Numerical Treatment On SEIR Problem With Temporal Fractal Fractional Derivatives

Jing Ye; Shengyi Zhou

doi:10.6180/jase.202308_26(8).0009

Numerical Treatment On SEIR Problem With Temporal Fractal Fractional Derivatives

Physics

Jing YeThis email address is being protected from spambots. You need JavaScript enabled to view it. and Shengyi Zhou

Sichuan Vocational College of Chemical Technology, Luzhou, Sichuan, 646300, China

Received: May 23, 2022
Accepted: September 8, 2022
Publication Date: November 2, 2022

Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

Download Citation: ||https://doi.org/10.6180/jase.202308_26(8).0009

ABSTRACT

In this work, a well-known epidemic SEIR model is considered with the fractal-fractional operator in the frame of the Atangana-Baleanu derivative. Moreover, Using the theorems of Schauders fixed point and Banach fixed, existence theory is practiced to guarantee that there are solutions to the model. Approximate solutions to the problem are presented using the Atangana-Toufik scheme. Also, 2D graphs of solutions for different fractional orders are shown. Along with chaotic behavior of results for each case are investigated.

Keywords: Mittag-Leffler kernel; numerical method; fractional SEIR epidemic model

REFERENCES

[1] S. Ahmad, A. Ullah, M. Partohaghighi, S. Saifullah, A. Akgul, and F. Jarad, (2021) “Oscillatory and complex behaviour of Caputo-Fabrizio fractional order HIV-1 infection model" AIMS Math 7(3): 4778–4792. DOI:10.3934/math.2022265.
[2] I. Zada, M. Naeem Jan, N. Ali, D. Alrowail, K. Sooppy Nisar, and G. Zaman, (2021) “Mathematical analysis of hepatitis B epidemic model with optimal control" Advances in Difference Equations 2021(1): 1–29. DOI:10.1186/s13662-021-03607-2.
[3] T. S. Shaikh, N. Fayyaz, N. Ahmed, N. Shahid, M. Rafiq, I. Khan, and K. S. Nisar, (2021) “Numerical study for epidemic model of hepatitis-B virus" The European Physical Journal Plus 136(4): 1–22. DOI: 10.1140/epjp/s13360-021-01248-8.
[4] L. Xuan, S. Ahmad, A. Ullah, S. Saifullah, A. Akgul, and H. Qu, (2022) “Bifurcations, stability analysis and complex dynamics of Caputo fractal-fractional cancer model" Chaos, Solitons & Fractals 159: 112113. DOI: 10.1016/j.chaos.2022.112113.
[5] R. T. Alqahtani, S. Ahmad, and A. Akgul, (2021) “Dynamical analysis of bio-ethanol production model under generalized nonlocal operator in Caputo sense" Mathematics 9(19): 2370. DOI: 10.3390/math9192370.
[6] C. Xu, S. Saifullah, A. Ali, et al., (2022) “Theoretical and numerical aspects of Rubella disease model involving fractal fractional exponential decay kernel" Results in Physics 34: 105287. DOI: 10.1016/j.rinp.2022.105287.
[7] M. Higazy, S. A. Alsallami, S. Abdel-Khalek, and A. El-Mesady, (2022) “Dynamical and structural study of a generalized Caputo fractional order Lotka-Volterra model" Results in Physics 37: 105478. DOI: 10.1016/j.rinp.2022.105478.
[8] O. J. Peter, A. S. Shaikh, M. O. Ibrahim, K. S. Nisar, D. Baleanu, I. Khan, and A. I. Abioye, (2021) “Analysis and dynamics of fractional order mathematical model of COVID-19 in Nigeria using atangana-baleanu operator": DOI: 10.32604/cmc.2020.012314.
[9] A. Mezouaghi, S. Djillali, A. Zeb, and K. S. Nisar, (2022) “Global proprieties of a delayed epidemic model with partial susceptible protection" Mathematical Biosciences and Engineering 19(1): 209–224. DOI: 10.3934/mbe.2022011.
[10] M. Hashemi, (2018) “Invariant subspaces admitted by fractional differential equations with conformable derivatives" Chaos, Solitons & Fractals 107: 161–169. DOI: 10.1016/j.chaos.2018.01.002.
[11] H. Rezazadeh, (2018) “New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity" Optik 167: 218–227. DOI: 10.1016/j.ijleo.2018.04.026.
[12] S. Kheybari, M. T. Darvishi, and M. S. Hashemi, (2020) “A semi-analytical approach to Caputo type timefractional modified anomalous sub-diffusion equations" Applied Numerical Mathematics 158: 103–122. DOI: 10.1016/j.apnum.2020.07.023.
[13] M. Hashemi, (2021) “A novel approach to find exact solutions of fractional evolution equations with non-singular kernel derivative" Chaos, Solitons & Fractals 152: 111367. DOI: 10.1016/j.chaos.2021.111367.
[14] M. Hashemi, M. ˙Inç, and M. Bayram, (2019) “Symmetry properties and exact solutions of the time fractional Kolmogorov-Petrovskii-Piskunov equation" Revista mexicana de fisica 65(5): 529–535. DOI: 10.31349/RevMexFis.65.529.
[15] H. Rezazadeh, A. Zafar, M. Hashemi, and E. Tala-Tebue, (2020) “New exact solution of the conformable Gilson–Pickering equation using the new modified Kudryashov’s method" International Journal of Modern Physics B 34(18): 2050161. DOI: 10.1142/S0217979220501611.
[16] H. Rezazadeh, D. Kumar, T. A. Sulaiman, and H. Bulut, (2019) “New complex hyperbolic and trigonometric solutions for the generalized conformable fractional Gardner equation" Modern Physics Letters B 33(17): 1950196. DOI: 10.1142/S0217984919501963.
[17] S. Pashayi, M. S. Hashemi, and S. Shahmorad, (2017) “Analytical lie group approach for solving fractional integro-differential equations" Communications in Nonlinear Science and Numerical Simulation 51: 66–77. DOI: 10.1016/j.cnsns.2017.03.023.
[18] H. Aminikhah, A. R. Sheikhani, and H. Rezazadeh, (2015) “Exact solutions for the fractional differential equations by using the first integral method" Nonlinear engineering 4(1): 15–22. DOI: 10.1515/nleng-2014-0018.
[19] M. Eslami and H. Rezazadeh, (2016) “The first integral method for Wu–Zhang system with conformable timefractional derivative" Calcolo 53(3): 475–485. DOI: 10.1007/s10092-015-0158-8.
[20] I. Pan and S. Das. Intelligent fractional order systems and control: an introduction. 438. Springer, 2012.
[21] M. S. Hashemi and D. Baleanu. Lie symmetry analysis of fractional differential equations. Chapman and Hall/CRC, 2020.
[22] M. Hashemi and D. Baleanu, (2016) “Lie symmetry analysis and exact solutions of the time fractional gas dynamics equation":
[23] A. Momoh, M. Ibrahim, I. Uwanta, and S. Manga, (2013) “Mathematical model for control of measles epidemiology" International Journal of Pure and Applied Mathematics 87(5): 707–718. DOI: 10.12732/ijpam.v87i5.4.
[24] O. A. Arqub and A. El-Ajou, (2013) “Solution of the fractional epidemic model by homotopy analysis method" Journal of King Saud University-Science 25(1): 73–81. DOI: 10.1016/j.jksus.2012.01.003.
[25] Ritu and Y. Gupta, (2021) “Numerical analysis approach for models of COVID-19 and other epidemics" International Journal of Modeling, Simulation, and Scientific Computing 12(03): 2041003. DOI: 10.1142/S1793962320410032.
[26] S. Saifullah, A. Ali, K. Shah, and C. Promsakon, (2022) “Investigation of fractal fractional nonlinear Drinfeld–Sokolov–Wilson system with non-singular operators" Results in Physics 33: 105145. DOI: 10.1016/j.rinp.2021.105145.
[27] Y.-K. Ma,W. K.Williams, V. Vijayakumar, K. S. Nisar, and A. Shukla, (2022) “Results on Atangana-Baleanu fractional semilinear neutral delay integro-differential systems in Banach space" Journal of King Saud University-Science 34(6): 102158. DOI: 10.1016/j.jksus.2022.102158.
[28] C. Dineshkumar, R. Udhayakumar, V. Vijayakumar, K. S. Nisar, A. Shukla, A.-H. Abdel-Aty, M. Mahmoud, and E. E. Mahmoud, (2022) “A note on existenceand approximate controllability outcomes of Atangana-Baleanu neutral fractional stochastic hemivariational inequality" Results in Physics: 105647. DOI: 10.1016/j.rinp.2022.105647.
[29] C. Dineshkumar, R. Udhayakumar, V. Vijayakumar, K. S. Nisar, and A. Shukla, (2022) “A note concerning to approximate controllability of Atangana-Baleanu fractional neutral stochastic systems with infinite delay" Chaos, Solitons & Fractals 157: 111916. DOI: 10.1016/j.chaos.2022.111916.
[30] M. Alesemi, N. Iqbal, and T. Botmart, (2022) “Novel analysis of the fractional-order system of non-linear partial differential equations with the exponential-decay kernel" Mathematics 10(4): 615. DOI: 10.3390/math10040615.
[31] N. H. Aljahdaly, R. P. Agarwal, R. Shah, and T. Botmart, (2021) “Analysis of the time fractional-order coupled burgers equations with non-singular kernel operators" Mathematics 9(18): 2326. DOI: 10.3390/math9182326.
[32] S. K. Sahoo, M. Tariq, H. Ahmad, B. Kodamasingh, A. A. Shaikh, T. Botmart, and M. A. El-Shorbagy, (2022) “Some novel fractional integral inequalities over a new class of generalized convex function" Fractal and Fractional 6(1): 42. DOI: 10.3390/fractalfract6010042.
[33] D. Ding, Q. Ma, and X. Ding, (2013) “A non-standard finite difference scheme for an epidemic model with vaccination" Journal of Difference Equations and Applications 19(2): 179–190. DOI: 10.1080/10236198.2011.614606.
[34] W. E. Eyaran, S. Osman, and M. Wainaina, (2019) “Modelling and analysis of seir with delay differential equation" Global Journal of Pure and Applied Mathematics 15(4): 365–382.
[35] A. Zeb, M. Khan, G. Zaman, S. Momani, and V. S. Ertürk, (2014) “Comparison of numerical methods of the SEIR epidemic model of fractional order" Zeitschrift für Naturforschung A 69(1-2): 81–89. DOI: 10.5560/ZNA.2013-0073.
[36] N. Piovella, (2020) “Analytical solution of SEIR model describing the free spread of the COVID-19 pandemic" Chaos, Solitons & Fractals 140: 110243. DOI: 10.1016/j.chaos.2020.110243.
[37] S. J. Weinstein, M. S. Holland, K. E. Rogers, and N. S. Barlow, (2020) “Analytic solution of the SEIR epidemic model via asymptotic approximant" Physica D: nonlinear phenomena 411: 132633. DOI: 10.1016/j.physd.2020.132633.
[38] K. Heng and C. L. Althaus, (2020) “The approximately universal shapes of epidemic curves in the Susceptible-Exposed-Infectious-Recovered (SEIR) model" Scientific Reports 10(1): 1–6. DOI: 10.1038/s41598-020-76563-8.
[39] A. Das, A. Dhar, S. Goyal, A. Kundu, and S. Pandey, (2021) “COVID-19: Analytic results for a modified SEIR model and comparison of different intervention strategies" Chaos, Solitons & Fractals 144: 110595. DOI: 10.1016/j.chaos.2020.110595.
[40] H. M. Youssef, N. A. Alghamdi, M. A. Ezzat, A. A. El-Bary, and A. M. Shawky, (2020) “A modified SEIR model applied to the data of COVID-19 spread in Saudi Arabia" AIP advances 10(12): 125210. DOI: 10.1063/5.0029698.
[41] A. Atangana, (2017) “Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system" Chaos, solitons & fractals 102: 396–406. DOI: 10.1016/j.chaos.2017.04.027.