A computational study of transmission dynamics for dengue fever with a fractional approach

Sunil Kumar^1, 2 ; R.P. Chauhan¹ ; Jagdev Singh³ ; Devendra Kumar⁴

¹ Department of Mathematics, National Institute of Technology, Jamshedpur 831014, Jharkhand, India.
² Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, United Arab Emirates.
³ Department of Mathematics, JECRC University, Jaipur 303905, Rajasthan, India.
⁴ Department of Mathematics, University of Rajasthan, Jaipur 302004, Rajasthan, India.

Mathematical modelling of natural phenomena, Tome 16 (2021), article no. 48.

Résumé

Fractional derivatives are considered an influential weapon in terms of analysis of infectious diseases because of their nonlocal nature. The inclusion of the memory effect is the prime advantage of fractional-order derivatives. The main objective of this article is to investigate the transmission dynamics of dengue fever, we consider generalized Caputo-type fractional derivative (GCFD) ( ) for alternate representation of dengue fever disease model. We discuss the existence and uniqueness of the solution of model by using fixed point theory. Further, an adaptive predictor-corrector technique is utilized to evaluate the considered model numerically.

Détail

DOI : 10.1051/mmnp/2021032

Bibliographie

[1] B. Ahmad, M. Alghanmi, S.K. Ntouyas, A. Alsaedi Fractional differential equations involving generalized derivative with Stieltjes and fractional integral boundary conditions Appl. Math. Lett 2018 111 117

[2] M. Al-Refai, T. Abdeljawad Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel Adv. Differ. Equ 2017 315

[3] D. Baleanu, H. Mohammadi, S. Rezapour A mathematical theoretical study of a particular system of Caputo–Fabrizio fractionaldifferential equations for the Rubella disease model Adv. Differ. Equ 2020 1 19

[4] D. Baleanu, G.-C. Wu, S.-D. Zeng Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations Chaos Solitons Fract 2017 99 105

[5] H.M. Baskonus, H. Bulut On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method Open Math 2015 547 556

[6] M. Benjemaa Taylor’s formula involving generalized fractional derivatives Appl. Math. Comput 2018 182 195

[7] M. Caputo, F. Mainardi A new dissipation model based on memory mechanism Pure Appl. Geophys 1971 134 147

[8] P. Chanprasopchai, I.M. Tang, P. Pongsumpun Sir model for dengue disease with effect of dengue vaccination Comput. Math. Methods Med 2018 9861572

[9] M. Derouich, A. Boutayeb, E. Twizell A model of dengue fever BioMedical Eng. OnLine 2003 4

[10] K. Diethelm, N.J. Ford, A.D. Freed A predictor-corrector approach for the numerical solution of fractional differential equations Nonlinear Dyn 2002 3 22

[11] H. El-Saka The fractional-order SIS epidemic model with variable population size J. Egypt. Math. Soc 2014 50 54

[12] A.N. Fall, S.N. Ndiaye, N. Sene Black–Scholes option pricing equations described by the Caputo generalized fractional derivative Chaos Solitons Fract 2019 108 118

[13] Z. Feng, J.X. Velasco-Hernández Competitive exclusion in a vector-host model for the dengue fever J. Math. Biol 1997 523 544

[14] J. Gómez-Aguilar Chaos and multiple attractors in a fractal–fractional Shinriki’s oscillator model Physica A 2020 122918

[15] J.F. Gómez-Aguilar, M.G. López-López, V.M. Alvarado-Martínez, D. Baleanu, H. Khan Chaos in a cancer model via fractional derivatives with exponential decay and Mittag-Leffler law Entropy 2017 681

[16] J.F. Gómez-Aguilar, V.F. Morales-Delgado, M.A. Taneco-Hernández, D. Baleanu, R.F. Escobar-Jiménez, M.M. Al Qurashi Analytical solutions of the electrical RLC circuit via Liouville–Caputo operators with local and non-local kernels Entropy 2016 402

[17] F. Jarad, T. Abdeljawad, D. Baleanu On the generalized fractional derivatives and their Caputo modification J. Nonlinear Sci. Appl 2017 2607 2619

[18] U.N. Katugampola New approach to a generalized fractional integral Appl. Math. Comput 2011 860 865

[19] U.N. Katugampola A new approach to generalized fractional derivatives Bull. Math. Anal. Appl 2014 1 15

[20] U.N. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations. Preprint arXiv:1411.5229 (2016).

[21] M. Khalid, M. Sultana and F.S. Khan, Numerical solution of SIR model of dengue fever. Int. J. Comput. Appl. 118 (21).

[22] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Vol. 204 of Theory and applications of fractional differential equations. Elsevier Science Limited (2006).

[23] I. Koca, P. Yaprakdal A new approach for nuclear family model with fractional order Caputo derivative Appl. Math. Nonlinear Sci 2020 393 404

[24] S. Kumar A new fractional modeling arising in engineering sciences and its analytical approximate solution Alexandria Eng. J 2013 813 819

[25] S. Kumar, R. Kumar, R.P. Agarwal, B. Samet A study of fractional Lotka-Volterra population model using Haar wavelet and Adams-Bashforth-Moulton methods Math. Methods Appl. Sci 2020 5564 5578

[26] S. Kumar, K.S. Nisar, R. Kumar, C. Cattani, B. Samet A new Rabotnov fractional-exponential function-based fractional derivative for diffusion equation under external force Math. Methods Appl. Sci 2020 4460 4471

[27] S. Kumar, A. Kumar, S. Abbas, M. Al Qurashi, D. Baleanu A modified analytical approach with existence and uniqueness for fractional Cauchy reaction–diffusion equations Adv. Differ. Equ 2020 1 18

[28] S. Kumar, A. Kumar, Z.M. Odibat A nonlinear fractional model to describe the population dynamics of two interacting species Math. Methods Appl. Sci 2017 4134 4148

[29] C. Li, D. Qian, Y. Chen On riemann-liouville and caputo derivatives Discrete Dyn. Nature Soc 2011

[30] Z. Odibat, D. Baleanu Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives Appl. Numer. Math 2020 94 105

[31] R. Ozarslan, E. Bas Kinetic model for drying in frame of generalized fractional derivatives Fractal Fract 2020 17

[32] I. Podlubny, Vol. 198 of Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier (1998).

[33] Y.M. Rangkuti, S. Side, M.S.M. Noorani Numerical analytic solution of SIR model of dengue fever disease in south Sulawesi using homotopy perturbation method and variational iteration method J. Math. Fund. Sci 2014 91 105

[34] K. Shah, F. Jarad, T. Abdeljawad On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizio derivative Alexandria Eng. J 2020 2305 2313

[35] A.S. Shaikh, K.S. Nisar Transmission dynamics of fractional order typhoid fever model using Caputo–Fabrizio operator Chaos Solitons Fract 2019 355 365

[36] Y. Shen, Mathematical models of dengue fever and measures to control it, Ph.D. dissertation, Florida State University Libraries (2014).

[37] S. Side, M.S.M. Noorani A sir model for spread of dengue fever disease (simulation for south Sulawesi, Indonesia and Selangor, Malaysia) World J. Model. Simul 2013 96 105

[38] J. Singh, D. Kumar, M. Al Qurashi, D. Baleanu A new fractional model for giving up smoking dynamics Adv. Differ. Equ 2017 88

[39] J. Singh, D. Kumar, Z. Hammouch, A. Atangana A fractional epidemiological model for computer viruses pertaining to a new fractional derivative Appl. Math. Comput 2018 504 515

[40] S. Ullah, M.A. Khan, M. Farooq A new fractional model for the dynamics of the hepatitis B virus using the Caputo-Fabrizio derivative Eur. Phys. J. Plus 2018 237