Abstract and Applied Analysis

On Fractional SIRC Model with Salmonella Bacterial Infection

Fathalla A. Rihan, Dumitru Baleanu, S. Lakshmanan, and R. Rakkiyappan

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We propose a fractional order SIRC epidemic model to describe the dynamics of Salmonella bacterial infection in animal herds. The infection-free and endemic steady sates, of such model, are asymptotically stable under some conditions. The basic reproduction number 0 is calculated, using next-generation matrix method, in terms of contact rate, recovery rate, and other parameters in the model. The numerical simulations of the fractional order SIRC model are performed by Caputo’s derivative and using unconditionally stable implicit scheme. The obtained results give insight to the modelers and infectious disease specialists.

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Abstr. Appl. Anal., Volume 2014 (2014), Article ID 136263, 9 pages.

First available in Project Euclid: 2 October 2014

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Rihan, Fathalla A.; Baleanu, Dumitru; Lakshmanan, S.; Rakkiyappan, R. On Fractional SIRC Model with Salmonella Bacterial Infection. Abstr. Appl. Anal. 2014 (2014), Article ID 136263, 9 pages. doi:10.1155/2014/136263. https://projecteuclid.org/euclid.aaa/1412276927

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  • F. A. Rihan, “Numerical modeling of fractional-order biological systems,” Abstract and Applied Analysis, vol. 2013, Article ID 816803, 11 pages, 2013.
  • E. Ahmed, A. Hashish, and F. A. Rihan, “On fractional order cancer model,” Fractional Calculus and Applied Analysis, vol. 3, no. 2, pp. 1–6, 2012.
  • F. Mainardi and R. Gorenflo, “On Mittag-Leffler-type functions in fractional evolution processes,” Journal of Computational and Applied Mathematics, vol. 118, no. 1-2, pp. 283–299, 2000.
  • W.-C. Chen, “Nonlinear dynamics and chaos in a fractional-order financial system,” Chaos, Solitons and Fractals, vol. 36, no. 5, pp. 1305–1314, 2008.
  • L. Debnath, “Recent applications of fractional calculus to science and engineering,” International Journal of Mathematics and Mathematical Sciences, no. 54, pp. 3413–3442, 2003.
  • A. M. A. El-Sayed, “Nonlinear functional-differential equations of arbitrary orders,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal A: Theory and Methods, vol. 33, no. 2, pp. 181–186, 1998.
  • A. M. A. El-Sayed, A. E. M. El-Mesiry, and H. A. A. El-Saka, “On the fractional-order logistic equation,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 20, no. 7, pp. 817–823, 2007.
  • E. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000.
  • D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus, Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, 2012, Models and numerical methods.
  • W. Lin, “Global existence theory and chaos control of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 709–726, 2007.
  • K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 29, no. 1–4, pp. 3–22, 2002, Fractional order calculus and its applications.
  • F. A. Rihan, “Computational methods for delay parabolic and time-fractional partial differential equations,” Numerical Methods for Partial Differential Equations. An International Journal, vol. 26, no. 6, pp. 1556–1571, 2010.
  • P. Bouvet, “Human Salmonellosis surveillance in france: recent data from the national reference center,” Salmonella and Salmonellosis, pp. 411–416, 2002.
  • L. Edelstein-Keshet, Mathematical Models in Biology, The Random House/Birkhäuser Mathematics Series, Random House, New York, NY, USA, 1988.
  • D. S. Jones, M. J. Plank, and B. D. Sleeman, Differential Equations and Mathematical Biology, Mathematical & Computational Biology, 2008.
  • D. Kaplan and L. Glass, Understanding nonlinear dynamics, vol. 19 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1995.
  • J. D. Murray, Mathematical Biology. II, Interdisciplinary Applied Mathematics, Springer, 3rd edition, 2003, Spatial models and biomedical applications.
  • M. Safan and F. A. Rihan, “Mathematical analysis of an SIS model with imperfect vaccination and backward bifurcation,” Mathematics and Computers in Simulation, vol. 96, pp. 195–206, 2014.
  • F. A. Rihan, Numerical Treatment of Delay Differential Equations in Bioscience [Ph.D. thesis], University of Manchester, 2000.
  • W. O. Kermack and A. G. McKendrick, “Contributions to the mathematical theory of epidemics,” Proceedings of Royal Society of London, vol. 115, no. 722, pp. 700–721, 1927.
  • R. Casagrandi, L. Bolzoni, S. A. Levin, and V. Andreasen, “The SIRC model and influenza A,” Mathematical Biosciences, vol. 200, no. 2, pp. 152–169, 2006.
  • M. El-Shahed and A. Alsaedi, “The fractional SIRC model and influenza A,” Mathematical Problems in Engineering, vol. 2011, Article ID 480378, 9 pages, 2011.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  • I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, 1999, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications.
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, 1993.
  • Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylor's formula,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 286–293, 2007.
  • L. Jódar, R. J. Villanueva, A. J. Arenas, and G. C. González, “Nonstandard numerical methods for a mathematical model for influenza disease,” Mathematics and Computers in Simulation, vol. 79, no. 3, pp. 622–633, 2008.
  • H. Xu, “Analytical approximations for a population growth model with fractional order,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1978–1983, 2009.
  • O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, “On the definition and the computation of the basic reproduction ratio ${R}_{0}$ in models for infectious diseases in heterogeneous populations,” Journal of Mathematical Biology, vol. 28, no. 4, pp. 365–382, 1990.
  • D. Matignon, “Stability results for fractional differential equations with applications to control processing,” Computational Engineering in System Application, vol. 2, pp. 963–968, 1996.
  • E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, “On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems,” Physics Letters A, vol. 358, no. 1, pp. 1–4, 2006.
  • T. Suzuki, “A generalized Banach contraction principle that characterizes metric completeness,” Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1861–1869, 2008.
  • R. Anguelov and J. M.-S. Lubuma, “Nonstandard finite difference method by nonlocal approximation,” Mathematics and Computers in Simulation, vol. 61, no. 3–6, pp. 465–475, 2003.
  • K. Diethelm, “An algorithm for the numerical solution of differential equations of fractional order,” Electronic Transactions on Numerical Analysis, vol. 5, pp. 1–6, 1997.
  • K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002.
  • P. Chapagain, J. S. Van Kessel, J. K. Karns et al., “A mathematical model of the dynamics of Salmonella cerro infection in a us dairy herd,” Epidemiology & Infection, vol. 136, pp. 263–272, 2008. \endinput