International Journal of Nonlinear Analysis and Applications

The asymptotic stability of a fractional epidemiological model "Covid 19 Variant Anglais" with Caputo derivative

Document Type : Research Paper

Authors

LMACS, FST of Beni Mellal, Sultan Moulay Slimane University, Morocco

Abstract

We have all been injured by corona and its mutations, not just us but the whole world.  The global impact of coronavirus (COVID-19) has been profound and the public health threat it represents is the most serious seen in a respiratory virus since 1918. This paper is concerned with a fractional order $S_{N}S_{C}IR$ model involving the Caputo fractional derivative. The effective methods to solve the fractional epidemic models we introduced to construct a simple and effective analytical technique that can be easily extended and applied to other fractional models and can help guide the concerned bodies in preventing or controlling, even predicting the infectious disease outbreaks.  The equilibrium points and the basic reproduction number are computed. An analysis of the local asymptotic stability at the disease-free equilibrium is given; Next, we study the stability of the equilibrium points in the sense of Mittag-Leffler. Moreover, some numerical simulations are included to verify the theoretical achievement. These results provide good evidence for the implications of the theoretical results corresponding to the model.

Keywords

[1] R. Almeida, A.M.C. Brito da Cruz, N. Martins, M. Teresa, and T. Monteiro, An epidemiological MSEIR model described by the Caputo fractional derivative, Int. J. Dyn. Control 7 (2019), 776–784.
[2] R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.
[3] P. Debnath, H.M. Srivastava, P. Kumam, and B. Hazarika, Fixed Point Theory and Fractional Calculus: Recent Advances and Applications, Springer, Singapore, 2022.
[4] B. Derdei, Study of epidemiological models: Stability, observation and estimation of parameters, University of Lorraine, 2013.
[5] K. Diethelm, A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dyn. 71 (2013), no. 4, 613—619.
[6] W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of epidemics, II—the problem of endemicity, Proc. R. Soc. Lond. A 138 (1932), 55-–83.
[7] W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of epidemics, III—further studies of the problem of endemicity, Proc. R. Soc. Lond. A 141 (1933), 94—122.
[8] A.E. Gorbalenya, S.C. Baker, and R.S. Baric, The species Severe acute respiratory syndrome-related coronavirus: Classifying 2019-nCoV and naming it SARS-CoV-2, Nature Microbio. 5 (2020), no. 4, 536–544.
[9] W.O. Kermack and A.G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A 115 (1927), 700—721.
[10] A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol. 68 (2006), no. 3, 615—626.
[11] Y. Li, Y.Q. Chen, and I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl. 59 (2010), no. 5, 1810–1821.
[12] M.Y. Li, J.R. Graef, L. Wang, and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci. 160 (1999), no. 2, 191-–213.
[13] S. Momani and S. Hadid, Lyapunov stability solutions of fractional integrodifferential equations, Int. J. Math. Math. Sci. 47 (2004), 25032507.
[14] F. Ndaırou, I. Area, J.J. Nieto, and D.F.M. Torres, Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan, Chaos Solitons Fractals 135 (2020), 109846.
[15] X. Wang, Z. Wang, H. Shen, Dynamical analysis of a discrete-time SIS epidemic model on complex networks, Appl. Math. Lett. 94 (2019) 292–299.
[16] L. Zhang, J. Li, and G. Chen, Extension of Lyapunov second method by fractional calculus, Pure Appl. Math. 3 (2005), 1008–5513.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 7
July 2023
Pages 35-43
  • Receive Date: 10 February 2023
  • Revise Date: 20 May 2023
  • Accept Date: 26 May 2023