. N. Alagaili, T. Briese, N. Mishra, V. Kapoor, S. C. Sameroff, E. de Wit, et al., Middle East respiratory syndrome coronavirus infection in dromedary camels in Saudi Arabia, MBio, 5 (2014), e00884–14, https://doi.org/10.1128/mBio.00884-14 doi: 10.1128/mBio.00884-14
[7]	D. Anwarud, Y. Li, M. A. Shah, The complex dynamics of hepatitis B infected individuals with optimal control, J. Syst. Sci. Complex., 34 (2021), 1301–1323. https://doi.org/10.1007/s11424-021-0053-0 doi: 10.1007/s11424-021-0053-0
[8]	C. Poletto, C. Pelat, D. Levy-Bruhl, Y. Yazdanpanah, P. Y. Boelle, V. Colizza, Assessment of the Middle East respiratory syndrome coronavirus (MERS-CoV) epidemic in the Middle East and risk of international spread using a novel maximum likelihood analysis approach, Eurosurveillance, 19 (2014), 20824. https://doi.org/10.2807/1560-7917.ES2014.19.23.20824 doi: 10.2807/1560-7917.ES2014.19.23.20824
[9]	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 (2019), 1–10. https://doi.org/10.1140/epjp/i2019-12854-0 doi: 10.1140/epjp/i2019-12854-0
[10]	M. Z. Ullah, A. K. Alzahrani, D. Baleanu, An efficient numerical technique for a new fractional tuberculosis model with nonsingular derivative operator, Taibah Univ. Sci., 13 (2019), 1147–1157. https://doi.org/10.1080/16583655.2019.1688543 doi: 10.1080/16583655.2019.1688543
[11]	H. Jafari, C. M. Khalique, M. Khan, M. A. Ghasemi, Two-step Laplace decomposition method for solving nonlinear partial differential equations, Int. J. Phys. Sci., 6 (2011), 4102–4109. DOI: 10.5897/IJPS11.146 doi: 10.5897/IJPS11.146
[12]	S. J. Johnston, H. Jafari, S. P. Moshokoa, V. M. Ariyan, D. Baleanu, Laplace homotopy perturbation method for Burgers equation with space and time-fractional order, Open Phys., 14 (2016), 247–252. https://doi.org/10.1515/phys-2016-0023 doi: 10.1515/phys-2016-0023
[13]	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 Soliton. Fract., 135 (2020), 109754. https://doi.org/10.1016/j.chaos.2020.109754 doi: 10.1016/j.chaos.2020.109754
[14]	R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. https://www.worldscientific.com/worldscibooks/10.1142/3779
[15]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, Amsterdam, 2006. https://www.elsevier.com/books/theory-and-applications-of-fractional-differential-equations/kilbas/978-0-444-51832-3
[16]	F. Liu, K. Burrage, Novel techniques in parameter estimation for fractional dynamical models arising from biological systems, Comput. Math. Appl., 62 (2011), 822–833. https://doi.org/10.1016/j.camwa.2011.03.002 doi: 10.1016/j.camwa.2011.03.002
[17]	F. Meral, T. Royston, R. Magin, Fractional calculus in viscoelasticity: An experimental study, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 939–945. https://doi.org/10.1016/j.cnsns.2009.05.004 doi: 10.1016/j.cnsns.2009.05.004
[18]	K. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Softw., 41 (2010), 9–12. https://doi.org/10.1016/j.advengsoft.2008.12.012 doi: 10.1016/j.advengsoft.2008.12.012
[19]	N. Anjum, J. H. He, Q. T. Ain, D. Tian, Li-He's modified homotopy perturbation method for doubly-clamped electrically actuated microbeams-based microelectromechanical system, Facta Univ. Ser.: Mech. Eng., 19 (2021), 601–612. https://doi.org/10.22190/FUME210112025A doi: 10.22190/FUME210112025A
[20]	Q. Ain, J. H. He, On two-scale dimension and its applications, Therm. Sci., 23 (2019), 1707–1712. https://doi.org/10.2298/TSCI190408138A doi: 10.2298/TSCI190408138A
[21]	Q. Ain, J. H. He, N. Anjum, M. Ali, The Fractional complex transform: A novel approach to the time-fractional Schrodinger equation, Fractals, 29 (2020). DOI:10.1142/S0218348X2150002X doi: 10.1142/S0218348X2150002X
[22]	S. C. Brailsford, P. R. Harper, B. Patel, N. Pitt, An analysis of the academic literature on simulation and modeling in health care, J. Simul., 3 (2009), 130–140. https://doi.org/10.1007/978-1-137-57328-5_11 doi: 10.1007/978-1-137-57328-5_11
[23]	J. Rappaz, R. Touzani, On a two-dimensional magnetohydrodynamic problem: Modeling and analysis, Esaim. Math. Model Numer. Anal., 26 (1992), 347–364. http://www.numdam.org/item?id=M2AN_1992_26_2_347_0
[24]	D. Baleanu, S. M. Aydogn, H. Mohammadi, S. Rezapour, On modeling of epidemic childhood diseases with the Caputo-Fabrizio derivative by using the Laplace Adomian decomposition method, Alex. Eng. J., 59 (2020), 3029–3039. https://doi.org/10.1016/j.aej.2020.05.007 doi: 10.1016/j.aej.2020.05.007
[25]	S. M. Aydogan, D. Baleanu, H. Mohammadi, S. Rezapour, On the mathematical model of Rabies by using the fractional Caputo-Fabrizio derivative, Adv. Diff. Equ., 2020 (2020), 1–21. https://doi.org/10.1186/s13662-020-02798-4 doi: 10.1186/s13662-020-02798-4
[26]	M. Rezapour, H. Mohammadi, M. E. Samei, SEIR epidemic model for COVID-19 transmission by Caputo derivative of fractional order, Adv. Diff. Equ., 2020 (2020), 1–19. https://doi.org/10.1186/s13662-020-02952-y doi: 10.1186/s13662-020-02952-y
[27]	V. Soukhovolsky, A. Kovalev, A. Pitt, B. Kessel, A new modeling of the COVID 19 pandemic, Chaos, Soliton. Fract., 139 (2020), 110039. https://doi.org/10.1016/j.chaos.2020.110039 doi: 10.1016/j.chaos.2020.110039
[28]	N. H. Aljahdaly, R. P. Agarwal, R. Shah, T. Botmart, Analysis of the time fractional-order coupled burgers equations with non-singular kernel operators, Mathematics, 9 (2021), p.2326. https://doi.org/10.3390/math9182326 doi: 10.3390/math9182326
[29]	M. Alesemi, N. Iqbal, T. Botmart, Novel analysis of the fractional-order system of non-linear partial differential equations with the exponential-decay kernel, Mathematics, 10 (2022), p.615. https://doi.org/10.3390/math10040615 doi: 10.3390/math10040615
[30]	N. Iqbal, T. Botmart, W. W. Mohammed, A. Ali, Numerical investigation of fractional-order Kersten–Krasil'shchik coupled KdV–mKdV system with Atangana–Baleanu derivative, Adv. Cont. Discr. Mod., 2022 (2022), 1–20. https://doi.org/10.1186/s13662-022-03709-5 doi: 10.1186/s13662-022-03709-5
[31]	Q. Ain, N. Anjum, A. Din, A. Zeb, S. Djilali, Z. A. Khan, On the analysis of Caputo fractional order dynamics of Middle East Lungs Coronavirus (MERS-CoV) Model, Alex. Eng. J., 61 (2022), 5123–5131. doi: https://doi.org/10.1016/j.aej.2021.10.016 doi: 10.1016/j.aej.2021.10.016
[32]	V. J. Prajapati, R. Meher, A robust analytical approach to the generalized Burgers Fisher equation with fractional derivatives including singular and non-singular kernels, J. Ocean Eng. Sci., (2022). https://doi.org/10.1016/j.joes.2022.06.035 doi: 10.1016/j.joes.2022.06.035
[33]	L. Verma, R. Meher, Z. Avazzadeh, O. Nikan, Solution for generalized fuzzy fractional Kortewege-de Varies equation using a robust fuzzy double parametric approach, J. Ocean Eng. Sci., (2022). https://doi.org/10.1016/j.joes.2022.04.026 doi: 10.1016/j.joes.2022.04.026
[34]	P. P. Sartanpara, R. Meher, S. K. Meher, The generalized time-fractional Fornberg-Whitham equation: An analytic approach, Partial Differential Equations Appl. Math., 5 (2022), p.100350. https://doi.org/10.1016/j.padiff.2022.100350 doi: 10.1016/j.padiff.2022.100350
[35]	K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Willy, New York, 1993. https://www.amazon.com/Introduction-Fractional-Calculus-Differential-Equations/dp/0471588849
[36]	J. Biazar, Solution of the epidemic model by Adomian decomposition method, Appl. Math. Comput., 137 (2006), 1101–1106. https://doi.org/10.1016/j.amc.2005.04.036 doi: 10.1016/j.amc.2005.04.036
[37]	K. Shah, H. Khalil, R. A. Khan, Analytical solutions of fractional order diffusion equations by natural transform method, Iran. J. Sci. Technol. Trans. A: Sci., 42 (2018), 1479–1490. https://doi.org/10.1007/s40995-016-0136-2 doi: 10.1007/s40995-016-0136-2
[38]	A. Abdelrazec, D. Pelinovsky, Convergence of the Adomian decomposition method for initial value problems, Numer. Methods Partial Differ. Equ., 27 (2011), 749–66.
[39]	A. Naghipour, J. Manafian, Application of the Laplace Adomian decomposition and implicit methods for solving Burgers' equation, TWMS J. Pure Appl. Math., 6 (2015), 68–77.
[40]	Z. Ahmad, S. A. El-Kafrawy, T. A. Alandijany, F. Giannino, A. A. Mirza, M. M. El-Daly, A. A. Faizo, L. Bajrai, M. A. Kamal, E. I. Azhar, A global report on the dynamics of COVID-19 with quarantine and hospitalization: A fractional order model with non-local kernel, Alex. Eng. J., 61 (2022), 8859–8874. https://doi.org/10.1016/j.compbiolchem.2022.107645 doi: 10.1016/j.compbiolchem.2022.107645
[41]	M. Rafei, D. D. Ganji, H. Daniali, Solution of the epidemic model by homotopy perturbation method, Appl. Math. Comput., 187 (2007), 1056–1062. https://doi.org/10.1016/j.amc.2006.09.019 doi: 10.1016/j.amc.2006.09.019
[42]	Y. Liu, Z. Li, Y. Zhang, Homotopy perturbation method to fractional biological population equation, Fract. Differ. Calc., 1 (2011), 117–124. https://doi.org/10.7153/fdc-01-07 doi: 10.7153/fdc-01-07
[43]	H. Afari Khalique, M. Nazari, Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion-wave equations, Appl. Math. Lett., 24 (2011), 1799–1805. https://doi.org/10.1016/j.aml.2011.04.037 doi: 10.1016/j.aml.2011.04.037
[44]	J. H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. Math. Comput., 151 (2004), 287–292. https://doi.org/10.1016/S0096-3003(03)00341-2 doi: 10.1016/S0096-3003(03)00341-2

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