Analysis of the fractional diarrhea model with Mittag-Leffler kernel

Muhammad Sajid Iqbal; Nauman Ahmed; Ali Akgül; Ali Raza; Muhammad Shahzad; Zafar Iqbal; Muhammad Rafiq; Fahd Jarad; Muhammad Sajid Iqbal; Nauman Ahmed; Ali Akgül; Ali Raza; Muhammad Shahzad; Zafar Iqbal; Muhammad Rafiq; Fahd Jarad

doi:10.3934/math.2022720

AIMS Mathematics

2022, Volume 7, Issue 7: 13000-13018. doi: 10.3934/math.2022720

Previous Article Next Article

Research article Special Issues

Analysis of the fractional diarrhea model with Mittag-Leffler kernel

1.
Department of Humanities and Basic Sciences, MCS, National University of Sciences and Technology, Islamabad, Pakistan
2.
Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
3.
Siirt University, Art and Science Faculty, Department of Mathematics, 56100 Siirt, Turkey
4.
Department of Mathematics, Govt. Maulana Zafar Ali Khan Graduate College Wazirabad, 52000, Punjab Higher Education Department (PHED), Lahore, 54000, Pakistan
5.
Department of Mathematics, University of Management and Technology, Lahore, Pakistan
6.
Department of Mathematics, Faculty of Science, University of Central Punjab, Lahore, Pakistan
7.
Department of Mathematics, Çankaya University, Etimesgut 06790, Ankara, Turkey
8.
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
9.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 21 December 2021 Revised: 03 April 2022 Accepted: 14 April 2022 Published: 09 May 2022
MSC : 65P99, 92D25

In this article, we have introduced the diarrhea disease dynamics in a varying population. For this purpose, a classical model of the viral disease is converted into the fractional-order model by using Atangana-Baleanu fractional-order derivatives in the Caputo sense. The existence and uniqueness of the solutions are investigated by using the contraction mapping principle. Two types of equilibrium points i.e., disease-free and endemic equilibrium are also worked out. The important parameters and the basic reproduction number are also described. Some standard results are established to prove that the disease-free equilibrium state is locally and globally asymptotically stable for the underlying continuous system. It is also shown that the system is locally asymptotically stable at the endemic equilibrium point. The current model is solved by the Mittag-Leffler kernel. The study is closed with constraints on the basic reproduction number $ R_{0} $ and some concluding remarks.
- fractal fractional derivative,
- existence and uniqueness,
- stability analysis,
- numerical simulations
Citation: Muhammad Sajid Iqbal, Nauman Ahmed, Ali Akgül, Ali Raza, Muhammad Shahzad, Zafar Iqbal, Muhammad Rafiq, Fahd Jarad. Analysis of the fractional diarrhea model with Mittag-Leffler kernel[J]. AIMS Mathematics, 2022, 7(7): 13000-13018. doi: 10.3934/math.2022720

Related Papers:

Abstract

In this article, we have introduced the diarrhea disease dynamics in a varying population. For this purpose, a classical model of the viral disease is converted into the fractional-order model by using Atangana-Baleanu fractional-order derivatives in the Caputo sense. The existence and uniqueness of the solutions are investigated by using the contraction mapping principle. Two types of equilibrium points i.e., disease-free and endemic equilibrium are also worked out. The important parameters and the basic reproduction number are also described. Some standard results are established to prove that the disease-free equilibrium state is locally and globally asymptotically stable for the underlying continuous system. It is also shown that the system is locally asymptotically stable at the endemic equilibrium point. The current model is solved by the Mittag-Leffler kernel. The study is closed with constraints on the basic reproduction number $ R_{0} $ and some concluding remarks.

References

[1]	N. Sykes, Constipation and diarrhea, In: Management of advanced disease, 4 Eds., CRC Press, 2004. http://doi.org/10.1093/med/9780199656097.003.0203
[2]	T. M. Sørensen, E. M. Vestergaard, S. K. Jensen, C. Lauridsen, S. Højsgaard, Performance and Diarrhea in piglets following weaning at seven weeks of age: challenge with E. coli O 149 and effect of dietary factors. Livest. Sci., 123 (2009), 314–321. http://doi.org/10.1016/j.livsci.2008.12.001 doi: 10.1016/j.livsci.2008.12.001
[3]	A. A. Sancak, H. C. Rutgers, C. A. Hart, R. M. Batt, Prevalence of enteropathic Escherichia coli in dogs with acute and chronic diarrhea, Vet. Rec., 154 (2004), 101–106. http://doi.org/10.1136/vr.154.4.101 doi: 10.1136/vr.154.4.101
[4]	H. Szajewska, A. Skorka, M. Dylag, Meta-analysis: Saccharomyces boulardii for treating acute Diarrhea in children, Alimentary Pharmacology and Therapeutics, 25 (2007), 257–264. http://doi.org/10.1111/j.1365-2036.2006.03202.x doi: 10.1111/j.1365-2036.2006.03202.x
[5]	A. Guarino, A. L. Vecchio, R. B. Canani, Chronic diarrhea in children, Best Pract. Res. Cl. Ga., 26 (2012), 649–661. http://doi.org/10.1016/j.bpg.2012.11.004 doi: 10.1016/j.bpg.2012.11.004
[6]	U. Navaneethan, R. A. Giannella, Definition, epidemiology, pathophysiology, clinical classification, and differential diagnosis of diarrhea, In: Diarrhea, Totowa, NJ: Humana Press, 2010, 1–31. http://doi.org/10.1007/978-1-60761-183-7_1
[7]	A. P. S. Hungin, L. Paxman, K. Koenig, J. Dalrymple, N. Wicks, J. Walmsley, Prevalence, symptom patterns and management of episodic Diarrhea in the community: a population-based survey in 11 countries, Alimentary Pharmacology and Therapeutics, 43 (2016), 586–595. http://doi.org/10.1111/apt.13513 doi: 10.1111/apt.13513
[8]	G. G. Gunsa, K. M. Rodamo, D. D. Dangiso, Determinants of acute diarrhea among children aged 6-59 months in Chiffre District, Southern Ethiopia: Unmatched case-control study, Journal of Gynecology and Obstetrics, 6 (2018), 15–25. http://doi.org/10.11648/j.jgo.20180602.11 doi: 10.11648/j.jgo.20180602.11
[9]	S. Kauchali, N. Rollins, J. Van den Broeck, Local beliefs about childhood diarrhea: importance for healthcare and research, J. Trop. Pediatrics, 50 (2004), 82–89. http://doi.org/10.1093/tropej/50.2.82 doi: 10.1093/tropej/50.2.82
[10]	E. Bonyah, G. Twagirumukiza, P. P. Gambrah, Mathematical analysis of diarrhea model with saturated incidence rate, Open J. Math. Sci., 3 (2019), 29–39. http://doi.org/10.30538/oms2019.0046 doi: 10.30538/oms2019.0046
[11]	L. Chola, J. Michalow, A. Tugendhaft, K. Hofman, Reducing diarrhea deaths in South Africa: costs and effects of scaling up essential interventions to prevent and treat Diarrhea in under-five children, BMC Public Health, 15 (2015), 394. http://doi.org/10.1186/s12889-015-1689-2 doi: 10.1186/s12889-015-1689-2
[12]	M. V. Jose, J. R. Bobadilla, Epidemiological model of diarrheal diseases and its application in prevention and control, Vaccine, 12 (1994), 109–116. http://doi.org/10.1016/0264-410x(94)90047-7 doi: 10.1016/0264-410x(94)90047-7
[13]	B. F. Iyun, E. A. Oke, Ecological and cultural barriers to treatment of childhood diarrhea in riverine areas of Ondo State, Nigeria, Social Science and Medicine, 50 (2000), 953–964. http://doi.org/10.1016/s0277-9536(99)00347-0 doi: 10.1016/s0277-9536(99)00347-0
[14]	A. Shaikh, K. S. Nisar, V. Jadhav, S. K. Elagan, M. Zakarya, Dynamical behaviour of HIV/AIDS model using fractional derivative with Mittag-Leffler kernel, Alex. Eng. J., 61 (2022), 2601–2610, http://doi.org/10.1016/j.aej.2021.08.030 doi: 10.1016/j.aej.2021.08.030
[15]	M. Farman, A. Akgül, K. S. Nisar, D. Ahmad, A. Ahmad, S. Kamangar, et al., Epidemiological analysis of fractional order COVID-19 model with Mittag-Leffler kernel, AIMS Mathematics, 7 (2022), 756–783. http://doi.org/10.3934/math.2022046 doi: 10.3934/math.2022046
[16]	K. Logeswari, C. Ravichandran, K. S. Nisar, Mathematical model for spreading of COVID-19 virus with the Mittag–Leffler kernel, Numer. Methods Partial Differential Equations, in press. http://doi.org/10.1002/num.22652
[17]	C. Ravichandran, K. Logeswari, S. K. Panda, K. S. Nisar, On new approach of fractional derivative by Mittag-Leffler kernel to neutral integro-differential systems with impulsive conditions, Chaos Soliton. Fract., 139 (2020), 110012. http://doi.org/10.1016/j.chaos.2020.110012 doi: 10.1016/j.chaos.2020.110012
[18]	S. Kumar, R. Kumar, J. Singh, K. S. Nisar, D. Kumar, An efficient numerical scheme for fractional model of HIV-1 infection of CD4+ T-cells with the effect of antiviral drug therapy, Alex. Eng. J., 59 (2020), 2053–2064. http://doi.org/10.1016/j.aej.2019.12.046 doi: 10.1016/j.aej.2019.12.046
[19]	I. Zada, M. N. Jan, N. Ali, D. Alrowail, K. S. Nisar, G. Zaman, Mathematical analysis of hepatitis B epidemic model with optimal control. Adv. Differ. Equ., 2021 (2021), 451. http://doi.org/10.1186/s13662-021-03607-2 doi: 10.1186/s13662-021-03607-2
[20]	I. Podlubny, Fractional differential equations of mathematics in science and engineering, New York, NY, USA: Academic Press, 1999.
[21]	A. Atangana, B. Dumitru, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. http://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
[22]	P. Ezanno, C. Fourichon, A. F. Viet, H. Seegers, Sensitivity analysis to identify key-parameters in modelling the spread of bovine viral diarrhea virus in a dairy herd, Prev. Vet. Med., 80 (2007), 49–64. http://doi.org/10.1016/j.prevetmed.2007.01.005 doi: 10.1016/j.prevetmed.2007.01.005
[23]	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. http://doi.org/10.1140/epjp/i2017-11717-0 doi: 10.1140/epjp/i2017-11717-0
[24]	T. A. Biala, A. Q. M. Khaliq, A fractional-order compartmental model for the spread of the COVID-19 pandemic, Commun. Nonlinear. Sci. Numer. Simulat., 98 (2021), 105764. http://doi.org/10.1016/j.cnsns.2021.105764 doi: 10.1016/j.cnsns.2021.105764
[25]	Y. Li, Y. Chen, I. Podlubny, Mittag–Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965–1969. http://doi.org/10.1016/j.automatica.2009.04.003 doi: 10.1016/j.automatica.2009.04.003

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)