Modeling and analysis of fractional order Zika model

Muhammad Farman; Ali Akgül; Sameh Askar; Thongchai Botmart; Aqeel Ahmad; Hijaz Ahmad; Muhammad Farman; Ali Akgül; Sameh Askar; Thongchai Botmart; Aqeel Ahmad; Hijaz Ahmad

doi:10.3934/math.2022216

AIMS Mathematics

2022, Volume 7, Issue 3: 3912-3938. doi: 10.3934/math.2022216

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Research article Special Issues

Modeling and analysis of fractional order Zika model

1.
Department of Mathematics and Statistics, University of Lahore, Lahore 54590, Pakistan
2.
Art and Science Faculty, Department of Mathematics, Siirt University, Siirt 56100, Turkey
3.
Department of Statistics and Operations Research, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
5.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy

Received: 26 September 2021 Accepted: 10 November 2021 Published: 10 December 2021
MSC : 37C75, 93B05, 65L07

We propose mathematical model for the transmission of the Zika virus for humans spread by mosquitoes. We construct a scheme for the Zika virus model with Atangna-Baleanue Caputo sense and fractal fractional operator by using generalized Mittag-Leffler kernel. The positivity and boundedness of the model are also calculated. The existence of uniquene solution is derived and stability analysis has been made for the model by using the fixed point theory. Numerical simulations are made by using the Atangana-Toufik scheme and fractal fractional operator with a different dimension of fractional values which support the theoretical outcome of the proposed system. Developed scheme including simulation will provide better understanding in future analysis and for control strategy regarding Zika virus.
- Zika virus,
- Atangana-Baleanu in Caputo sense,
- Atangana-Toufik,
- fractal fractional,
- Mittag-Leffler kernel
Citation: Muhammad Farman, Ali Akgül, Sameh Askar, Thongchai Botmart, Aqeel Ahmad, Hijaz Ahmad. Modeling and analysis of fractional order Zika model[J]. AIMS Mathematics, 2022, 7(3): 3912-3938. doi: 10.3934/math.2022216

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Abstract

We propose mathematical model for the transmission of the Zika virus for humans spread by mosquitoes. We construct a scheme for the Zika virus model with Atangna-Baleanue Caputo sense and fractal fractional operator by using generalized Mittag-Leffler kernel. The positivity and boundedness of the model are also calculated. The existence of uniquene solution is derived and stability analysis has been made for the model by using the fixed point theory. Numerical simulations are made by using the Atangana-Toufik scheme and fractal fractional operator with a different dimension of fractional values which support the theoretical outcome of the proposed system. Developed scheme including simulation will provide better understanding in future analysis and for control strategy regarding Zika virus.

References

[1]	V. Sikka, V. K. Chattu, R. K. Popli, S. C. Galwankar, D. Kelkar, S. G. Sawicki, et al., The emergence of Zika virus as a global health security threat: A review and a consensus statement of the INDUSEM Joint Working Group (JWG), J. Glob. Infect. Dis., 8 (2016), 3-15. doi: 10.4103/0974-777X.176140. doi: 10.4103/0974-777X.176140
[2]	M. Z. Mehrjardi, Is Zika virus an Emerging TORCH agent? An invited commentary, Virology: Res. Treat., 8 (2017), 1-3. doi: 10.1177/1178122X17708993. doi: 10.1177/1178122X17708993
[3]	D. M. Knipe, P. M. Howley, Fields virology, 5 Eds., Lippincott Williams & Wilkins, 1156 (2017), 1199.
[4]	E. B. Hayes, Zika virus outside Africa, Emerg. Infect. Dis., 15 (2009), 1347-1350. doi: 10.3201/eid1509.090442. doi: 10.3201/eid1509.090442
[5]	D. Baleanu, A. Mousalou, S. Rezapour, On the existence of solutions for some infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential equations, Bound. Value Probl., 145 (2017). doi: 10.1186/s13661-017-0867-9. doi: 10.1186/s13661-017-0867-9
[6]	D. Baleanu, S. Etemad, S. Rezapour, On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators, Alex. Eng. J., 59 (2020), 3019-3027. doi: 10.1016/j.aej.2020.04.053. doi: 10.1016/j.aej.2020.04.053
[7]	D. Baleanu, Z. Nazemi, S. Rezapour, Attractivity for a k-dimensional system of fractional functional differential equations and global attractively for a k-dimensional system of nonlinear fractional differential equations, J. Inequal. Appl., 31 (2014). doi: 10.1186/1029-242X-2014-31. doi: 10.1186/1029-242X-2014-31
[8]	F. Mainardi, Fractional calculus: Theory and applications, Mathematics, 6 (2018), 145. doi: 10.3390/math6090145. doi: 10.3390/math6090145
[9]	M. A. C. Pinto, J. A. T. Machado, Fractional dynamics of computer virus propagation, Math. Probl. Eng., 2014 (2014). doi: 10.1155/2014/476502. doi: 10.1155/2014/476502
[10]	E. K. Akgul, Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives, Chaos, 29 (2019), 023108. doi: 10.1063/1.5084035. doi: 10.1063/1.5084035
[11]	R. Hilfer, Applications of fractional calculus in physics, World Scientific, USA, 2001.
[12]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
[13]	H. Bulut, H. M. Baskonus, F. B. M. Belgacem, The analytical solutions of some fractional ordinary differential equations by sumudu transform method, Abst. Appl. Anal., 2013 (2013). doi: 10.1155/2013/203875. doi: 10.1155/2013/203875
[14]	A. Atangana, B. T. Alkahtani, Analysis of the Keller-Segel model with a fractional derivative without singular kernel, Entropy, 17 (2015), 4439-4453. doi: 10.3390/e17064439. doi: 10.3390/e17064439
[15]	A. Atangana, B. T. Alkahtani, Analysis of non-homogenous heat model with new trend of derivative with fractional order, Chaos Soliton. Fract., 89 (2016), 566-571. doi: 10.1016/j.chaos.2016.03.027. doi: 10.1016/j.chaos.2016.03.027
[16]	A. Atangana, A. Akgul, Can transfer function and Bode diagram be obtained from Sumudu transform, Alex. Eng. J., 59 (2020), 1971-1984. doi: 10.1016/j.aej.2019.12.028. doi: 10.1016/j.aej.2019.12.028
[17]	D. Kumar, J. Singh, D. Baleanu, A hybrid computational approach for Klein-Gordon equations on Cantor sets, Nonlinear Dyn., 87 (2017), 511-517. doi: 10.1007/s11071-016-3057-x. doi: 10.1007/s11071-016-3057-x
[18]	M. Farman, A. Ahmad, A. Akgul, M. U. Saleem, M. Naeem, D. Baleanue, Epidemiological analysis of the coronavirus disease outbreak with random effects, CMC-Comput. Mater. Con., 67 (2021), 3215-3227.
[19]	H. Ahmad, N. Alam, M. Omri, New computational results for a prototype of an excitable system, Results Phys., 28 (2021). doi: 10.1016/j.rinp.2021.104666. doi: 10.1016/j.rinp.2021.104666
[20]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 1-13.
[21]	S. Javeed, S. Anjum, K. S. Alimgeer, M. Atif, S. W. Yao, W. A. Farooq, et al., A novel mathematical model for COVID-19 with remedial strategies, Results Phys., 27 (2021). doi: 10.1016/j.rinp.2021.104248. doi: 10.1016/j.rinp.2021.104248
[22]	M. Farman, A. Ahmad, A. Akgul, M. U. Saleem, M. Rizwan, M. O Ahmad, A mathematical analysis and simulation for Zika virus model with time fractional derivative, Math. Method. Appl. Sci., 2020 (2020), 1-12. doi: 10.1002/mma.6891. doi: 10.1002/mma.6891
[23]	I. E. Kibona, C. H. Yang, SIR model of spread of Zika virus infections: Zikv linked to microcephaly simulations, Health, 9 (2017), 1190-1210. doi: 10.4236/health.2017.98086. doi: 10.4236/health.2017.98086
[24]	A. Maysaroh, S. B. Waluya, W. Wuryanto, Analysis and simulation model mathematical model of Zika disease with one serotype virus Zika, Unnes J. Math., 8 (2019), 56-71. doi: 10.15294/ujm.v8i1.23297. doi: 10.15294/ujm.v8i1.23297
[25]	B. S. T. Alkahtani, A. Atangana, I. Koca, Novel analysis of the fractional Zika model using the Adams typepredictor-corrector rule for non-singular and non-local fractional operators, J. Nonlinear Sci. Appl., 10 (2017), 3191-3200. doi: 10.22436/jnsa.010.06.32. doi: 10.22436/jnsa.010.06.32
[26]	S. Rezapour, H. Mohammadi, A. Jajarmi, A new mathematical model for Zika virus transmission, Adv. Differ. Equ., 589 (2020). doi: 10.1186/s13662-020-03044-7. doi: 10.1186/s13662-020-03044-7
[27]	M. Toufik, A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, Eur. Phys. J. Plus, 132 (2017), 444. doi: 10.1140/epjp/i2017-11717-0. doi: 10.1140/epjp/i2017-11717-0
[28]	M. Higazy, F. M. Allehiany, E. E. Mahmoud, Numerical study of fractional order COVID-19 pandemic transmission model in context of ABO blood group, Results Phys., 22 (2021), 103852. doi: 10.1016/j.rinp.2021.103852. doi: 10.1016/j.rinp.2021.103852
[29]	M. Higazy, E. E. Mahmoud, E. M. Khalil, S. Abdel-Khalek, S. M. Abo-Dahab, H. Alotaibi, Dynamics and robust control of a new realizable chaotic nonlinear model, Complexity, 17 (2021). doi: 10.1155/2021/6692369. doi: 10.1155/2021/6692369
[30]	A. M. S. Mahdy, M. S. Mohamed, K. A. Gepreel, A. AL-Amiri, M. Higazy, Dynamical characteristics and signal flow graph of nonlinear fractional smoking mathematical model, Chaos Soliton. Fract., 141 (2020), 110308. doi: 10.1016/j.chaos.2020.110308. doi: 10.1016/j.chaos.2020.110308
[31]	E. E. Mahmoud, M. Higazy, O. A. Althagafi, A novel strategy for complete and phase robust synchronizations of chaotic nonlinear systems, Symmetry, 12 (2020), 1765. doi: 10.3390/sym12111765. doi: 10.3390/sym12111765
[32]	K. A. Gepreel, M. Higazy, A. M. S. Mahdy, Optimal control, signal flow graph, and system electronic circuit realization for nonlinear Anopheles mosquito model, Int. J. Mod. Phys. C, 31 (2020), 2050130. doi: 10.1142/S0129183120501302. doi: 10.1142/S0129183120501302
[33]	M. Higazy, M. A. Alyami, New Caputo-Fabrizio fractional order SEIASqEqHR model for COVID-19 epidemic transmission with genetic algorithm based control strategy, Alex. Eng. J., 59 (2020), 4719-4736. doi: 10.1016/j.aej.2020.08.034. doi: 10.1016/j.aej.2020.08.034
[34]	M. Higazy, Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic, Chaos Soliton. Fract., 138 (2020), 110007. doi: 10.1016/j.chaos.2020.110007. doi: 10.1016/j.chaos.2020.110007
[35]	A. Mahdy, M. Higazy, Numerical different methods for solving the nonlinear biochemical reaction model, Int. J. Appl. Comput. Math., 5 (2019), 148. doi: 10.1007/s40819-019-0740-x. doi: 10.1007/s40819-019-0740-x
[36]	E. E Mahmoud, M. Higazy, T. M. Al-Harthi, A new nine-dimensional chaotic Lorenz system with quaternion variables: Complicated dynamics, electronic circuit design, anti-anticipating synchronization, and chaotic masking communication application, Mathematics, 7 (2019), 877. doi: 10.3390/math7100877. doi: 10.3390/math7100877
[37]	E. E. Mahmoud, M. Higazy, A. Hammad, S. M. Abo-Dahab, S. Abdel-Khalek, E. M. Khalil, Quaternion anti-synchronization of a novel realizable fractional chaotic model, Chaos Soliton. Fract., 144 (2021), 110715. doi: 10.1016/j.chaos.2021.110715. doi: 10.1016/j.chaos.2021.110715
[38]	Z. Memon, S. Qureshi, B. R. Memon, Assessing the role of quarantine and isolation as control strategies for COVID-19 outbreak: A case study, Chaos Soliton. Fract., 144 (2021), 110655. doi: 10.1016/j.chaos.2021.110655. doi: 10.1016/j.chaos.2021.110655
[39]	S. Qureshi, M. M. Chang, A. A. Shaikh, Analysis of series RL and RC circuits with time-invariant source using truncated M, atangana beta and conformable derivatives, J. Ocean Eng. Sci., 6 (2021), 217-227. doi: 10.1016/j.joes.2020.11.006. doi: 10.1016/j.joes.2020.11.006
[40]	S. Qureshi, R. Jan, Modeling of measles epidemic with optimized fractional order under Caputo differential operator, Chaos Soliton. Fract., 145 (2021), 110766. doi: 10.1016/j.chaos.2021.110766. doi: 10.1016/j.chaos.2021.110766

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