An efficient numerical approach for solving fractional model of nonautonomous seasonal eco-epidemic system by using Bernoulli wavelets
-
Nagendra Kumar Yadav
,
Rajesh Kumar Sinha
Abstract
In this study, the Bernoulli wavelet method is employed to solve a nonlinear fractional seasonal eco-epidemic model. This paper introduces an operational matrix using Bernoulli wavelets to solve a system of nonautonomous fractional differential equations involving time-dependent disease transmission rates between predator and preys, mortality rate of sick predators, predation rate on healthy preys, and an extra food supply. This approach involves transforming the differential equation system into a system of algebraic equations, simplifying the solution process. We analyzed the convergence and stability analysis of the proposed iterative method. We have also examined the solution of aforementioned differential equations system using the Toufik–Atangana scheme to assess the accuracy and suitability of the Bernoulli wavelet approach. Additionally, several numerical simulations have been conducted to validate our findings.
-
Research ethics: As there was no formal ethics approval process required for this study, no specific committee approval was obtained. We assure you that the research was conducted ethically and in compliance with relevant regulations and standards.
-
Informed consent: Not applicable.
-
Author contributions: This manuscript is an original work conducted by the undersigned authors and has not been published or submitted elsewhere for publication. All listed authors have made significant contributions to the research.
-
Use of Large Language Models, AI and Machine Learning Tools: None declared.
-
Conflict of interest: The authors declare that they have no conflict of interest.
-
Research funding: The research presented in our manuscript was conducted without external funding or financial support. We confirm that there are no conflicts of interest regarding the publication of this research.
-
Data availability: All data analyzed or produced during this study are provided.
References
[1] H. Rudolf, Applications of Fractional Calculus in Physics, Singapore, World Scientific, 2000.Suche in Google Scholar
[2] K. Oldham and J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 111, New York, Elsevier, 1974.Suche in Google Scholar
[3] V. S. Kiryakova, Generalized Fractional Calculus and Applications, Boca Raton, CRC Press, 1993.Suche in Google Scholar
[4] J. Sabatier, O. P. Agrawal, and J. T. Machado, Advances in Fractional Calculus, vol. 4, Springer, 2007.10.1007/978-1-4020-6042-7Suche in Google Scholar
[5] O. A. Arqub, S. Tayebi, D. Baleanu, M. S. Osman, W. Mahmoud, and H. Alsulami, “A numerical combined algorithm in cubic B-spline method and finite difference technique for the time-fractional nonlinear diffusion wave equation with reaction and damping terms,” Results Phys., vol. 41, p. 105912, 2022, https://doi.org/10.1016/j.rinp.2022.105912.Suche in Google Scholar
[6] S. W. Yao et al.., “A novel collective algorithm using cubic uniform spline and finite difference approaches to solving fractional diffusion singular wave model through damping-reaction forces,” Fractals, vol. 31, no. 04, p. 2340069, 2023, https://doi.org/10.1142/s0218348x23400698.Suche in Google Scholar
[7] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science Limited, 2006.Suche in Google Scholar
[8] X. J. Yang, General Fractional Derivatives: Theory, Methods and Applications, Boca Raton, Chapman and Hall/CRC, 2019.10.1201/9780429284083Suche in Google Scholar
[9] O. A. Arqub, “Computational algorithm for solving singular Fredholm time-fractional partial integrodifferential equations with error estimates,” J. Appl. Math. Comput., vol. 59, no. 1, pp. 227–243, 2019, https://doi.org/10.1007/s12190-018-1176-x.Suche in Google Scholar
[10] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198, San Diego, Elsevier, 1999.Suche in Google Scholar
[11] O. A. Arqub, H. Alsulami, and M. Alhodaly, “Numerical Hilbert space solution of fractional Sobolev equation in 1+1-dimensional space,” Math. Sci., vol. 18, no. 2, pp. 217–228, 2024, https://doi.org/10.1007/s40096-022-00495-9.Suche in Google Scholar
[12] X. Niu, T. Zhang, and Z. Teng, “The asymptotic behavior of a nonautonomous eco-epidemic model with disease in the prey,” Appl. Math. Modell., vol. 35, no. 1, pp. 457–470, 2011, https://doi.org/10.1016/j.apm.2010.07.010.Suche in Google Scholar
[13] Y. Lu, X. Wang, and S. Liu, “A non-autonomous predator-prey model with infected prey,” Discrete Contin. Dyn. Syst. B, vol. 23, no. 9, pp. 3817–3836, 2018, https://doi.org/10.3934/dcdsb.2018082.Suche in Google Scholar
[14] O. Misra, J. Dhar, and O. Sisodiya, “Dynamical study of svirb epidemic model for water-borne disease with seasonal variability,” Dyn. Contin. Discrete Impulsive Syst., Ser. A Math. Anal, vol. 27, pp. 351–374, 2020.Suche in Google Scholar
[15] C. Saad-Roy, P. Van den Driessche, and A. A. Yakubu, “A mathematical model of anthrax transmission in animal populations,” Bull. Math. Biol., vol. 79, no. 2, pp. 303–324, 2017, https://doi.org/10.1007/s11538-016-0238-1.Suche in Google Scholar PubMed
[16] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42, Berlin–Heidelberg, Springer, 2013.Suche in Google Scholar
[17] X. Li, J. Ren, S. A. Campbell, G. S. Wolkowicz, and H. Zhu, “How seasonal forcing influences the complexity of a predator-prey system,” Discrete Contin. Dyn. Syst. Ser. B, vol. 23, no. 2, pp. 785–807, 2018.10.3934/dcdsb.2018043Suche in Google Scholar
[18] B. Sahoo and S. Poria, “Dynamics of a predator-prey system with seasonal effects on additional food,” Int. J. Ecosyst., vol. 1, no. 1, pp. 10–13, 2011, https://doi.org/10.5923/j.ije.20110101.02.Suche in Google Scholar
[19] C. Packer, R. D. Holt, P. J. Hudson, K. D. Lafferty, and A. P. Dobson, “Keeping the herds healthy and alert: Implications of predator control for infectious disease,” Ecol. Lett., vol. 6, no. 9, pp. 797–802, 2003, https://doi.org/10.1046/j.1461-0248.2003.00500.x.Suche in Google Scholar
[20] K. pada Das, K. Kundu, and J. Chattopadhyay, “A predator-prey mathematical model with both the populations affected by diseases,” Ecol. Complexity, vol. 8, no. 1, pp. 68–80, 2011, https://doi.org/10.1016/j.ecocom.2010.04.001.Suche in Google Scholar
[21] M. A. Wild, N. T. Hobbs, M. S. Graham, and M. W. Miller, “The role of predation in disease control: A comparison of selective and nonselective removal on prion disease dynamics in deer,” J. Wildlife Dis., vol. 47, no. 1, pp. 78–93, 2011, https://doi.org/10.7589/0090-3558-47.1.78.Suche in Google Scholar PubMed
[22] E. Numfor, F. M. Hilker, and S. Lenhart, “Optimal culling and biocontrol in a predator-prey model,” Bull. Math. Biol., vol. 79, pp. 88–116, 2017, https://doi.org/10.1007/s11538-016-0228-3.Suche in Google Scholar PubMed
[23] E. Beltrami and T. Carroll, “Modeling the role of viral disease in recurrent phytoplankton blooms,” J. Math. Biol., vol. 32, pp. 857–863, 1994, https://doi.org/10.1007/bf00168802.Suche in Google Scholar
[24] P. Roy and R. K. Upadhyay, “Assessment of rabbit hemorrhagic disease in controlling the population of red fox: A measure to preserve endangered species in Australia,” Ecol. Complexity, vol. 26, pp. 6–20, 2016, https://doi.org/10.1016/j.ecocom.2016.01.002.Suche in Google Scholar
[25] S. Bera, A. Maiti, and G. Samanta, “A prey-predator model with infection in both prey and predator,” Filomat, vol. 29, no. 8, pp. 1753–1767, 2015, https://doi.org/10.2298/fil1508753b.Suche in Google Scholar
[26] K. Hadeler and H. Freedman, “Predator-prey populations with parasitic infection,” J. Math. Biol., vol. 27, no. 6, pp. 609–631, 1989, https://doi.org/10.1007/bf00276947.Suche in Google Scholar
[27] X. Gao, Q. Pan, M. He, and Y. Kang, “A predator-prey model with diseases in both prey and predator,” Physica A, vol. 392, no. 23, pp. 5898–5906, 2013, https://doi.org/10.1016/j.physa.2013.07.077.Suche in Google Scholar
[28] H. Singh, J. Dhar, and H. S. Bhatti, “Dynamics of a prey-generalized predator system with disease in prey and gestation delay for predator,” Model. Earth Syst. Environ., vol. 2, pp. 1–9, 2016, https://doi.org/10.1007/s40808-016-0096-8.Suche in Google Scholar
[29] S. Biswas, S. K. Sasmal, S. Samanta, M. Saifuddin, Q. J. A. Khan, and J. Chattopadhyay, “A delayed eco-epidemiological system with infected prey and predator subject to the weak Allee effect,” Math. Biosci., vol. 263, pp. 198–208, 2015, https://doi.org/10.1016/j.mbs.2015.02.013.Suche in Google Scholar PubMed
[30] S. Belvisi and E. Venturino, “An ecoepidemic model with diseased predators and prey group defense,” Simul. Modell. Pract. Theory, vol. 34, pp. 144–155, 2013, https://doi.org/10.1016/j.simpat.2013.02.004.Suche in Google Scholar
[31] M. Haque and E. Venturino, “An ecoepidemiological model with disease in predator: the ratio-dependent case,” Math. Methods Appl. Sci., vol. 30, no. 14, pp. 1791–1809, 2007, https://doi.org/10.1002/mma.869.Suche in Google Scholar
[32] C. Tannoia, E. Torre, and E. Venturino, “An incubating diseased-predator ecoepidemic model,” J. Biol. Phys., vol. 38, pp. 705–720, 2012, https://doi.org/10.1007/s10867-012-9281-9.Suche in Google Scholar PubMed PubMed Central
[33] M. Haque and D. Greenhalgh, “When a predator avoids infected prey: A model-based theoretical study,” Math. Med. Biol., vol. 27, no. 1, pp. 75–94, 2010, https://doi.org/10.1093/imammb/dqp007.Suche in Google Scholar PubMed
[34] B. Sahoo and S. Poria, “Disease control in a food chain model supplying alternative food,” Appl. Math. Modell., vol. 37, no. 8, pp. 5653–5663, 2013, https://doi.org/10.1016/j.apm.2012.11.017.Suche in Google Scholar PubMed PubMed Central
[35] B. Sahoo and S. Poria, “Effects of additional food on an ecoepidemic model with time delay on infection,” Appl. Math. Comput., vol. 245, pp. 17–35, 2014, https://doi.org/10.1016/j.amc.2014.07.066.Suche in Google Scholar
[36] M. Razzaghi and S. Yousefi, “The Legendre wavelets operational matrix of integration,” Int. J. Syst. Sci., vol. 32, no. 4, pp. 495–502, 2001, https://doi.org/10.1080/002077201300080910.Suche in Google Scholar
[37] M. Shamsi and M. Razzaghi, “Solution of Hallen’s integral equation using multiwavelets,” Comput. Phys. Commun., vol. 168, no. 3, pp. 187–197, 2005, https://doi.org/10.1016/j.cpc.2005.01.016.Suche in Google Scholar
[38] M. Lakestani, M. Razzaghi, and M. Dehghan, “Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations,” Math. Probl. Eng., vol. 2006, p. 096184, 2006, https://doi.org/10.1155/mpe/2006/96184.Suche in Google Scholar
[39] G. Beylkin, R. Coifman, and V. Rokhlin, “Fast wavelet transforms and numerical algorithms I,” Commun. Pure Appl. Math., vol. 44, no. 2, pp. 141–183, 1991, https://doi.org/10.1002/cpa.3160440202.Suche in Google Scholar
[40] N. K. Yadav, R. K. Sinha, R. Kumar, and R. Ranjan, “An efficient numerical algorithm to solve the chaotic behaviour of fractional financial model using Bernstein polynomials with convergence and bifurcation analysis,” Comput. Econ., vol. 75, pp. 1–39, 2025. https://doi.org/10.1007/s10614-025-10928-x.Suche in Google Scholar
[41] S. Shiralashetti and S. Kumbinarasaiah, “Laguerre wavelets collocation method for the numerical solution of the Benjamina-Bona-Mohany equations,” J. Taibah Univ. Sci., vol. 13, no. 1, pp. 9–15, 2019, https://doi.org/10.1080/16583655.2018.1515324.Suche in Google Scholar
[42] N. K. Yadav and R. K. Sinha, “Bernstein wavelets based numerical algorithm for solving fractional order lumpy skin disease model,” J. Nonlinear, Complex Data Sci., vol. 7, pp. 415–436, 2025.10.1515/jncds-2024-0090Suche in Google Scholar
[43] S. Shiralashetti and S. Kumbinarasaiah, “Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear Lane-Emden type equations,” Appl. Math. Comput., vol. 315, pp. 591–602, 2017, https://doi.org/10.1016/j.amc.2017.07.071.Suche in Google Scholar
[44] J. Wang, T. Z. Xu, Y. Q. Wei, and J. Q. Xie, “Numerical simulation for coupled systems of nonlinear fractional order integro-differential equations via wavelets method,” Appl. Math. Comput., vol. 324, pp. 36–50, 2018, https://doi.org/10.1016/j.amc.2017.12.010.Suche in Google Scholar
[45] P. Rahimkhani and Y. Ordokhani, “A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions,” Numer. Methods Partial Differ. Equations, vol. 35, no. 1, pp. 34–59, 2019, https://doi.org/10.1002/num.22279.Suche in Google Scholar
[46] B. Yuttanan and M. Razzaghi, “Legendre wavelets approach for numerical solutions of distributed order fractional differential equations,” Appl. Math. Modell., vol. 70, pp. 350–364, 2019, https://doi.org/10.1016/j.apm.2019.01.013.Suche in Google Scholar
[47] S. Kumar, R. Kumar, R. P. Agarwal, and B. Samet, “A study of fractional Lotka-Volterra population model using Haar wavelet and Adams-Bashforth-Moulton methods,” Math. Methods Appl. Sci., vol. 43, no. 8, pp. 5564–5578, 2020, https://doi.org/10.1002/mma.6297.Suche in Google Scholar
[48] D. Kumar, J. Singh, M. Al Qurashi, and D. Baleanu, “A new fractional sirs-si malaria disease model with application of vaccines, antimalarial drugs, and spraying,” Adv. Differ. Equations, vol. 2019, no. 1, pp. 1–19, 2019, https://doi.org/10.1186/s13662-019-2199-9.Suche in Google Scholar
[49] J. Singh, D. Kumar, and D. Baleanu, “On the analysis of fractional diabetes model with exponential law,” Adv. Differ. Equations, vol. 2018, p. 231, 2018, https://doi.org/10.1186/s13662-018-1680-1.Suche in Google Scholar
[50] A. Goswami, J. Singh, D. Kumar, and Sushila, “An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma,” Physica A, vol. 524, pp. 563–575, 2019, https://doi.org/10.1016/j.physa.2019.04.058.Suche in Google Scholar
[51] J. Gupta, J. Dhar, and P. Sinha, “An eco-epidemic model with seasonal variability: A nonautonomous model,” Arabian J. Math., vol. 11, no. 3, pp. 521–538, 2022, https://doi.org/10.1007/s40065-022-00375-z.Suche in Google Scholar
[52] A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, vol. 378, Berlin–Heidelberg, Springer, 2014.Suche in Google Scholar
[53] E. Keshavarz, Y. Ordokhani, and M. Razzaghi, “Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations,” Appl. Math. Modell., vol. 38, no. 24, pp. 6038–6051, 2014, https://doi.org/10.1016/j.apm.2014.04.064.Suche in Google Scholar
[54] A. Soltanpour Moghadam, M. Arabameri, D. Baleanu, and M. Barfeie, “Numerical solution of variable fractional order advection-dispersion equation using Bernoulli wavelet method and new operational matrix of fractional order derivative,” Math. Methods Appl. Sci., vol. 43, no. 7, pp. 3936–3953, 2020.10.1002/mma.6164Suche in Google Scholar
[55] F. Mehrdoust, A. H. Refahi Sheikhani, M. Mashoof, and S. Hasanzadeh, “Block-pulse operational matrix method for solving fractional Black-Scholes equation,” J. Econ. Stud., vol. 44, no. 3, pp. 489–502, 2017, https://doi.org/10.1108/jes-05-2016-0107.Suche in Google Scholar
[56] A. Ebadian and A. A. Khajehnasiri, “Block-pulse functions and their applications to solving systems of higher-order nonlinear Volterra integro-differential equations,” Electron. J. Differ. Equations, vol. 54, pp. 1–9, 2014.Suche in Google Scholar
[57] Y. Li and N. Sun, “Numerical solution of fractional differential equations using the generalized block pulse operational matrix,” Comput. Math. Appl., vol. 62, no. 3, pp. 1046–1054, 2011, https://doi.org/10.1016/j.camwa.2011.03.032.Suche in Google Scholar
[58] M. Toufik and A. Atangana, “New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models,” Eur. Phys. J. Plus, vol. 132, pp. 1–16, 2017, https://doi.org/10.1140/epjp/i2017-11717-0.Suche in Google Scholar
[59] J. Sol’is-P’erez, J. F. G’omez-Aguilar, and A. Atangana, “A fractional mathematical model of breast cancer competition model,” Chaos, Solitons Fractals, vol. 127, pp. 38–54, 2019, https://doi.org/10.1016/j.chaos.2019.06.027.Suche in Google Scholar
[60] A. Akgül, “A novel method for a fractional derivative with non-local and non-singular kernel,” Chaos, Solitons Fractals, vol. 114, pp. 478–482, 2018, https://doi.org/10.1016/j.chaos.2018.07.032.Suche in Google Scholar
[61] A. Akgül, “Reproducing kernel method for fractional derivative with non-local and non-singular kernel,” in Fractional Derivatives with Mittag-Leffler Kernel: Trends and Applications in Science and Engineering, Cham, Springer International Publishing, 2019, pp. 1–12.10.1007/978-3-030-11662-0_1Suche in Google Scholar
[62] S. Kumar, A. Kumar, and M. Jleli, “A numerical analysis for fractional model of the spread of pests in tea plants,” Numer. Methods Partial Differ. Equations, vol. 38, no. 3, pp. 540–565, 2022.Suche in Google Scholar
[63] D. Prakasha, P. Veeresha, and H. M. Baskonus, “Analysis of the dynamics of hepatitise virus using the Atangana-Baleanu fractional derivative,” Eur. Phys. J. Plus, vol. 134, pp. 1–11, 2019, https://doi.org/10.1140/epjp/i2019-12590-5.Suche in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
