Qualitative Theory and Numerical Simulation of SIRC Model Corresponding to Nonlocal Fractional Order derivative

Document Type : Original Article

Authors

1 Department of Mathematics, Govt P.G Jahanzeb College Swat, Khyber Pakhtunkhwa, Pakistan

2 Department of Mathematics, Islamia College University, Peshawar, Khyber Pakhtunkhwa, Pakistan.

Abstract

In this article, the existence theory and numerical solutions for fractional order SIRC model via nonlocal fractional order derivative are developed. Using the tools of analysis, the conditions for the existence and stability of the proposed model are established. With the help of Laplace Adomain Decomposition method, we obtain the approximate solutions for the underlying model. In the last part, using Matlab, we plotted various graphs to discuss the proposed model for different fractional order values of $xi$.

Keywords


[1] R. Hilfer, Applications of Fractional Calculus in Physics",World Scientific, Singapore, 2000. 

[2] A.A. Kilbas, O.I. Marichev, S.G. Samko, Fractional Integrals and Derivatives (Theory and Applications)",
Gordon and Breach, Switzerland, 1993. 
[3] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations", North
Holland Mathematics Studies, vol. 204, Elseveir, Amsterdam, 2006 
[4] K.S. Miller, B. Ross, \An Introduction to the Fractional Calculus and Fractional Differential Equations", Wiley,
New York, 1993. 
[5] C. Goodrich, Existence of a positive solution to a class of Fractional differential equations," Comput. Math.
Appl., 59 (2010) 3889-3999. 
[6] P. Rahimkhani, Y. Ordokhani, E. Babolian, Numerical solution of fractional pantograph differential equations
by using generalized fractional-order Bernoulli wavelet", J. Comput. Appl. Math., 493-510 (2017). 
[7] U. Saeed, M.ur. Rehman, Hermite wavelet method for fractional delay differential equations", J. Differ. Eqn.
(2014). 
[8] Y. Yang, Y. Huang, Spectral-collocation methods for fractional pantograph delay integro differential equations",
Adv. Math. Phys. (2013). 
[9] M.K. Ishteva, properties and application of the Caputo Fractional operator", Deptt. Math. Univ. Karlsruhe,
2005. 
[10] Y. Zhou, Basic theory of fractional di erential equations", world scientific publishing. USA, 1964. 
[11] A. Ali, K. Shah, R.A. Khan Existence of positive solution to a class of boundary value problems of fractional
di erential equations", Compu. Methods Diff. Equ. 19-29 (2016). 
[12] A. Ali et al. Existence and stability of solution to a toppled systems of differential equations of non-integer
order", Bound. Value. Probl. 2017 (2017). 
[13] M. Caputo, Linear Models of dissipation whose Q is almost frequency independent", Int. Jou. Geo. Sci. 529-539.
(1967) 
[14] V. Lakshmikantham, S. Leela, J. Vasundhara, Theory of Fractional Dynamic Systems", Cambridge Academic
Publishers, Cambridge, UK, 2009. 
[15] Cai, L. Wu, Analysis of an HIV/AIDS treatment model with a nonlinear incidence rate", Chaos. Soliton. Frat.
175-182. (2009) 
[16] R.C. Wu, X.D. Hei, L.P. Chen, Finite-time stability of fractional-order neural networks with delay", Commun.
Theor. Phys. 189-193 2013. 
[17] A. Nanware, D.B. Dhaigude, Existence and uniqueness of solutions of differential equations of fractional order
with integral boundary conditions", J. Nonlinear Sci. Appl. 246-254 2014. 
[18] R.P. Agarwal, M. Belmekki, M. Benchohra, A survey on semilinear differential equations and inclusions involving
Riemann-Liouville fractional derivative", Adv. Differ. Equ. 2009. 
[19] J.C. Trigeassou, A Lyapunov approach to the stability of fractional differential equations", Signal Process,
437-445 (2011). 
[20] G. Lijun, D.Wang, G.Wang, Further results on exponential stability for impulsive switched nonlinear time-delay
systems with delayed impulse effects", Appl. Math. Comput. 186-200 2015. 
[21] I. Stamova, Mittag-Leffer stability of impulsive di erential equations of fractional order", Q. Appl. Math. 525-
535 2015. 
[22] S.M. Ullam, Problems in Modern Mathematics", Science Editors, Wiley, New York 1940. 
[23] D.H. Hyers, On the stability of the linear functional equation", Proc. Natl. Acad. Sci. 222-224 (1941). 
[24] S.M. Ulam, Problems in Modern Mathematics", Wiley, New York, 1940. 1
[25] S.M. Ulam, A Collection of Mathematical Problems", Interscience, New York, 1960. 1
[26] T.M. Rassias, On the stability of the linear mapping in Banach spaces", Proc. Am. Math. Soc. 297-300 (1978).
1
[27] M. Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential equations with fractional
order and nonlocal conditions", J. Nonl. Anal. 2391-2396 (2009). 1, 2.3
[28] K. Shah, R.A. Khan, Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional
order di erential equations with anti periodic boundary conditions", Di er. Equ. Appl. 245-262 (2015). 1
[29] R.A. Khan, K. Shah, Existence and uniqueness of solutions to fractional order multi-point boundary value
problems", Commun. Appl. Anal. 515-526 (2015). 1
[30] A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel", Theor, Appl. Heat
Transfer Model, Thermal Science. 763-769. (2016). 1, 2.1
[31] J.D. Djida, A. Atangana, I. Area, Numerical computation of a fractional derivative with non-local and non-
singular kernel", Math. Model. Nat. Phenomena. 4-13 (2017). 1, 2.1
[32] O. Algahtani, Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order Allen
Cahn model", Chaos Solitons Fractals (2016). 1, 2.1
[33] D. Kumar, J. Singh, S. Kumar, Numerical computation of fractional multi-dimensional diffusion equations by
using a modiffed homotopy perturbation method", J. Assoc. Arab. Univ. Basic. Appl. Sci., 20-26 (2015). 1
[34] S. Yanga, A. Xiao, H. Su, Convergence of the variational iteration method for solving multi-order fractional
di erential equations", Comput. Math. Appl., 2871-2879 (2010). 1
[35] Z. Odibat and S. Momani, A generalized differential transform method for linear partial differential equations
of fractional order", Appl. Math. Lett. , 194-199 (2008). 
[36] M. Deghan, Y.A. Yousef, A. Lotfi, The use of He's variational iteration method for solving the telegraph and
fractional telegraph equations", Int. J. Numer. Methods Biomed. Eng., 219-231 (2011). 1
[37] P. Zhou, et al. A pneumonia outbreak associated with a new coronavirus of probable bat origin", Nature 270-273
(2020). 1
[38] World Health Organization, Coronavirus disease 2019 (COVID-19): Situation Report", 21 April, 2020. 1
[39] L. Edelstein-Keshet, \Mathematical models in biology. Society for Industrial and Applied Mathematics", 2005.
1
[40] C.A.A. Beauchemin , H. Andreas, A review of mathematical models of in uenza A infections within a host or
cell culture: lessons learned and challenges ahead", BMC public health (2011). 1
[41] Brauer, Fred, Ven den Driessche, J. Wu, Lecture notes in mathematical epidemiology", Berlin, Germany,
Springer, 2008. 1
[42] LA. Rvachev, M. Ira, Jr. Longini, A mathematical model for the global spread of in uenza", Mathematical
biosciences 3-22 (1985). 1
[43] J.D Murray, Mathematical biology: An Introduction", Springer Science and Business Media, Vol. 17. 2007. 1
[44] J. Biazar, Solution of the epidemic model by Adomian decomposition method", App. Math. comput., 1101-1106
(2006). 1
[45] K. Shah et al, Semi-analytical study of Pine Wilt Disease model with convex rate under Caputo-Fabrizio
fractional order derivative", Chaos. Sol. Frac. (2020). 1
[46] A. Abdilraze, D. Pelinosky, Convergence of the Adomian Decomposition method for initial value problems",
Num. Methods. Part. Di . Equ. 749-766 (2009). 1
[47] A. Naghipour, J. Manafan, Application of the Laplace adomian decomposition method and implicit methods
for solving Burger's equation", TWMS J. Pure. Apple. Math. 68-77 (2015). 1
[48] K. Shah, H. Khalil, R.A. Khan, Analytical solutions of fractional order diffusion equations by Natural transform
method", Iran J. Sci. Technol. Trans. Sci. (2016). 1
[49] D.W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations", third ed., Oxford University Press, (1999).
1
[50] P. Palese, J.F. Young, Variation of in uenza A, B, and C viruses", Science, 215, 1468-1474 (1982). 1
[51] R. Anderson, R. May, Infectious Disease of Humans", Dynamics and Control, Oxford University Press, Oxford,
UK, (1995). 1
[52] R.G. Webster, W.J. Bean, O.T. Gorman, T.M. Chambers, Y. Kawaoka, \Evolution and ecology of in uenza A
viruses", Microbiological Reviews, 56, 152-179 (1992). 1
[53] R. Casagrandi, L. Bolzoni, S.A. Levin, V. Andreasen, The SIRC model and the in uenza A", Mathematical
Biosciences, 2002, 152-169 2006. 1
[54] G.P. Samanta, Global dynamic of nonautonomous SIRC model and in uenza A with distributed time delay",
Di erential Equations and Dynamical Systems, 18(4), 341-362 (2010). 1
[55] W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics", Proceedings of
Royal Society of London, 115, 700-721 (1927). 1
[56] H. Li, S. Guo, Dynamic of a SIRC epidemiological model", Electronic journal of Differential equations, 2017(121),
1-18 (2017). 1
[57] K. Shah, F. Jarad and T. Abdeljawad, On a nonlinear fractional order model of dengue fever disease under
Caputo-Fabrizo derivative", Alexandria Engineering Journal (2020). 5.1