TY - JOUR AU - Du, Wenju AU - Zhang, Jiangang AU - Qin, Shuang AU - Yu, Jianning PY - 2016 TI - Bifurcation analysis in a discrete SIR epidemic model with the saturated contact rate and vertical transmission JO - Journal of Nonlinear Sciences and Applications SP - 4976--4989 VL - 9 IS - 7 AB - 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. SN - ISSN 2008-1901 UR - http://dx.doi.org/10.22436/jnsa.009.07.02 DO - 10.22436/jnsa.009.07.02 ID - Du2016 ER - TY - JOUR TI - Some discrete-time SI, SIR and SIS epidemic models AU - L. J. Allen JO - Math. Biosci. PY - 1994 DA - 1994// VL - 124 ID - Allen1994 ER - TY - BOOK TI - Infectious diseases of humans: dynamics and control AU - R. M. Anderson AU - R. M. May AU - B. Anderson PB - Oxford Univ. Press PY - 1992 DA - 1992// CY - Oxford ID - Anderson1992 ER - TY - JOUR TI - Seasonality and period-doubling bifurcations in an epidemic model AU - J. L. Aron AU - I. B. Schwartz JO - J. Theoret. Biol. PY - 1984 DA - 1984// VL - 110 ID - Aron1984 ER - TY - JOUR TI - Dynamics of measles epidemics: estimating scaling of transmission rates using a time series SIR mode AU - O. N. Bjørnstad AU - B. F. Finkenstaedt AU - B. T. Greenfell JO - Ecol. Monogr. PY - 2002 DA - 2002// VL - 72 ID - Bjørnstad2002 ER - TY - JOUR TI - Complex dynamics of discrete SEIS models with simple demography AU - H. Cao AU - Y. Zhou AU - B. Song JO - Discrete Dyn. Nat. Soc. PY - 2011 DA - 2011// VL - 2011 ID - Cao2011 ER - TY - JOUR TI - Discrete-time SIS models with complex dynamics AU - C. Castillo-Chavez AU - A. A. Yakubu JO - Proceedings of the Third World Congress of Nonlinear Analysts, Nonliear Anal. PY - 2001 DA - 2001// VL - 47 ID - Castillo-Chavez2001 ER - TY - JOUR TI - Bifurcation and chaotic behavior of a discrete singular biological economic system AU - B. Chen AU - J. Chen JO - Appl. Math. Comput. PY - 2012 DA - 2012// VL - 219 ID - Chen2012 ER - TY - JOUR TI - Bifurcation analysis and chaos in a discrete reduced Lorenz system AU - E. M. Elabbasy AU - A. A. Elsadany AU - Y. Zhang JO - Appl. Math. Comput. PY - 2014 DA - 2014// VL - 228 ID - Elabbasy2014 ER - TY - JOUR TI - Resonance and bifurcation in a discrete-time predator-prey system with Holling functional response AU - R. K. Ghaziani AU - W. Govaerts AU - C. Sonck JO - Nonlinear Anal. Real World Appl. PY - 2012 DA - 2012// VL - 13 ID - Ghaziani2012 ER - TY - BOOK TI - Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences AU - J. Guckenheimer AU - P. Holmes PB - Springer-Verlag PY - 1983 DA - 1983// CY - New York ID - Guckenheimer1983 ER - TY - JOUR TI - Bifurcation and chaotic behavior of a discrete-time predator-prey system AU - Z. He AU - X. Lai JO - Nonlinear Anal. Real World Appl. PY - 2011 DA - 2011// VL - 12 ID - He2011 ER - TY - JOUR TI - Global analysis of SIR epidemic model with the saturated contact rate and vertical transmission AU - X. L. Hu AU - F. G. Sun AU - C. X. Wang JO - Basic Sci. J. textile Univ. PY - 2010 DA - 2010// VL - 23 ID - Hu2010 ER - TY - JOUR TI - Stability analysis in a class of discrete SIRS epidemic models AU - Z. Hu AU - Z. Teng AU - H. Jiang JO - Nonlinear Anal. Real World Appl. PY - 2012 DA - 2012// VL - 13 ID - Hu2012 ER - TY - JOUR TI - Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response AU - Z. Hu AU - Z. Teng AU - L. Zhang JO - Nonlinear Anal. Real World Appl. PY - 2011 DA - 2011// VL - 12 ID - Hu2011 ER - TY - JOUR TI - Stability and bifurcation analysis in a discrete SIR epidemic model AU - Z. Hu AU - Z. Teng AU - L. Zhang JO - Math. Comput. Simulation PY - 2014 DA - 2014// VL - 97 ID - Hu2014 ER - TY - JOUR TI - A contribution to the mathematical theory of epidemics AU - W. O. Kermack AU - A. G. McKendrick JO - Proc. A., Math. Phys. Eng. Sci. PY - 1927 DA - 1927// VL - 115 ID - Kermack1927 ER - TY - JOUR TI - Dynamics of a discrete food-limited population model with time delay AU - Y. Li JO - Appl. Math. Comput. PY - 2012 DA - 2012// VL - 218 ID - Li2012 ER - TY - BOOK TI - Dynamical models, stability, symbolic dynamics and chaos AU - C. Robinson PB - CRC-Press PY - 1999 DA - 1999// CY - London ID - Robinson1999 ER - TY - JOUR TI - Seasonal dynamics of recurrent epidemics AU - L. Stone AU - R. Olinky AU - A. Huppert JO - Nature PY - 2007 DA - 2007// VL - 446 ID - Stone2007 ER - TY - JOUR TI - Further investigations into the stability and bifurcation of a discrete predator-prey model AU - C. Wang AU - X. Li JO - J. Math. Anal. Appl. PY - 2015 DA - 2015// VL - 422 ID - Wang2015 ER - TY - JOUR TI - Bifurcations analysis and tracking control of an epidemic model with nonlinear incidence rate AU - N. Yi AU - P. Liu AU - Q. Zhang JO - Appl. Math. Model. PY - 2012 DA - 2012// VL - 36 ID - Yi2012 ER - TY - JOUR TI - Bifurcation analysis in a discrete differential-algebraic predator-prey system AU - G. Zhang AU - Y. Shen AU - B. Chen JO - Appl. Math. Model. PY - 2014 DA - 2014// VL - 38 ID - Zhang2014 ER - TY - JOUR TI - Stability and bifurcation analysis of reactiondiffusion neural networks with delays AU - H. Zhao AU - J. Yuan AU - X. Zhang JO - Neurocomputing PY - 2015 DA - 2015// VL - 147 ID - Zhao2015 ER -