A Nonstandard Finite Difference Scheme for SIS Epidemic Model with Delay: Stability and Bifurcation Analysis

Author(s)
Agus Suryanto

Affiliation(s)

Department of Mathematics, Faculty of Sciences, Brawijaya University, Malang, Indonesia.

Department of Mathematics, Faculty of Sciences, Brawijaya University, Malang, Indonesia.

ABSTRACT

A numerical scheme for a SIS epidemic model with a delay is constructed by applying a nonstandard finite difference (NSFD) method. The dynamics of the obtained discrete system is investigated. First we show that the discrete system has equilibria which are exactly the same as those of continuous model. By studying the distribution of the roots of the characteristics equations related to the linearized system, we can provide the stable regions in the appropriate parameter plane. It is shown that the conditions for those equilibria to be asymptotically stable are consistent with the continuous model for any size of numerical time-step. Furthermore, we also establish the existence of Neimark-Sacker bifurcation (also called Hopf bifurcation for map) which is controlled by the time delay. The analytical results are confirmed by some numerical simulations.

A numerical scheme for a SIS epidemic model with a delay is constructed by applying a nonstandard finite difference (NSFD) method. The dynamics of the obtained discrete system is investigated. First we show that the discrete system has equilibria which are exactly the same as those of continuous model. By studying the distribution of the roots of the characteristics equations related to the linearized system, we can provide the stable regions in the appropriate parameter plane. It is shown that the conditions for those equilibria to be asymptotically stable are consistent with the continuous model for any size of numerical time-step. Furthermore, we also establish the existence of Neimark-Sacker bifurcation (also called Hopf bifurcation for map) which is controlled by the time delay. The analytical results are confirmed by some numerical simulations.

Cite this paper

A. Suryanto, "A Nonstandard Finite Difference Scheme for SIS Epidemic Model with Delay: Stability and Bifurcation Analysis,"*Applied Mathematics*, Vol. 3 No. 6, 2012, pp. 528-534. doi: 10.4236/am.2012.36080.

A. Suryanto, "A Nonstandard Finite Difference Scheme for SIS Epidemic Model with Delay: Stability and Bifurcation Analysis,"

References

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[2] W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, “Nicholson’s Blowflies (Revisited),” Nature, Vol. 287, No. 5777, 1980, pp. 17-21. doi:10.1038/287017a0

[3] J. Wei and M. Y. Li, “Hopf Bifurcation Analysis in a Delayed Nicholson Blowflies Equation,” Nonlinear Analysis, Vol. 60, No. 7, 2005, pp. 1351-1367. doi:10.1016/j.na.2003.04.002

[4] X. Ding and W. Li, “Stability and Bifurcation of Numerical Discretization Nicholson Blowflies Equation with Delay,” Discrete Dynamics in Nature and Society, Vol. 2006, 2006, pp. 1-12. http://www.hindawi.com/journals/ddns/2007/092959/ref/

[5] J. Wei and X. Zou, “Bifurcation Analysis of a Population Model and the Resulting SIS Epidemic Model with Delay,” Journal of Computational and Applied Mathematics, Vol. 197, No. 1, 2006, pp. 169-187. doi:10.1016/j.cam.2005.10.037

[6] Q. Chi, Y. Qu and J. Wei, “Bifurcation Analysis of an Epidemic Model with Delay as the Control Variable,” Applied Mathematics E-Notes, Vol. 9, 2009, pp. 307-319. http://www.math.nthu.edu.tw/~amen/2009/091020-2.pdf

[7] E. Kunnawuttipreechachan, “Stability of a Numerical Discretization Scheme for the SIS Epidemic Model with a Delay,” Proceedings of the World Congress on Engineering, London, Vol. 3, 30 June-2 July 2010, pp. 19231930. http://www.iaeng.org/publication/WCE2010/WCE2010_pp1923-1930.pdf

[8] A. Suryanto, “Stability and Bifurcation of a Discrete SIS Epidemic Model with a Delay,” Proceedings of the 2nd International Conference on Basic Sciences, Malang, 24-25 February 2012, pp. 1-6.

[9] R. E. Mickens, “Application of Nonstandard Finite Difference Schemes,” World Scientific Publishing Co Pte. Ltd., Singapore City, 2000. doi:10.1142/9789812813251

[10] 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

[11] D. T. Dimitrov and H. V. Kojouharov, “Nonstandard Finite Difference Method for Predator-Prey Models with General Functional Response,” Mathematics and Computers in Simulation, Vol. 78, No. 1, 2008, pp. 1-11. doi:10.1016/j.matcom.2007.05.001

[12] A. J. Arenas, J. A. Morano and J. C. Cortés, “Non-Standard Numerical Method for a Mathematical Model of RSV Epidemiological Transmission,” Computers and Mathematics with Applications, Vol. 56, No. 3, 2008, pp. 670-678. doi:10.1016/j.camwa.2008.01.010

[13] 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, 2008, pp. 622633. doi:10.1016/j.matcom.2008.04.008

[14] A. J. Arenas, G. González-Parra and B. M. Chen-Charpentier, “A Nonstandard Numerical Scheme of Predictor-Corrector Type for Epidemic Models,” Computers and Mathematics with Applications, Vol. 59, No. 12, 2010, pp. 3740-3749. doi:10.1016/j.camwa.2010.04.006

[15] A. Suryanto, “A Dynamically Consistent Nonstandard Numerical Scheme for Epidemic Model with Saturated Incidence Rate,” International Journal of Mathematics and Computation, Vol. 13, No. D11, 2011, pp. 112-123. http://ceser.in/ceserp/index.php/ijmc/article/view/1151

[16] A. Suryanto, “A Dynamically Consistent Numerical Method for SIRS Epidemic Model with Non-Monotone Incidence Rate,” Proceedings of the 7th International Conference on Mathematics, Statistics and Its Application, Bangkok, 7-10 December 2011, pp. 273-282.

[17] C. Zhang, Y. Zu and B. Zheng, “Stability and Bifurcation of a Discrete Red Blood Cell Survival Model,” Chaos, Solitons and Fractal, Vol. 28, No. 2, 2006, pp. 386-394. doi:10.1016/j.chaos.2005.05.042.

[1] K. L. Cooke, P. V. D. Driessche and X. Zou, “Interaction of Maturation Delay and Nonlinear Birth in Population and Epidemic Model,” Journal of Mathematical Biology, Vol. 39, No. 4, 1999, pp. 332-352. doi:10.1007/s002850050194

[2] W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, “Nicholson’s Blowflies (Revisited),” Nature, Vol. 287, No. 5777, 1980, pp. 17-21. doi:10.1038/287017a0

[3] J. Wei and M. Y. Li, “Hopf Bifurcation Analysis in a Delayed Nicholson Blowflies Equation,” Nonlinear Analysis, Vol. 60, No. 7, 2005, pp. 1351-1367. doi:10.1016/j.na.2003.04.002

[4] X. Ding and W. Li, “Stability and Bifurcation of Numerical Discretization Nicholson Blowflies Equation with Delay,” Discrete Dynamics in Nature and Society, Vol. 2006, 2006, pp. 1-12. http://www.hindawi.com/journals/ddns/2007/092959/ref/

[5] J. Wei and X. Zou, “Bifurcation Analysis of a Population Model and the Resulting SIS Epidemic Model with Delay,” Journal of Computational and Applied Mathematics, Vol. 197, No. 1, 2006, pp. 169-187. doi:10.1016/j.cam.2005.10.037

[6] Q. Chi, Y. Qu and J. Wei, “Bifurcation Analysis of an Epidemic Model with Delay as the Control Variable,” Applied Mathematics E-Notes, Vol. 9, 2009, pp. 307-319. http://www.math.nthu.edu.tw/~amen/2009/091020-2.pdf

[7] E. Kunnawuttipreechachan, “Stability of a Numerical Discretization Scheme for the SIS Epidemic Model with a Delay,” Proceedings of the World Congress on Engineering, London, Vol. 3, 30 June-2 July 2010, pp. 19231930. http://www.iaeng.org/publication/WCE2010/WCE2010_pp1923-1930.pdf

[8] A. Suryanto, “Stability and Bifurcation of a Discrete SIS Epidemic Model with a Delay,” Proceedings of the 2nd International Conference on Basic Sciences, Malang, 24-25 February 2012, pp. 1-6.

[9] R. E. Mickens, “Application of Nonstandard Finite Difference Schemes,” World Scientific Publishing Co Pte. Ltd., Singapore City, 2000. doi:10.1142/9789812813251

[10] 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

[11] D. T. Dimitrov and H. V. Kojouharov, “Nonstandard Finite Difference Method for Predator-Prey Models with General Functional Response,” Mathematics and Computers in Simulation, Vol. 78, No. 1, 2008, pp. 1-11. doi:10.1016/j.matcom.2007.05.001

[12] A. J. Arenas, J. A. Morano and J. C. Cortés, “Non-Standard Numerical Method for a Mathematical Model of RSV Epidemiological Transmission,” Computers and Mathematics with Applications, Vol. 56, No. 3, 2008, pp. 670-678. doi:10.1016/j.camwa.2008.01.010

[13] 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, 2008, pp. 622633. doi:10.1016/j.matcom.2008.04.008

[14] A. J. Arenas, G. González-Parra and B. M. Chen-Charpentier, “A Nonstandard Numerical Scheme of Predictor-Corrector Type for Epidemic Models,” Computers and Mathematics with Applications, Vol. 59, No. 12, 2010, pp. 3740-3749. doi:10.1016/j.camwa.2010.04.006

[15] A. Suryanto, “A Dynamically Consistent Nonstandard Numerical Scheme for Epidemic Model with Saturated Incidence Rate,” International Journal of Mathematics and Computation, Vol. 13, No. D11, 2011, pp. 112-123. http://ceser.in/ceserp/index.php/ijmc/article/view/1151

[16] A. Suryanto, “A Dynamically Consistent Numerical Method for SIRS Epidemic Model with Non-Monotone Incidence Rate,” Proceedings of the 7th International Conference on Mathematics, Statistics and Its Application, Bangkok, 7-10 December 2011, pp. 273-282.

[17] C. Zhang, Y. Zu and B. Zheng, “Stability and Bifurcation of a Discrete Red Blood Cell Survival Model,” Chaos, Solitons and Fractal, Vol. 28, No. 2, 2006, pp. 386-394. doi:10.1016/j.chaos.2005.05.042.