Square-Root Dynamics of a SIR-Model in Fractional Order

Affiliation(s)

National Fisheries Research and Development Institute, Busan, South Korea.

Department of Mathematics, University of Malakand, Chakdara, Pakistan.

Department of Mathematics, Pusan National University, Busan, South Korea.

National Fisheries Research and Development Institute, Busan, South Korea.

Department of Mathematics, University of Malakand, Chakdara, Pakistan.

Department of Mathematics, Pusan National University, Busan, South Korea.

ABSTRACT

In this paper, we consider an SIR-model for which the interaction term is the square root of the susceptible and infected individuals in the form of fractional order differential equations. First the non-negative solution of the model in fractional order is presented. Then the local stability analysis of the model in fractional order is discussed. Finally, the general solutions are presented and a discrete-time finite difference scheme is constructed using the nonstandard finite difference (NSFD) method. A comparative study of the classical Runge-Kutta method and ODE45 is presented in the case of integer order derivatives. The solutions obtained are presented graphically.

In this paper, we consider an SIR-model for which the interaction term is the square root of the susceptible and infected individuals in the form of fractional order differential equations. First the non-negative solution of the model in fractional order is presented. Then the local stability analysis of the model in fractional order is discussed. Finally, the general solutions are presented and a discrete-time finite difference scheme is constructed using the nonstandard finite difference (NSFD) method. A comparative study of the classical Runge-Kutta method and ODE45 is presented in the case of integer order derivatives. The solutions obtained are presented graphically.

Cite this paper

Y. Seo, A. Zeb, G. Zaman and I. Jung, "Square-Root Dynamics of a SIR-Model in Fractional Order,"*Applied Mathematics*, Vol. 3 No. 12, 2012, pp. 1882-1887. doi: 10.4236/am.2012.312257.

Y. Seo, A. Zeb, G. Zaman and I. Jung, "Square-Root Dynamics of a SIR-Model in Fractional Order,"

References

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[2] R. M. Anderson and R. M. May, “Infectious Disease of Humans, Dynamics and Control,” Oxford University Press, Oxford, 1991.

[3] C. Castillo-Garsow, G. Jordan-Salivia and A. Rodriguez Herrera, “Mathematical Models for Dynamics of Tobacco Use, Recovery and Relapse,” Technical Report Series BU-1505-M, Cornell Uneversity, Ithaca, 2000.

[4] M. Choisy, J. F. Guezan and P. Rohani, “Dynamics of Infectious Diseases and Pulse Vaccination: Teasing Apart the Embedded Resonance Effects,” Journal of Physics D, Vol. 223, No. 1, 2006, pp. 26-35. doi:10.1016/j.physd.2006.08.006

[5] M. G. Gomes, L. J. White and G. F. Medley, “Infection, Reinfection, and Vaccination under Suboptimal Immune Protection: Epidemiological Perspectives,” Journal of Theoretical Biology, Vol. 228, No. 4, 2004, pp. 539-549. doi:10.1016/j.jtbi.2004.02.015

[6] G. Zaman and I. H. Jung, “Optimal Vaccination and Treatment in the SIR Epidemic Model,” Proceedings of KSIAM, Vol. 3, No. 2, 2007, pp. 31-33.

[7] R. E. Mickens, “A SIR-Model with Square-Root Dynamics: An NSFD Scheme,” Journal of Difference Equations and Applications, Vol. 16, No. 2-3, 2009, pp. 209-216.

[8] Z. M. Odibat and N. T. Shawafeh, “Generalized Taylor’s Formula,” Computers & Mathematics with Applications, Vol. 186, No. 1, 2007, pp. 286-293. doi:10.1016/j.amc.2006.07.102

[9] W. Lin, “Global Existence Theory and Chaos Control of Fractional Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 332, No. 1, 2007, pp. 709-726.

[10] L. Debnath, “Recent Applications of Fractional Calculus to Science and Engineering,” International Journal of Mathematics and Mathematical Sciences, Vol. 2003, No. 54, 2003, pp. 3413-3442.

[11] I. Podlubny, “Fractional Differential Equations,” Academic Presss, London, 1999.

[12] D. Matignon, “Stability Results for Fractional Differential Equations with Applications to Control Processing,” Computational Engineering in System Application, Vol. 2 1996, p. 963.

[13] V. S. Erturk, Z. M. Odibait and S. Momani, “An approximate Solution of a Fractional Order Differential Equation Model of Human T-Cell Lymphotropic Virus I HTLV-I, Infection of CD4 T-Cells,” Computers & Mathematics with Applications, Vol. 62, No. 3, 2011, pp. 996-1002. doi:10.1016/j.camwa.2011.03.091

[14] S. Miller and B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations,” Willey, New York, 1993.

[15] N. Ozalp and E. Demirci, “A Fractional Order SEIR Model with Vertical Transmission,” Mathematical and Computer Modelling, Vol. 54, No. 1, 2011, pp. 1-6. doi:10.1016/j.mcm.2010.12.051

[16] R. E. Mickens, “Nonstandard Finite Difference Models of Differential Equations,” World Scientific, Singapore, 1994.

[17] R. E. Mickens, “Calculation of Denominator Functions for NSFD Schemes for Differential Equations Satisfying a Positivity Condition,” Numerical Methods for Partial Differential Equations, Vol. 23, No. 3, 2007, pp. 672-691. doi:10.1002/num.20198

[18] R. E. Mickens, “Numerical Integration of Population Models Satisfying Conservation Laws: NSFD Methods,” Journal of Biological Dynamics, Vol. 1, No. 4, 2007, pp. 427-436. doi:10.1080/17513750701605598

[19] R. E. Mickens, R. Buckmire and K. McMurtry, “Numerical Studies of a Nonlinear Heat Equation with Square Root Reaction Term,” Numerical Methods for Partial Differential Equations, Vol. 25, No. 3, 2009, pp. 598-609. doi:10.1002/num.20361

[1] B. Adams and M. Boots, “The Influence of Immune Cross-Reaction on Phase Structure in Resonant Solutions of a Multi-Strain Seasonal SIR Model,” Journal of Theoretical Biology, Vol. 248, No. 1, 2007, pp. 202-211. doi:10.1016/j.jtbi.2007.04.023

[2] R. M. Anderson and R. M. May, “Infectious Disease of Humans, Dynamics and Control,” Oxford University Press, Oxford, 1991.

[3] C. Castillo-Garsow, G. Jordan-Salivia and A. Rodriguez Herrera, “Mathematical Models for Dynamics of Tobacco Use, Recovery and Relapse,” Technical Report Series BU-1505-M, Cornell Uneversity, Ithaca, 2000.

[4] M. Choisy, J. F. Guezan and P. Rohani, “Dynamics of Infectious Diseases and Pulse Vaccination: Teasing Apart the Embedded Resonance Effects,” Journal of Physics D, Vol. 223, No. 1, 2006, pp. 26-35. doi:10.1016/j.physd.2006.08.006

[5] M. G. Gomes, L. J. White and G. F. Medley, “Infection, Reinfection, and Vaccination under Suboptimal Immune Protection: Epidemiological Perspectives,” Journal of Theoretical Biology, Vol. 228, No. 4, 2004, pp. 539-549. doi:10.1016/j.jtbi.2004.02.015

[6] G. Zaman and I. H. Jung, “Optimal Vaccination and Treatment in the SIR Epidemic Model,” Proceedings of KSIAM, Vol. 3, No. 2, 2007, pp. 31-33.

[7] R. E. Mickens, “A SIR-Model with Square-Root Dynamics: An NSFD Scheme,” Journal of Difference Equations and Applications, Vol. 16, No. 2-3, 2009, pp. 209-216.

[8] Z. M. Odibat and N. T. Shawafeh, “Generalized Taylor’s Formula,” Computers & Mathematics with Applications, Vol. 186, No. 1, 2007, pp. 286-293. doi:10.1016/j.amc.2006.07.102

[9] W. Lin, “Global Existence Theory and Chaos Control of Fractional Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 332, No. 1, 2007, pp. 709-726.

[10] L. Debnath, “Recent Applications of Fractional Calculus to Science and Engineering,” International Journal of Mathematics and Mathematical Sciences, Vol. 2003, No. 54, 2003, pp. 3413-3442.

[11] I. Podlubny, “Fractional Differential Equations,” Academic Presss, London, 1999.

[12] D. Matignon, “Stability Results for Fractional Differential Equations with Applications to Control Processing,” Computational Engineering in System Application, Vol. 2 1996, p. 963.

[13] V. S. Erturk, Z. M. Odibait and S. Momani, “An approximate Solution of a Fractional Order Differential Equation Model of Human T-Cell Lymphotropic Virus I HTLV-I, Infection of CD4 T-Cells,” Computers & Mathematics with Applications, Vol. 62, No. 3, 2011, pp. 996-1002. doi:10.1016/j.camwa.2011.03.091

[14] S. Miller and B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations,” Willey, New York, 1993.

[15] N. Ozalp and E. Demirci, “A Fractional Order SEIR Model with Vertical Transmission,” Mathematical and Computer Modelling, Vol. 54, No. 1, 2011, pp. 1-6. doi:10.1016/j.mcm.2010.12.051

[16] R. E. Mickens, “Nonstandard Finite Difference Models of Differential Equations,” World Scientific, Singapore, 1994.

[17] R. E. Mickens, “Calculation of Denominator Functions for NSFD Schemes for Differential Equations Satisfying a Positivity Condition,” Numerical Methods for Partial Differential Equations, Vol. 23, No. 3, 2007, pp. 672-691. doi:10.1002/num.20198

[18] R. E. Mickens, “Numerical Integration of Population Models Satisfying Conservation Laws: NSFD Methods,” Journal of Biological Dynamics, Vol. 1, No. 4, 2007, pp. 427-436. doi:10.1080/17513750701605598

[19] R. E. Mickens, R. Buckmire and K. McMurtry, “Numerical Studies of a Nonlinear Heat Equation with Square Root Reaction Term,” Numerical Methods for Partial Differential Equations, Vol. 25, No. 3, 2009, pp. 598-609. doi:10.1002/num.20361