Bifurcation analysis in a discrete SIR epidemic model with the saturated contact rate and vertical transmission

Volume 9, Issue 7, pp 4976--4989 http://dx.doi.org/10.22436/jnsa.009.07.02

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Authors

Wenju Du - School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, China. Jiangang Zhang - Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, China. Shuang Qin - Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, China. Jianning Yu - School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, China.

Abstract

The aim of paper is dealing with the dynamical behaviors of a discrete SIR epidemic model with the saturated contact rate and vertical transmission. More precisely, we investigate the local stability of equilibriums, the existence, stability and direction of flip bifurcation and Neimark-Sacker bifurcation of the model by using the center manifold theory and normal form method. Finally, the numerical simulations are provided for justifying the validity of the theoretical analysis.

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Keywords

• Discrete SIR epidemic model
• stability
• flip bifurcation
• Neimark-Sacker bifurcation.

MSC

•  92D30
•  37N25
•  39A28

References

• [1] L. J. Allen, Some discrete-time SI, SIR and SIS epidemic models, Math. Biosci., 124 (1994), 83--105


• [2] R. M. Anderson, R. M. May, B. Anderson, Infectious diseases of humans: dynamics and control, Oxford Univ. Press, Oxford (1992)


• [3] J. L. Aron, I. B. Schwartz, Seasonality and period-doubling bifurcations in an epidemic model, J. Theoret. Biol., 110 (1984), 665--679


• [4] O. N. Bjørnstad, B. F. Finkenstaedt, B. T. Greenfell, Dynamics of measles epidemics: estimating scaling of transmission rates using a time series SIR mode, Ecol. Monogr., 72 (2002), 169--184


• [5] H. Cao, Y. Zhou, B. Song, Complex dynamics of discrete SEIS models with simple demography, Discrete Dyn. Nat. Soc., 2011 (2011), 21 pages


• [6] C. Castillo-Chavez, A. A. Yakubu, Discrete-time SIS models with complex dynamics, Proceedings of the Third World Congress of Nonlinear Analysts, Nonliear Anal., 47 (2001), 4753--4762


• [7] B. Chen, J. Chen, Bifurcation and chaotic behavior of a discrete singular biological economic system, Appl. Math. Comput., 219 (2012), 2371--2386


• [8] E. M. Elabbasy, A. A. Elsadany, Y. Zhang, Bifurcation analysis and chaos in a discrete reduced Lorenz system, Appl. Math. Comput., 228 (2014), 184--194


• [9] R. K. Ghaziani, W. Govaerts, C. Sonck, Resonance and bifurcation in a discrete-time predator-prey system with Holling functional response, Nonlinear Anal. Real World Appl., 13 (2012), 1451--1465


• [10] J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, Springer-Verlag, New York (1983)


• [11] Z. He, X. Lai, Bifurcation and chaotic behavior of a discrete-time predator-prey system, Nonlinear Anal. Real World Appl., 12 (2011), 403--417


• [12] X. L. Hu, F. G. Sun, C. X. Wang, Global analysis of SIR epidemic model with the saturated contact rate and vertical transmission, Basic Sci. J. textile Univ., 23 (2010), 120--122


• [13] Z. Hu, Z. Teng, H. Jiang, Stability analysis in a class of discrete SIRS epidemic models, Nonlinear Anal. Real World Appl., 13 (2012), 2017--2033


• [14] Z. Hu, Z. Teng, L. Zhang, Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response, Nonlinear Anal. Real World Appl., 12 (2011), 2356--2377


• [15] Z. Hu, Z. Teng, L. Zhang, Stability and bifurcation analysis in a discrete SIR epidemic model, Math. Comput. Simulation, 97 (2014), 80--93


• [16] W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. A., Math. Phys. Eng. Sci., 115 (1927), 700--721


• [17] Y. Li, Dynamics of a discrete food-limited population model with time delay, Appl. Math. Comput., 218 (2012), 6954--6962


• [18] C. Robinson, Dynamical models, stability, symbolic dynamics and chaos, CRC-Press, London (1999)


• [19] L. Stone, R. Olinky, A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533--536


• [20] C. Wang, X. Li, Further investigations into the stability and bifurcation of a discrete predator-prey model, J. Math. Anal. Appl., 422 (2015), 920--939


• [21] N. Yi, P. Liu, Q. Zhang, Bifurcations analysis and tracking control of an epidemic model with nonlinear incidence rate, Appl. Math. Model., 36 (2012), 1678--1696


• [22] G. Zhang, Y. Shen, B. Chen, Bifurcation analysis in a discrete differential-algebraic predator-prey system, Appl. Math. Model., 38 (2014), 4835--4848


• [23] H. Zhao, J. Yuan, X. Zhang, Stability and bifurcation analysis of reactiondiffusion neural networks with delays, Neurocomputing, 147 (2015), 280--290