The Stability of Highly Pathogenic Avian Influenza Epidemic Model with Saturated Contact Rate

Yakui  Xue; Shuqin  Che; Likang  Ma

doi:10.4236/am.2014.521313

Shuqin Che¹, Yakui Xue¹, Likang Ma²

Abstract: In this paper we present a highly pathogenic Avian influenza epidemic model with saturated contact rate. According to study of the dynamics, we calculated the basic reproduction number of the model. Through the analysis of this model, we have the following conclusion: if R0 ≤ 1, there is only one disease-free equilibrium which is globally stable, the disease will die; if R0 > 1, there is only one endemic equilibrium which is globally stable, disease will be popular.

Keywords: Avian Influenza, The Basic Reproduction Number, Stability

Cite this paper: Che, S. , Xue, Y. and Ma, L. (2014) The Stability of Highly Pathogenic Avian Influenza Epidemic Model with Saturated Contact Rate. Applied Mathematics, 5, 3365-3371. doi: 10.4236/am.2014.521313.

References

[1] Iwami, S., Takeuchi, Y. and Liu, X.N. (2007) Avian-Human Influenza Epidemic Model. Mathematical Biosciences, 207, 1-25.
http://dx.doi.org/10.1016/j.mbs.2006.08.001

[2] Feng, Z.J. (2013) Human Infect with Avian Influenza H7N9 Treatment Program. Encyclopedia of Medical Network, 4.
http://www.baikemy.com/jiankangkepu/10339395895809

[3] Iwami, S., Takeuchi, Y., Korobeinikov, A. and Liu, X. (2008) Prevention of Avian Influenza Epidemic: What Policy Should We Choose. Journal of Theoretical Biology, 252, 732-741.
http://dx.doi.org/10.1016/j.mbs.2006.08.001

[4] Rao, D.M., Chernyakhovsky, A. and Rao, V. (2009) Modeling and Analysis of Global Epidemiology of Avian Influenza. Environmental Modeling & Software, 24, 124-134.
http://dx.doi.org/10.1016/j.envsoft.2008.06.011

[5] Wang, W. (2006) Backward Bifurcation of an Epidemic Treatment. Mathermatical Biosciences, 201, 58-71.
http://dx.doi.org/10.1016/j.mbs.2005.12.022

[6] Zhang, X. and Liu, X. (2008) Backford Bifurcation of an Epidemic Model with Saturated Treatment Function. Journal of Mathematical Analysis and Applications, 348, 433-443.
http://dx.doi.org/10.1016/j.jmaa.2008.07.042

[7] Capasso, V. and Serio, G. (1978) A Generalization of the Kermack-Mckendrick Deterministic Epidemic Model. Mathematical Biosciences, 42, 43-61.
http://dx.doi.org/10.1016/0025-5564(78)90006-8

[8] Guo, S.M., Guo, L.N. and Li, X.Z. (2010) Analysis of an Avian Influenza Epidemic Model with Saturation Treatment. Mathematics in Practice and Theory, 40, 134-137.

[9] Li, Y.-L. (2009) Study on SI Transmission Model of Highly Pathogenic Avian Influenza. Journal of Anhui Agricultural Sciences, 37, 13603-13605.

[10] Liang, R.H. (2007) The Construction and Application of SEID Model for the Disseminate Mechanism of Highly Pathogenic Avian Influenza (HPAI). Journal of Xinyang Normal University: Natural Science Edition, 20, 262-265.

[11] Lasalle, J.P. (1986) The Stability and Control of Discrete Process. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-1076-4

[12] van den Driessche, P. and Watmough, J. (2007) Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission. Mathematical Biosciences, 196, 1679-1684.

[13] Ma, Z.-E. and Zhou, Y.-C. (2001) The Theory of Ordinary Differential Equation Qualitative and Stability. Science Press, Beijing.

[14] Lasalle, J.P. (1976) The Stability of Dynamical Systems. Regional Conference Series in Applied Mathematics Philadelphia, SIAM, 21, 418-420.