A VARIABLE FRACTIONAL ORDER NETWORK MODEL OF ZIKA VIRUS
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Date
2018
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Journal Title
Journal ISSN
Volume Title
Type
Article
Publisher
Journal of Fractional Calculus and Applications
Series Info
Journal of Fractional Calculus and Applications;Vol. 9(1) Jan. 2018, pp. 204-221.
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Abstract
A variable fractional-order network model of Zika is presented in
this paper. We also carry out a detailed analysis on the equilibrium points
and its stability. Numerical solutions are obtained using a predictor-corrector
method to handle the fractional derivatives. The fractional derivatives are de-
scribed in the Caputo sense. Numerical simulations are presented to illustrate
the results. Also, the numerical simulations show that, modeling the Zika vari-
able fractional order model has more advantages than classical integer-order
modeling.
Description
Keywords
University for ZIKA VIRUS, FRACTIONAL ORDER
Citation
[1] R. L. J. S. Allen, \An Introduction to Mathematical Biology": New Jersey, Pearson Prentice Hall, Upper Saddle River, 2006. [2] A. A. M. Arafa, S.Z. Rida, and M. Khalil," The effect of anti-viral drug treatment of human immunode ciency", Appl. Math. Model., vol.37, no.4, pp. 2189{2196, 2013. [3] A. Atangana, and A.H. Cloot, \Stability and convergence of the space fractional variable- order Schrodinger equation", Adv. Differ. Equ., vol.2013, no.80, 2013. [4] A. H. Bhrawy, and M. A. Zaky, \Numerical simulation for two- dimensional variable-order fractional nonlinear cable equation", Nonlinear Dyn., vol.80, no. (1-2), pp.101-116, 2015. [5] A. V. Chechkin, R. Goren o, and I.M. Sokolov, "Fractional diffusion in inhomogeneous me- dia", J. Phys. A: Math. Gen., vol.38, no. 42, 2005. [6] A. V. Chechkin, V. Yu. Gonchar, R. Goren o, N. Korabel, and I.M. Sokolov, \Generalized fractional diffusion equations for accelerating sub diffusion and truncated Levy ights", Phys. Rev., vol. 78, no.2, 2008. [7] G. Diaz, and C.F.M Coimbra, \Nonlinear dynamics and control of a variable order oscilla- tor with application to the van del Pol equation", Nonlinear Dyn., vol. 56,no.(1-2),pp.145- 157,2009. [8] K. Diethelm, \Analysis of Fractional Differential Equation: An Application-Oriented Expo- sition Using Differential Operators of Caputo Type", Springer-Verlag, Berlin, Heidelberg, 2010. [9] H. El saka, E. Ahmed," a fractional order network model of Zika" ,BioRxiv,2016. [10] H. A. El-Saka and A. El-Sayed, \Fractional Order Equations and Dynamical Systems", Lam- bert Academic Publishing, Germany, 2013. [11] H. A. El-Saka, \The fractional-order SIR and SIRS epidemic models with variable population size", Math. Sci. Lett., vol.2, No. 3, pp. 1-6, 2013. [12] S. Funk, A. J. Kucharski, A. Camacho, R. M. Eggo, L. Yakob, L. M. Murray, and W. John Edmunds," Comparative analysis of dengue and Zika outbreaks reveals differences by setting and virus", PLOS Negl.Trop. Dis., 2016. [13] K. Gao, and D.Y. Hua," Effects of immunity on global oscillations in epidemic spreading in small-world networks", Physics Procedia, vol.3, no.5, pp.1801{1809, 2010. [14] D. Gao, Y. Lou, D. He, T. C. Porco, Y. Kuang, G. Chowell, and S. Ruan," Prevention and control of Zika as a mosquito-borne and sexually transmitted disease: A mathematical modeling analysis", Scienti c Reports, vol.6, 2016. [15] W. G. Glockle, T. F. Nonnenmacher,"A fractional calculus approach of self-similar protein dynamics", Biophys. J., vol.68, no.1, pp. 46{53, 1995. [16] H. W. Hethcote," The mathematics of infectious diseases". SIAM Rev., vol. 42, no.4, pp.599{ 653, 2000. [17] Y. Hua, Lequan Minab and Y. Kuang, \Modeling the dynamics of epidemic spreading on homogenous and heterogeneous networks", Applicable Analysis: An International Journal, vol. 94, no.11, pp.1308-2330, 2015. [18] R. Isea, and E. Karl Lonngren, \A Preliminary Mathematical Model for the Dynamic Trans- mission of Dengue, Chikungunya and Zika", American Journal of Modern Physics and Ap- plication, Vol. 3, No. 2, pp.11-15, 2016. [19] V. Isham, S. Harden, and M. Nekovee, \Stochastic epidemics and rumors on nite random networks," Physica A, vol. 389, no. 3, pp. 561{576, 2010. [20] M. J. Keeling, and P. Rohani," Modeling Infectious Diseases in Humans and Animals", NJ, Princeton University Press, 2008. [21] M. Khalid, and Fareeha Sami Khan, \Stability Analysis of Deterministic Mathematical Model for Zika Virus", British Journal of Mathematics & Computer Science, vol.19, no.4, pp.1-10, 2016. [22] G. C. Korobeinikov Wake, "Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models", Appl. Math. Lett., vol.15, no.8, pp. 955{960, 2002. [23] A. j. Kucharski, S. Funk, R.M. Eggo, H.P. Mallet, W. J. Edmunds, E. J. Nilles, \Transmission dynamics of Zika virus in island populations: a modeling analysis of the 2013-14 French Polynesia outbreak", PLOS Negl. Trop. Dis., vol. 10, no. 5, pp. e0004726, 2016. [24] C. Li, and W. Deng, "Remarks on fractional derivatives", Appl. Math. Comput., vol.187, no.2, pp.777-784, 2007. [25] C. F. Lorenzo, and T.T. Hartley, "Variable order and distributed order fractional operators", Nonlinear Dyn., vol.29, no.1, pp.57-98, 2002. [26] D. Matignon, \Stability results for fractional differential equations with applications to con- trol processing", In: IEEE-SMC Proceedings, Computational Engineering in Systems Appli- cations Multi-Conference ,IMACS , vol. 2, pp. 963{968,1996. [27] C. A. Monje, Y. Q. Chen , B. M. Vinagre, D. Xue, and V. Feliu , \Fractional-order Systems and Controls": Fundamentals and Applications. Springer-Verlag, London, 2010. [28] Z. Odibat, and N. Shawagfeh, \Generalized Taylor's formula", Appl. Math. Comput., vol.186, no.1, pp.286-293, 2007. [29] M. D. Ortigueira, F.J. V. Coito and J.J. Trujillo," Discrete-time differential systems. Signal Processing", vol. 107, no.22, pp.198{217, 2015. [30] L. M. Petrovic, D.T. Spasic, and T. M. Atanackovic, \On a mathematical model of a human root dentin", Dental Materials, vol. 21, no.2, pp. 125-128, 2005. [31] C. M. A. Pinto and J. A. T. Machado,"Fractional model for malaria transmission under control strategies", Computers and Mathematics with Applications, vol. 66, no.5,pp. 908{ 916, 2013. [32] I. Podlubny, \Fractional Differential Equations", Academic Press, San Diego. California, USA, 1999. [33] S. G. Samko, \Fractional integration and differentiation of variable order", Analysis Mathe- matics, vol. 21, no.3, p.213-236, 1995. [34] S. Samko, \Fractional integration and differentiation of variable order: an overview", Non- linear Dyn., vol.71, no.4, pp.653-662, 2013. [35] S. G. Samko, and B. Ross,"Integration and differentiation to a variable fractional order", Integral Transform. Spec. Funct., vol.1, no.4, pp. 277-300, 1993. [36] H. Sheng, H. G. Sun, C. Coopmans, Y. Q. Chen, and G. W. Bohannan ,"A Physical exper- imental study of variable-order fractional integrator and differentiator", Eur. Phys. J. Spec. Top.,vol.193,no.1,pp.93-104 ,2011. [37] Li. Shujuan , D.H. Gouge, K. Walker, Al Fournier, Shaku Nair, M. Wierda, and J. Hurley, "The Zika Virus", University of Arizona Cooperative Extension, Arizona pest management center,2016. [38] D. Sierociuk, I. Podlubny, and I. Petras," Experimental evidence of variable-order behavior of ladders and nested ladders", IEEE Trans. Control Syst. Technol., vol. 21, no.2, pp.459-466, 2013. [39] W. Smit, and H. de Vries, "Rheological models containing fractional derivatives", Rheol Acta,vol.9,no.4,pp.525-534,1970. [40] H. G. Sun, W. Chen, H. Wei, and Y. Q. Chen, "A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems", Eur. Phys. J. Special Topics,vol.193,no.1,pp.185-192,2011. [41] N. H. Sweilam, S.M.AL-Mekhla?, \Numerical study for multi-strain tuberculosis (TB) model of variable-order fractional derivatives", J.Adv. Res., vol.7, no.2, pp.271-283, 2015. [42] D. Tavares, R. Almeida, D. F. M. Torres, \Caputo derivatives of fractional variable order: numerical approximations". Commun. Nonlinear. Sci. Numer. Simul., vol.35, pp.69-87, 2016. [43] Yi. Xiao, Yicang Zhou, Sanyi Tang," Modeling disease spread in dispersal networks at two levels". Math Med Biol, vol.28, no.3,pp.22-244,2011. [44] M. Xu, J. Yang, D. Zhao and H. Zhao, \An image-enhancement method based on variable- order fractional differential operators", Bio-Medical Materials and Engineering, vol. 26, no.s1, pp.S1325-S1333, 2015. [45] Xu. Yufeng and He. Zhimin,"Existence and uniqueness results for Cauchy problem of variable- order fractional differential equations", J. Appl. Math. Comput vol.43, no.(1-2), pp.295-306, 2013.