Stability Analysis and Dynamics Preserving Nonstandard Finite Difference Schemes for a Malaria Model

Mathematical Population Studies

Volumn 20, Issue 2, 2013, Pages 101-122

  Anguelov, Roumen ^a   Dumont, Yves ^b   Lubuma, Jean ^a   Mureithi, Eunice ^a  

^a UNIVERSITY OF PRETORIA   (South Africa)

^b UMR AMAP   (France)

Author keywords

bifurcation analysis; dynamic consistency; global asymptotic stability; malaria; nonstandard finite difference

Indexed keywords

BIFURCATION; FINITE DIFFERENCE METHOD; MALARIA; MORTALITY; NUMERICAL MODEL; STABILITY ANALYSIS;

EID: 84877299952     PISSN: 08898480     EISSN: 1547724X     Source Type: Journal    
DOI: 10.1080/08898480.2013.777240     Document Type: Article

References (30)

1. Anguelov, R., Dumont, Y., Lubuma, J.M.-S., and Shillor, M. (2009a). Comparison of some standard and nonstandard numerical methods for the MSEIR epidemiological model. In T. Simos, G. Psihoyios, Ch. Tsitouras, (Eds.), Proceedings of the International Conference of Numerical Analysis and Applied Mathematics (Crete, Greece, 18-22, September 2009), American Institute of Physics Conference Proceedings-AIP 1168, 2: 1209-1212.
2. Anguelov, R., Dumont, Y., Lubuma, J.M.-S., and Shillor, M. (2009b). Dynamically consistent non-standard finite difference schemes for the MSEIR epidemiological model. In T. Simos, G. Psihoyios, Ch. Tsitouras (Eds.), Proceedings of the International Conference of Numerical Analysis and Applied Mathematics, American Institute of Physics Conference Proceedings-AIP 1168, 2: 1213-1216.
3. Anguelov, R., Dumont, Y., Lubuma, J.M.-S., and Shillor, M. (2010a). Dynamically consistent nonstandard finite difference schemes for epidemiological models. University of Pretoria Technical Report, UPWT/2009/22. Submitted for publication.
4. Anguelov, R. and Lubuma, J.M.-S. (2001). Contributions to the mathematics of the nonstandard finite difference method and applications. Numerical Methods for Partial Differential Equations, 17: 518 - 543.
5. Anguelov, R. and Lubuma, J.M.-S. (2003). Nonstandard finite difference method by nonlocal approximation. Mathematics and Computers in Simulation, 61 (3-6): 465 - 475.
6. Anguelov, R., Lubuma, J.M.-S., and Shillor, M. (2011). Topological dynamic consistency of nonstandard finite difference schemes for dynamical systems. Journal of Difference Equations and Applications, 17: 1769 - 1791.
7. Anguelov, R., Dumont, Y., and Lubuma, J.M.-S. (2012). On nonstandard finite difference schemes in biosciences. Proceedings of the 4th International Conference on Application of Mathematics in Technical and Natural Sciences, American Institute of Physics Conference Proceedings-AIP 1487: 212-223.
8. Auslander, J. and Seibert, P. (1964). Prolongations and stability in dynamical systems. Annales de l'Institut Fourier, 14: 237 - 267.
9. Berman, A. and Plemmons, R.J. (1979). Nonnegative Matrices in the Mathematical Sciences. New York, NY: Academic Press.
10. Carr, J. (1981). Applications of Centre Manifold Theory. New York, NY: Springer-Verlag.
11. 0 and its role on global stability. In C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschne, and A.A. Yakubu, (Eds.), Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction. New York: Springer, pp. 229 - 250.
12. Castillo-Chavez, C. and Song, B. (2004). Dynamical model of tuberculosis and their applications. Mathematical Biosciences and Engineering 1 (2): 361 - 404.
13. Chitnis, N. (2005). Using mathematical models in controlling the spread of Malaria. (Doctoral dissertation). Program in Applied Mathematics, University of Arizona, Tucson, AZ.
14. Chitnis, N., Cushing, J.M., and Hyman, J.M. (2006). Bifurcation analysis of a mathematical model for Malaria transmission. SIAM Journal of Applied Mathematics, 67(1): 24-45.
15. Chitnis, N., Hyman, J.M., and Cushing, J.M. (2008). Determining important parameters in the Spread through the sensitivity analysis of a mathematical model. Bulletin of Mathematical Biology, 70: 1272 - 1296.
16. Diekmann, O. and Heesterbeek, J.A. (2000). Mathematical epidemiology of infectious diseases. New York, NY: Wiley.
17. Dumont, Y., Chiroleu, F., and Domerg, C. (2008). On a temporal model for the Chikungunya disease: Modeling, theory and numerics. Mathematical Biosciences, 213: 70 - 81.
18. Dumont, Y. and Chiroleu, F. (2010). Vector-control for the Chikungunya disease. Mathematical Biosciences and Engineering, 7 (2): 315 - 348.
19. Dumont, Y. and Tchuenche, J.M. (2012). Mathematical studies on the sterile insect technique for the chikungunya disease and dedes albopictus. Journal of Mathematical Biology, 65: 809 - 854.
20. Garba, S.M., Gumel, A.B., and Lubuma, J.M.-S. (2011). Dynamically-consistent nonstandard finite difference method for an epidemic model. Mathematical and Computer Modelling, 53: 131 - 150.
21. Jacquez, J. and Simon, C. (1993). Qualitative theory of compartmental systems. SIAM Review, 35 (1): 43 - 79.
22. Kamgang, J.C. and Sallet, G. (2008). Computation of threshold conditions for epidemiological models and global stability of the disease free equilibrium (DFE). Mathematical Biosciences, 213: 1 - 12.
23. LaSalle, J.P. (1968). Stability theory for ordinary differential equations. Journal of Differential Equations, 4: 57 - 65.
24. LaSalle, J.P. (1976). The Stability of Dynamical Systems. Philadelphia, PA: SIAM.
25. Macdonald, G. (1957). The Epidemiology and Control of Malaria. London, England: Oxford University Press.
26. Mickens, R.E. (1994). Nonstandard Finite Difference Models of Differential Equations. Singapore: World Scientific.
27. Mickens, R.E. (2007). Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition. Numerical Methods for Partial Differential Equations, 23: 672 - 691.
28. Ross, R. (1911). The Prevention of Malaria., 2nd ed. London, England: Murray.
29. Van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180 (1-2): 29 - 48.
30. Walter, W. (1970). Differential and Integral Inequalities. New York, NY: Springer-Verlag.