Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A NONSTANDARD FINITE DIFFERENCE METHOD APPLIED TO A MATHEMATICAL CHOLERA MODEL

  • Liao, Shu (School of Mathematics and Statistics Chongqing Technology and Business University) ;
  • Yang, Weiming (School of Mathematics and Statistics Chongqing Technology and Business University)
  • Received : 2016.03.21
  • Accepted : 2017.06.26
  • Published : 2017.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we aim to construct a nonstandard finite difference (NSFD) scheme to solve numerically a mathematical model for cholera epidemic dynamics. We first show that if the basic reproduction number is less than unity, the disease-free equilibrium (DFE) is locally asymptotically stable. Moreover, we mainly establish the global stability analysis of the DFE and endemic equilibrium by using suitable Lyapunov functionals regardless of the time step size. Finally, numerical simulations with different time step sizes and initial conditions are carried out and comparisons are made with other well-known methods to illustrate the main theoretical results.

Keywords

References

  1. L. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci. 124 (1994), no. 1, 83-105. https://doi.org/10.1016/0025-5564(94)90025-6
  2. L. Allen and A. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biosci. 63 (2000), no. 1, 1-33. https://doi.org/10.1016/0025-5564(83)90047-0
  3. L. Allen, D. Flores, R. Ratnayake, and J. Herbold, Discrete-time deterministic and stochastic models for the spread of rabies, Appl. Math. Comput. 132 (2002), no. 2-3, 271-292. https://doi.org/10.1016/S0096-3003(01)00192-8
  4. R. Anguelov and J. M. S. Lubuma, Contributions to the mathematics of the nonstandard nite di erence method and applications, Numer. Methods Partial Differential Equations 17 (2001), no. 5, 518-543. https://doi.org/10.1002/num.1025
  5. R. Anguelov, Nonstandard finite difference method by nonlocal approximation, Math. Comput. Simulation 61 (2003), no. 3-6, 465-475. https://doi.org/10.1016/S0378-4754(02)00106-4
  6. A. J. Arenas, G. Gonzalez-Parra, and B. M. Chen-Charpentier, A nonstandard numer- ical scheme of predictor-corrector type for epidemic models, Comput. Math. Appl. 59 (2010), no. 12, 3740-3749. https://doi.org/10.1016/j.camwa.2010.04.006
  7. F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, New York, NY, 2001.
  8. V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev Epidemiol Sante Publique 27 (1979), 121-132.
  9. C. Castillo-Chavez and A. A. Yakubu, Discrete-time SIS models with complex dynamics, Nonlinear Anal. 47 (2001), no. 7, 4753-4762. https://doi.org/10.1016/S0362-546X(01)00587-9
  10. C. T. Codeco, Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMC Infectious Diseases (2001), 1.1.
  11. S. M. Garba, A. B. Gumel, and J. M. S. Lubuma, Dynamically-consistent non-standard finite difference method for an epidemic model, Math. Comput. Modelling 53 (2011), no. 1-2, 131-150. https://doi.org/10.1016/j.mcm.2010.07.026
  12. F. Guerrero, G. Gonzalez-Parra, and A. J. Arenas, A nonstandard finite difference numerical scheme applied to a mathematical model of the prevalence of smoking in Spain: a case study, Comput. Appl. Math. 33 (2013), no. 1, 1-13. https://doi.org/10.1007/s40314-013-0039-1
  13. A. B. Gumel, C. Connell McCluskey, and J. Watmough, An SVEIR model for assessing potential impact of an imperfect anti-SARS vaccine, Math. Biosci. Eng. 3 (2006), no. 3, 485-512. https://doi.org/10.3934/mbe.2006.3.485
  14. D. M. Hartley, J. G. Morris, and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLoS Medicine 3 (2006), 63-69. https://doi.org/10.1371/journal.pmed.0030063
  15. S. D. Hove-Musekwa, F. Nyabadza, C. Chiyaka, P. Das, A. Tripathi, and Z. Mukan- davire, Modelling and analysis of the effects of malnutrition in the spread of cholera, Math. Comput. Modelling 53 (2011), no. 9-10, 1583-1595. https://doi.org/10.1016/j.mcm.2010.11.060
  16. Z. Hu, Z. Teng, and H. Jiang, Stability analysis in a class of discrete SIRS epidemic models, Nonlinear Anal. Real World Appl. 13 (2012), no. 5, 2017-2033. https://doi.org/10.1016/j.nonrwa.2011.12.024
  17. Z. Hu, Z. Teng, and L. Zhang, Stability and bifurcation analysis in a discrete SIR epidemic model, Math. Comput. Simulation 97 (2014), no. 2, 80-93. https://doi.org/10.1016/j.matcom.2013.08.008
  18. L. Jodar, R. J. Villanueva, A. J. Arenas, and G. C. Gonzalez, Nonstandard numerical methods for a mathematical model for in uenza disease, Math. Comput. Simulation 79 (2008), no. 3, 622-633. https://doi.org/10.1016/j.matcom.2008.04.008
  19. S. Liao and J. Wang, Stability analysis and application of a mathematical cholera model, Math. Biosci. Eng. 8 (2011), no. 3, 733-752. https://doi.org/10.3934/mbe.2011.8.733
  20. S. Liao and W. Yang, On the dynamics of a vaccination model with multiple transmission way, Int. J. Appl. Math. Comput. Sci. 23 (2013), no. 4, 761-772. https://doi.org/10.2478/amcs-2013-0057
  21. X. Ma, Y. Zhou, and H. Cao, Global stability of the endemic equilibrium of a discrete SIR epidemic model, Adv. Di erence Equ. 2013 (2013), 19 pp. https://doi.org/10.1186/1687-1847-2013-19
  22. R. E. Mickens, Exact solutions to a finite difference model of a nonlinear reaction- advection equation: implications for numerical analysis, Numer. Methods Partial Differential Equations 5 (1989), no. 4, 313-325. https://doi.org/10.1002/num.1690050404
  23. R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.
  24. R. E. Mickens, Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations, J. Difference Equ. Appl. 11 (2005), no. 7, 645-653. https://doi.org/10.1080/10236190412331334527
  25. R. E. Mickens, Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Wiley Inter. Sci. 23 (2006), no. 3, 672-691.
  26. R. E. Mickens, Discretizations of nonlinear differential equations using explicit nonstandard methods, J. Comput. Appl. Math. 110 (1999), no. 1, 181-185. https://doi.org/10.1016/S0377-0427(99)00233-2
  27. A. K. Misra and V. Singh, A delay mathematical model for the spread and control of water borne diseases, J. Theoret. Biol. 301 (2012), 49-56. https://doi.org/10.1016/j.jtbi.2012.02.006
  28. Z. Mukandavire and W. Garira, Sex-structured HIV/AIDS model to analyse the effects of condom use with application to Zimbabwe, J. Math. Biol. 54 (2007), 669-699. https://doi.org/10.1007/s00285-006-0063-5
  29. Z. Mukandavire, S. Liao, J. Wang, H. Ga , D. L. Smith, and J. G. Morris, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proceedings of the National Academy of Sciences of the United States of America 108 (2011), no. 21, 8767-8772.
  30. Y. Muroya, A. Bellen, Y. Enatsu, and Y. Nakata, Global stability for a discrete epi-demic model for disease with immunity and latency spreading in a heterogeneous host population, Nonlinear Anal. Real World Appl. 13 (2012), no. 1, 258-274. https://doi.org/10.1016/j.nonrwa.2011.07.031
  31. A. Ramani, A. S. Carstea, R. Willox, and B. Grammaticos, Oscillating epidemics: a discrete-time model, Phys. A 333 (2004), no. 1, 278-292. https://doi.org/10.1016/j.physa.2003.10.051
  32. J. Satsuma, R. Willox, A. Ramani, B. Grammaticos, and A. S. Carstea, Extending the SIR epidemic model, Phys. A 336 (2004), no. 3, 369-375. https://doi.org/10.1016/j.physa.2003.12.035
  33. M. D. La Sen and S. Alonso-Quesada, Some equilibrium, stability, instability and os- cillatory results for an extended discrete epidemic model with evolution memory, Adv. Difference Equ. 2013 (2013), 29 pp. https://doi.org/10.1186/1687-1847-2013-29
  34. M. Sekiguchi and E. Ishiwata, Global dynamics of a discretized SIRS epidemic model with time delay, J. Math. Anal. Appl. 371 (2010), no. 1, 195-202. https://doi.org/10.1016/j.jmaa.2010.05.007
  35. A. Suryanto, W. M. Kusumawinahyu, I. Darti, and I. Yanti, Dynamically consistent discrete epidemic model with modi ed saturated incidence rate, Appl. Math. Comput. 32 (2013), no. 2, 373-383. https://doi.org/10.1007/s40314-013-0026-6
  36. J. H. Tien and D. J. D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull. Math. Biol. 72 (2010), no. 6, 1502-1533.
  37. R. Villanueva, A. Arenas, and G. Gonzalez-Parra, A nonstandard dynamically consistent numerical scheme applied to obesity dynamics, J. Appl. Math. 2008 (2008), Art. ID 640154, 14 pp.
  38. J. Wang and S. Liao, A generalized cholera model and epidemic-endemic analysis, J. Biol. Dyn. 6 (2012), no. 2, 568-589. https://doi.org/10.1080/17513758.2012.658089
  39. L. Wang, Z. Teng, and H. Jiang, Global attractivity of a discrete SIRS epidemic model with standard incidence rate, Math. Methods Appl. Sci. 36 (2013), no. 5, 601-619. https://doi.org/10.1002/mma.2734
  40. World Health Organization. Available from: http://www.who.org.
  41. X. Zhou, J. Cui, and Z. Zhang, Global results for a cholera model with imperfect vaccination, J. Franklin Inst. 349 (2012), no. 3, 770-791. https://doi.org/10.1016/j.jfranklin.2011.09.013