The Dynamic Behavior of a Discrete Vertical and Horizontal Transmitted Disease Model under Constant Vaccination

Full-Text HTML XML Download Download as PDF (Size:384KB) PP. 171-184
DOI: 10.4236/ijmnta.2016.54017    389 Downloads   535 Views  

ABSTRACT

In this paper, a class of discrete vertical and horizontal transmitted disease model under constant vaccination is researched. Under the hypothesis of population being constant size, the model is transformed into a planar map and its equilibrium points and the corresponding eigenvalues are solved out. By discussing the influence of coefficient parameters on the eigenvalues, the hyperbolicity of equilibrium points is determined. By getting the equations of flows on center manifold, the direction and stability of the transcritical bifurcation and flip bifurcation are discussed.

Cite this paper

Li, M. , Liu, X. and Zhou, X. (2016) The Dynamic Behavior of a Discrete Vertical and Horizontal Transmitted Disease Model under Constant Vaccination. International Journal of Modern Nonlinear Theory and Application, 5, 171-184. doi: 10.4236/ijmnta.2016.54017.

References

[1] Piyawong, W., Twizell, E.H. and Gumel, A.B. (2003) An Unconditionally Convergent Finite-Difference Scheme for the SIR Model. Applied Mathematics and Computation, 146, 611-625.
https://doi.org/10.1016/S0096-3003(02)00607-0
[2] Pourabbas, E., d’Onofrio, A. and Rafanelli, M. (2001) A Method to Estimate the Incidence of Communicable Diseases under Seasonal Fluctuations with Application to Cholera. Applied Mathematics and Computation, 118, 161-174.
https://doi.org/10.1016/S0096-3003(99)00212-X
[3] Beretta, E. and Takeuchi, Y. (1997) Convergence Results in SIR Epidemic Model with Varying Population Sizes. Nonlinear Analysis, 28, 1909-1921.
https://doi.org/10.1016/S0362-546X(96)00035-1
[4] Meng, X., Chen, L. and Song, Z. (2007) The Global Dynamics Behaviors for a New Delay SEIR Epidemic Disease Model with Vertical Transmission and Pulse Vaccination. Applied Mathematics and Computation, 28, 1259-1271.
https://doi.org/10.1007/s10483-007-0914-x
[5] Allen, L.J.S. (1994) Some Discrete-Time SI, SIR and SIS Epidemic Models. Mathematical Biosciences, 124, 83-105.
https://doi.org/10.1016/0025-5564(94)90025-6
[6] Ma, Z. Zhou, Y. Wang, W. and Jin, Z. (2004) Mathematical Modelling and Research of Epidemic Dynamical Systems (in Chinese). Science Press, Beijing.
[7] Zhou, X., Li, X. and Wang, W.S. (2014) Bifurcations for a Deterministic SIR Epidemic Model in Discrete Time. Advances in Difference Equations, 168.
https://doi.org/10.1186/1687-1847-2014-168
[8] Liao, X., Wang, H., Huang, X., Zeng, W. and Zhou, X. (2015) The Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete Time. Applied Mathematics, 6, 1665-1675.
https://doi.org/10.4236/am.2015.610148
[9] Meng, X., Chen, L. and Cheng, H. (2007) Two Profitless Delays for the SEIRS Epidemic Disease Model with Nonlinear Incidence and Pulse Vaccination. Applied Mathematics and Computation, 186, 516-529.
https://doi.org/10.1016/j.amc.2006.07.124
[10] Agur, Z.L., et al. (1993) Pulse Mass Measles Vaccination across Age Cohorts. Proceedings of the National Academy of Sciences of the USA, 90, 11698-11702.
https://doi.org/10.1073/pnas.90.24.11698
[11] Shulgin, B., et al. (1998) Pulse Vaccination Strategy in the SIR Epidemic Model. Bulletin of Mathematical Biology, 60, 1-26.
https://doi.org/10.1016/S0092-8240(98)90005-2
[12] Meng, X., Chen, L. and Song, Z. (2007) The Global Dynamics Behaviors for a New Delay SEIR Epidemic Disease Model with Vertical Transmission and Pulse Vaccination. Applied Mathematics and Mechanics (English Edition), 28, 1259-1271.
https://doi.org/10.1007/s10483-007-0914-x
[13] Meng, X. and Chen, L. (2008) The Dynamics of a New SIR Epidemic Model Concerning Pulse Vaccination Strategy. Applied Mathematics and Computation, 197, 582-597.
https://doi.org/10.1016/j.amc.2007.07.083
[14] Anderson, R.M. and May, R.M. (1991) Infections Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford.
[15] Wiggins, S. (1990) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York.
https://doi.org/10.1007/978-1-4757-4067-7
[16] Guckenheimer, J. and Holmes, P. (1983) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York.
https://doi.org/10.1007/978-1-4612-1140-2

  
comments powered by Disqus

Copyright © 2017 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.