On mathematical modelling of measles disease via collocation approach

Shahid Ahmed; Shah Jahan; Kamal Shah; Thabet Abdeljawad; Shahid Ahmed; Shah Jahan; Kamal Shah; Thabet Abdeljawad

doi:10.3934/publichealth.2024032

AIMS Public Health

2024, Volume 11, Issue 2: 628-653. doi: 10.3934/publichealth.2024032

Previous Article Next Article

Research article Special Issues

On mathematical modelling of measles disease via collocation approach

1.
Department of Mathematics, Central University of Haryana, Mohindergarh-123031, India
2.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia

Received: 04 February 2024 Accepted: 11 April 2024 Published: 06 May 2024

Measles, a highly contagious viral disease, spreads primarily through respiratory droplets and can result in severe complications, often proving fatal, especially in children. In this article, we propose an algorithm to solve a system of fractional nonlinear equations that model the measles disease. We employ a fractional approach by using the Caputo operator and validate the model's by applying the Schauder and Banach fixed-point theory. The fractional derivatives, which constitute an essential part of the model can be treated precisely by using the Broyden and Haar wavelet collocation methods (HWCM). Furthermore, we evaluate the system's stability by implementing the Ulam-Hyers approach. The model takes into account multiple factors that influence virus transmission, and the HWCM offers an effective and precise solution for understanding insights into transmission dynamics through the use of fractional derivatives. We present the graphical results, which offer a comprehensive and invaluable perspective on how various parameters and fractional orders influence the behaviours of these compartments within the model. The study emphasizes the importance of modern techniques in understanding measles outbreaks, suggesting the methodology's applicability to various mathematical models. Simulations conducted by using MATLAB R2022a software demonstrate practical implementation, with the potential for extension to higher degrees with minor modifications. The simulation's findings clearly show the efficiency of the proposed approach and its application to further extend the field of mathematical modelling for infectious illnesses.
- fractional SEIR modeling,
- fixed point theory,
- Haar wavelet,
- numerical analysis,
- Ulam-Hyers stability
Citation: Shahid Ahmed, Shah Jahan, Kamal Shah, Thabet Abdeljawad. On mathematical modelling of measles disease via collocation approach[J]. AIMS Public Health, 2024, 11(2): 628-653. doi: 10.3934/publichealth.2024032

Related Papers:

Abstract

Measles, a highly contagious viral disease, spreads primarily through respiratory droplets and can result in severe complications, often proving fatal, especially in children. In this article, we propose an algorithm to solve a system of fractional nonlinear equations that model the measles disease. We employ a fractional approach by using the Caputo operator and validate the model's by applying the Schauder and Banach fixed-point theory. The fractional derivatives, which constitute an essential part of the model can be treated precisely by using the Broyden and Haar wavelet collocation methods (HWCM). Furthermore, we evaluate the system's stability by implementing the Ulam-Hyers approach. The model takes into account multiple factors that influence virus transmission, and the HWCM offers an effective and precise solution for understanding insights into transmission dynamics through the use of fractional derivatives. We present the graphical results, which offer a comprehensive and invaluable perspective on how various parameters and fractional orders influence the behaviours of these compartments within the model. The study emphasizes the importance of modern techniques in understanding measles outbreaks, suggesting the methodology's applicability to various mathematical models. Simulations conducted by using MATLAB R2022a software demonstrate practical implementation, with the potential for extension to higher degrees with minor modifications. The simulation's findings clearly show the efficiency of the proposed approach and its application to further extend the field of mathematical modelling for infectious illnesses.

Acknowledgments

Authors are thankful to Prince Sultan University for APC and support through the TAS research lab. We would also like to thank Central University of Haryana for providing necessary facilities to carry out this research.

Conflict of interest

There are no conflicts of interest in this study.

References

[1]	Shah K, Khan ZA, Ali A, et al. (2020) Haar wavelet collocation approach for the solution of fractional order COVID-19 model using Caputo derivative. Alex Eng J 59: 3221-3231. https://doi.org/10.1016/j.aej.2020.08.028
[2]	Ullah S, Khan MA, Farooq M (2018) A fractional model for the dynamics of TB virus. Chaos Solit Fractals 116: 63-71. https://doi.org/10.1016/j.chaos.2018.09.001
[3]	Altaf KM, Atangana A (2019) Dynamics of Ebola disease in the framework of different fractional derivatives. Entropy 21: 303. https://doi.org/10.3390/e21030303
[4]	Khan H, Alzabut J, Shah A, et al. (2023) On fractal-fractional waterborne disease model: A study on theoretical and numerical aspects of solutions via simulations. Fractals 31: 2340055. https://doi.org/10.1142/S0218348X23400558
[5]	Zarin R, Ahmed I, Kumam P, et al. (2021) Fractional modeling and optimal control analysis of rabies virus under the convex incidence rate. Results Phys 28: 104665. https://doi.org/10.1016/j.rinp.2021.104665
[6]	Khan M, Khan T, Ahmad I, et al. (2022) Modeling of Hepatitis B virus transmission with fractional analysis. Math Probl Eng 2022: 1-14. https://doi.org/10.1155/2022/6202049
[7]	Magin R. L. (2012) Fractional calculus in bioengineering: A tool to model complex dynamics. In Proceedings of the 13th International Carpathian Control Conference, High Tatras, Slovakia: IEEE : 464-469. https://doi.org/10.1109/CarpathianCC.2012.6228688
[8]	Ahmed S, Shah K, Jahan S, et al. (2023) An efficient method for the fractional electric circuits based on Fibonacci wavelet. Results Phys 52: 106753. https://doi.org/10.1016/j.rinp.2023.106753
[9]	Caputo M, Mainardi F (1971) A new dissipation model based on memory mechanism. Pure Appl Geophys 91: 134-147. https://doi.org/10.1007/BF00879562
[10]	Caputo M, Fabricio M (2015) A new definition of fractional derivative without singular kernel. Prog Fract Differ Appl 1: 73-85. http://dx.doi.org/10.12785/pfda/010201
[11]	Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and Applications of Fractional Differential Equations. Amester Dam: Elsevier 1-523. https://doi.org/10.1016/S0304-0208(06)80001-0
[12]	Magin R, Podlubny I, et al. (2011) On the fractional signals and systems. Signal Proc 91: 350-371. https://doi.org/10.1016/j.sigpro.2010.08.003
[13]	Rahman MU, Ahmad S, Arfan M, et al. (2022) Fractional order mathematical model of serial killing with different choices of control strategy. Fractal Fract 6: 162. https://doi.org/10.3390/fractalfract6030162
[14]	Shah K, Wenqi L, Raezah AA, et al. (2024) Unraveling pine wilt disease: Comparative study of stochastic and deterministic model using spectral method. Expert Syst Appl 240: 122407. https://doi.org/10.1016/j.eswa.2023.122407
[15]	Ahmad Z, Bonanomi G, di Serafino D, et al. (2023) Transmission dynamics and sensitivity analysis of pine wilt disease with asymptomatic carriers via fractal-fractional differential operator of Mittag-Leffler kernel. Appl Numer Math 185: 446-465. https://doi.org/10.1016/j.apnum.2022.12.004
[16]	Malik A, Alkholief M, Aldakheel FM, et al. (2022) Sensitivity analysis of COVID-19 with quarantine and vaccination: A fractal-fractional model. Alex Eng J 61: 8859-8874. https://doi.org/10.1016/j.aej.2022.02.024
[17]	Ahmad Z, El-Kafrawy SA, Alandijany TA, et al. (2022) A global report on the dynamics of COVID-19 with quarantine and hospitalization: A fractional order model with non-local kernel. Comput Biol Chem 98: 107645. https://doi.org/10.1016/j.compbiolchem.2022.107645
[18]	Sinan M, Ahmad H, Ahmad Z, et al. (2022) Fractional mathematical modeling of malaria disease with treatment & insecticides. Results Phys 34: 105220. https://doi.org/10.1016/j.rinp.2022.105220
[19]	Sinan M, Kamal S H A H, Abdeljawad T, et al. (2023) Analysis of Nonlinear Mathematical Model of COVID-19 via Fractional-Order Piecewise Derivative. Chaos Theory and Appl 5: 27-33. https://doi.o rg/10.51537/chaos.1210461
[20]	Aghdaoui H, Alaoui AL, Nisar KS, et al. (2021) On analysis and optimal control of a SEIRI epidemic model with general incidence rate. Results Phys 20: 103681. https://doi.org/10.1016/j.rinp.2020.103681
[21]	Hadi F, Amin R, Khan I, et al. (2023) Numerical solutions of nonlinear delay integro-differential equations using Haar wavelet collocation method. Fractals 31: 2340039. https://doi.org/10.1142/S0218348X2340039X
[22]	Amin R, Hafsa, Hadi F, et al. (2023) Solution of Variable-Order Nonlinear Fractional Differential Equations Using Haar Wavelet Collocation Technique. Fractals 31: 2340022. https://doi.org/10.1142/S0218348X23400224
[23]	Scindia PS, Nisar KS (2023) Ulam's type stability of impulsive delay integrodifferential equations in Banach spaces. Int J Nonlin Sci Num 24: 1311-1326. https://doi.org/10.1515/ijnsns-2021-0261
[24]	Berhe HW, Makinde OD (2020) Computational modelling and optimal control of measles epidemic in human population. Biosyst 190: 104102. https://doi.org/10.1016/j.biosystems.2020.104102
[25]	Berhe HW, Makinde OD, Theuri DM (2019) Co-dynamics of measles and dysentery diarrhea diseases with optimal control and cost-effectiveness analysis. Appl Math Comput 347: 903-921. https://doi.org/10.1016/j.amc.2018.11.049
[26]	Yano TK, Makinde OD, Malonza DM (2016) Modelling childhood disease outbreak in a community with inflow of susceptible and vaccinated new-born. Global J Pure Appl Math 12: 3895-3916.
[27]	Yang Y, Xu L (2019) Stability of a fractional order SEIR model with general incidence. Appl Math Lett 105: 106303. https://doi.org/10.1016/j.aml.2020.106303
[28]	Hashim I, Abdulaziz O, Momani S (2009) Homotopy analysis method for fractional IVPs. Commun Nonlinear Sci Numer Simul 14: 674-684. https://doi.org/10.1016/j.cnsns.2007.09.014
[29]	Kumar M (2022) Spreading behavior of biological SIR system of a COVID-19 disease through a fast Taylor wavelet based numerical algorithm. Results Control Optim 13: 100316. https://doi.org/10.1016/j.rico.2023.100316
[30]	Sazmand A, Behroozifar M (2018) Application Jacobi spectral method for solving the time-fractional differential equation. J Comput Appl Math 339: 49-68. https://doi.org/10.1016/j.cam.2018.02.018
[31]	Walnut DF (2013) An introduction to wavelet analysis. New York: Birkhäuser Boston 161-268. https://doi.org/10.1007/978-1-4612-0001-7
[32]	Raizah Z, Zarin R (2023) Advancing COVID-19 understanding: Simulating omicron variant spread using fractional-order models and haar wavelet collocation. Math 11: 1925. https://doi.org/10.3390/math11081925
[33]	Ahmed S, Jahan S, Nisar KS (2023) Hybrid fibonacci wavelet method to solve fractional-order logistic growth model. Math Methods Appl Sci 46: 16218-16231. https://doi.org/10.1002/mma.9446
[34]	Lepik Ü (2009) Solving fractional integral equations by the Haar wavelet method. Appl Math Comput 214: 468-478. https://doi.10.1016/j.amc.2009.04.015
[35]	Li Y, Zhao W (2010) Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl Math Comput 216: 2276-2285. https://doi.org/10.1016/j.amc.2010.03.063
[36]	Kumar S, Kumar R, Momani S, et al. (2023) A study on fractional COVID-19 disease model by using Hermite wavelets. Math Methods Appl Sci 46: 7671-7687. https://doi.org/10.1002/mma.7065
[37]	Kumar S, Ahmadian A, Kumar R, et al. (2022) An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets. Math 8: 558. https://doi.org/10.3390/math8040558
[38]	Amin R, Yúzbaşı S, Nazir S (2022) Efficient Numerical Scheme for the Solution of HIV Infection CD4+ T-Cells Using Haar Wavelet Technique. CMES-Comput Model Eng Sci 131. https://doi.org/10.32604/cmes.2022.019154
[39]	Prakash B, Setia A, Alapatt D (2017) Numerical solution of nonlinear fractional SEIR epidemic model by using Haar wavelets. J Comput Sci 22: 109-118. https://doi.org/10.1016/j.jocs.2017.09.001
[40]	Momoh AA, Ibrahim MO, Uwanta IJ, et al. (2013) Mathematical model for control of measles epidemiology. Int J Pure Appl Math 87: 707-718. http://dx.doi.org/10.12732/ijpam.v87i5.4
[41]	Kumar S, Kumar R, Osman MS, et al. (2021) A wavelet based numerical scheme for fractional order SEIR epidemic of measles by using Genocchi polynomials. Numer Methods Partial Differ Equ 37: 1250-1268. https://doi.org/10.1002/num.22577
[42]	Al-Smadi MH, Gumah GN (2014) On the homotopy analysis method for fractional SEIR epidemic model. J Appl Sci Eng Tech 7: 3809-3820. https://doi.org/10.19026/rjaset.7.738
[43]	Zeidler E (2013) Nonlinear Functional Analysis and Its Applications, IV: Applications to Mathematical Physics. Berlin Heidelberg, Germany: Springer Science & Business Media 158-289. https://doi.org/10.1007/978-1-4612-4566-7
[44]	Khan A, Khan H, Gómez-Aguilar JF, et al. (2019) Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel. Chaos Solit Fractals 127: 422-427. https://doi.org/10.1016/j.chaos.2019.07.026
[45]	Jamal N, Sarwar M, Khashan MM (2021) Hyers-Ulam stability and existence criteria for the solution of second-order fuzzy differential equations. Journal of Function Spaces 2021: 1-13. https://doi.org/10.1155/2021/6664619
[46]	Khan H, Chen W, Khan A, et al. (2018) Hyers–Ulam stability and existence criteria for coupled fractional differential equations involving p-Laplacian operator. Adv Differ Equ 2018: 1-16. https://doi.org/10.1186/s13662-018-1899-x

Reader Comments

Your name:*

Email:*
© 2024 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)