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Bifurcation analysis of a discrete SIS model with bilinear incidence depending on new infection
1. | Department of Mathematics, Shaanxi University of Science & Technology, Xi'an, 710021, China |
2. | Department of Mathematics, Xi'an Jiaotong University, Xi'an, 710049 |
3. | Department of Mathematics, Xi’an Jiaotong University, Xi’an, 710049 |
References:
[1] |
L. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci., 124 (1994), 83-105.
doi: 10.1016/0025-5564(94)90025-6. |
[2] |
L. Allen and A. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biosci., 163 (2000), 1-33.
doi: 10.1016/S0025-5564(99)00047-4. |
[3] |
L. Allen and P. van den Driessche, The basic reproduction number in some discrete-time epidemic models, J. Differ. Equ. Appl., 14 (2008), 1127-1147.
doi: 10.1080/10236190802332308. |
[4] |
R. Arreola, A. Crossa, M. C. Velasco and A. A. Yakubu, Discrete-time SEIS models with exogenous re-infection and dispersal between two patches., Available from: \url{http://mtbi.asu.edu/files/Discrete_time_SEIS_Models_with_Exogenous_Reinfection_and_Dispersal_between_Two_Patches.pdf}., ().
|
[5] |
W. J. Beyn and J. Lorenz, Center manifolds of dynamical systems under discretization, Numer. Funct. Anal. Optimiz., 9 (1987), 381-414.
doi: 10.1080/01630568708816239. |
[6] |
H. Cao, Z. Dou, X. Liu, F. Zhang, Y. Zhou and Z. Ma, The impact of antiretroviral therapy on the basic reproductive number of HIV transmission, Math. Model. Appl., 1 (2012), 33-37. |
[7] |
H. Cao, Y. Xiao and Y. Zhou, The dynamics of a discrete SEIT model with age and infection-age structures, INT. J. Bio., 5 (2012), 61-76.
doi: 10.1142/S1793524512600042. |
[8] |
H. Cao and Y. Zhou, The discrete age-structured SEIT model with application to tuberculosis transmission in China, Math. Comput. Model., 55 (2012), 385-395.
doi: 10.1016/j.mcm.2011.08.017. |
[9] |
H. Cao and Y. Zhou, The basic reproduction number of discrete SIR and SEIS models with periodic parameters, Discrete Cont. Dyn. Sys. B, 18 (2013), 37-56. |
[10] |
H. Cao, Y. Zhou and B. Song, Complex dynamics of discrete SEIS models with simple demography, Discrete Dyn. Nat. Soc., (2011), Art. ID 653937, 21 pp.
doi: 10.1155/2011/653937. |
[11] |
C. Castillo-Chavez and A. A. Yakubu, Discrete-time SIS models with complex dynamics, Nonliear Anal. TMA, 47 (2001), 4753-4762.
doi: 10.1016/S0362-546X(01)00587-9. |
[12] |
C. Castillo-Chavez and A. A. Yakubu, Discrete-time SIS models with simple and complex population dynamics, in "Mathematical Approaches for Emerging and Reemerging Infectious Diseases: A introduction" (eds. C. Castillo-Chavez with S. Blower, P. van den Driessche, D. Kirschner and A. A. Yakubu), Springer-Verlag, New York, (2002), 153-163. |
[13] |
C. Celik and O. Duman, Allee effect in a discrete-time predator-prey system, Chaos Soliton. Fract., 40 (2009), 1956-1962.
doi: 10.1016/j.chaos.2007.09.077. |
[14] |
J. E. Franke and A. A. Yakubu, Discrete-time SIS epidemic model in a seasonal environment, SIAM J. Appl. Math., 66 (2006), 1563-1587.
doi: 10.1137/050638345. |
[15] |
P. A. Gonzalez, R. A. Saenz, B. N. Sanchez, C. Castillo-Chavez and A. A. Yakubu, Dispersal between two patches in a discrete time SEIS model, MTBI technical Report, 2000. |
[16] |
J. M. Grandmonet, Nonlinear difference equations, bifurcations and chaos: An introduction, Research in Economics, 62 (2008), 120-177. |
[17] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscilations, Dynamical Systems, and Bifurcations of Vector Fields," Springer, New York, 1983. |
[18] |
M. P. Hassell, Density dependence in single-species populations, J. Anim. Ecol., 44 (1975), 283-289.
doi: 10.2307/3863. |
[19] |
Z. Hu, Z. Teng and H. Jiang, Stability analysis in a class of discrete SIRS epidemic models, Nonlinear Anal. RWA, 13 (2012), 2017-2033.
doi: 10.1016/j.nonrwa.2011.12.024. |
[20] |
Z. Hu, Z. Teng and L. Zhang, Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response, Nonlinear Anal. RWA, 12 (2011), 2356-2377.
doi: 10.1016/j.nonrwa.2011.02.009. |
[21] |
L. Li, G. Sun and Z. Jin, Bifurcation and chaos in an epidemic model with nonlinear incidence rates, Appl. Math. Comput., 216 (2010), 1226-1234.
doi: 10.1016/j.amc.2010.02.014. |
[22] |
X. Li and W. Wang, A discrete epidemic model with stage structure, Chaos Solution. Fract., 26 (2005), 947-958.
doi: 10.1016/j.chaos.2005.01.063. |
[23] |
R. M. May, Biological population obeying difference equations: Stable points, stable cycles, and chaos, J. Theor. Biol., 51 (1975), 511-524.
doi: 10.1016/0022-5193(75)90078-8. |
[24] |
R. M. May, Deterministic models with chaotic dynamics, Nature, 256 (1975), 165-166. |
[25] |
R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467. |
[26] |
H. R. Thieme, Covergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
doi: 10.1007/BF00173267. |
[27] |
X. Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, Comm. Appl. Nonl. Anal., 3 (1996), 43-66. |
[28] |
Y. Zhou and H. Cao, Discrete tuberculosis transmission models and their application, in "A Survey of Mathematical Biology, Fields Communications Series" (ed. S. Sivaloganathan), 57, A co-publication of the AMS and Fields Institute, Canada, (2010), 83-112. |
[29] |
Y. Zhou, K. Khan, Z. Feng and J. Wu, Projection of tuberculosis incidence with increasing immigration trends, J. Theor. Biol., 254 (2008), 215-228.
doi: 10.1016/j.jtbi.2008.05.026. |
[30] |
Y. Zhou and Z. Ma, Global stability of a class of discrete age-structured SIS models with immigration, Math. Biosci. Eng., 6 (2009), 409-425.
doi: 10.3934/mbe.2009.6.409. |
[31] |
Y. Zhou, Z. Ma and F. Brauer, A discrete epidemicmodel for SARS transmission and control in China, Math. Comput. Model., 40 (2004), 1491-1506.
doi: 10.1016/j.mcm.2005.01.007. |
[32] |
Y. Zhou and F. Paolo, Dynamics of a discrete age-structured SIS models, Discrete Cont. Dyn. Sys. B, 4 (2004), 843-852.
doi: 10.3934/dcdsb.2004.4.841. |
show all references
References:
[1] |
L. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci., 124 (1994), 83-105.
doi: 10.1016/0025-5564(94)90025-6. |
[2] |
L. Allen and A. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biosci., 163 (2000), 1-33.
doi: 10.1016/S0025-5564(99)00047-4. |
[3] |
L. Allen and P. van den Driessche, The basic reproduction number in some discrete-time epidemic models, J. Differ. Equ. Appl., 14 (2008), 1127-1147.
doi: 10.1080/10236190802332308. |
[4] |
R. Arreola, A. Crossa, M. C. Velasco and A. A. Yakubu, Discrete-time SEIS models with exogenous re-infection and dispersal between two patches., Available from: \url{http://mtbi.asu.edu/files/Discrete_time_SEIS_Models_with_Exogenous_Reinfection_and_Dispersal_between_Two_Patches.pdf}., ().
|
[5] |
W. J. Beyn and J. Lorenz, Center manifolds of dynamical systems under discretization, Numer. Funct. Anal. Optimiz., 9 (1987), 381-414.
doi: 10.1080/01630568708816239. |
[6] |
H. Cao, Z. Dou, X. Liu, F. Zhang, Y. Zhou and Z. Ma, The impact of antiretroviral therapy on the basic reproductive number of HIV transmission, Math. Model. Appl., 1 (2012), 33-37. |
[7] |
H. Cao, Y. Xiao and Y. Zhou, The dynamics of a discrete SEIT model with age and infection-age structures, INT. J. Bio., 5 (2012), 61-76.
doi: 10.1142/S1793524512600042. |
[8] |
H. Cao and Y. Zhou, The discrete age-structured SEIT model with application to tuberculosis transmission in China, Math. Comput. Model., 55 (2012), 385-395.
doi: 10.1016/j.mcm.2011.08.017. |
[9] |
H. Cao and Y. Zhou, The basic reproduction number of discrete SIR and SEIS models with periodic parameters, Discrete Cont. Dyn. Sys. B, 18 (2013), 37-56. |
[10] |
H. Cao, Y. Zhou and B. Song, Complex dynamics of discrete SEIS models with simple demography, Discrete Dyn. Nat. Soc., (2011), Art. ID 653937, 21 pp.
doi: 10.1155/2011/653937. |
[11] |
C. Castillo-Chavez and A. A. Yakubu, Discrete-time SIS models with complex dynamics, Nonliear Anal. TMA, 47 (2001), 4753-4762.
doi: 10.1016/S0362-546X(01)00587-9. |
[12] |
C. Castillo-Chavez and A. A. Yakubu, Discrete-time SIS models with simple and complex population dynamics, in "Mathematical Approaches for Emerging and Reemerging Infectious Diseases: A introduction" (eds. C. Castillo-Chavez with S. Blower, P. van den Driessche, D. Kirschner and A. A. Yakubu), Springer-Verlag, New York, (2002), 153-163. |
[13] |
C. Celik and O. Duman, Allee effect in a discrete-time predator-prey system, Chaos Soliton. Fract., 40 (2009), 1956-1962.
doi: 10.1016/j.chaos.2007.09.077. |
[14] |
J. E. Franke and A. A. Yakubu, Discrete-time SIS epidemic model in a seasonal environment, SIAM J. Appl. Math., 66 (2006), 1563-1587.
doi: 10.1137/050638345. |
[15] |
P. A. Gonzalez, R. A. Saenz, B. N. Sanchez, C. Castillo-Chavez and A. A. Yakubu, Dispersal between two patches in a discrete time SEIS model, MTBI technical Report, 2000. |
[16] |
J. M. Grandmonet, Nonlinear difference equations, bifurcations and chaos: An introduction, Research in Economics, 62 (2008), 120-177. |
[17] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscilations, Dynamical Systems, and Bifurcations of Vector Fields," Springer, New York, 1983. |
[18] |
M. P. Hassell, Density dependence in single-species populations, J. Anim. Ecol., 44 (1975), 283-289.
doi: 10.2307/3863. |
[19] |
Z. Hu, Z. Teng and H. Jiang, Stability analysis in a class of discrete SIRS epidemic models, Nonlinear Anal. RWA, 13 (2012), 2017-2033.
doi: 10.1016/j.nonrwa.2011.12.024. |
[20] |
Z. Hu, Z. Teng and L. Zhang, Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response, Nonlinear Anal. RWA, 12 (2011), 2356-2377.
doi: 10.1016/j.nonrwa.2011.02.009. |
[21] |
L. Li, G. Sun and Z. Jin, Bifurcation and chaos in an epidemic model with nonlinear incidence rates, Appl. Math. Comput., 216 (2010), 1226-1234.
doi: 10.1016/j.amc.2010.02.014. |
[22] |
X. Li and W. Wang, A discrete epidemic model with stage structure, Chaos Solution. Fract., 26 (2005), 947-958.
doi: 10.1016/j.chaos.2005.01.063. |
[23] |
R. M. May, Biological population obeying difference equations: Stable points, stable cycles, and chaos, J. Theor. Biol., 51 (1975), 511-524.
doi: 10.1016/0022-5193(75)90078-8. |
[24] |
R. M. May, Deterministic models with chaotic dynamics, Nature, 256 (1975), 165-166. |
[25] |
R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467. |
[26] |
H. R. Thieme, Covergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
doi: 10.1007/BF00173267. |
[27] |
X. Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, Comm. Appl. Nonl. Anal., 3 (1996), 43-66. |
[28] |
Y. Zhou and H. Cao, Discrete tuberculosis transmission models and their application, in "A Survey of Mathematical Biology, Fields Communications Series" (ed. S. Sivaloganathan), 57, A co-publication of the AMS and Fields Institute, Canada, (2010), 83-112. |
[29] |
Y. Zhou, K. Khan, Z. Feng and J. Wu, Projection of tuberculosis incidence with increasing immigration trends, J. Theor. Biol., 254 (2008), 215-228.
doi: 10.1016/j.jtbi.2008.05.026. |
[30] |
Y. Zhou and Z. Ma, Global stability of a class of discrete age-structured SIS models with immigration, Math. Biosci. Eng., 6 (2009), 409-425.
doi: 10.3934/mbe.2009.6.409. |
[31] |
Y. Zhou, Z. Ma and F. Brauer, A discrete epidemicmodel for SARS transmission and control in China, Math. Comput. Model., 40 (2004), 1491-1506.
doi: 10.1016/j.mcm.2005.01.007. |
[32] |
Y. Zhou and F. Paolo, Dynamics of a discrete age-structured SIS models, Discrete Cont. Dyn. Sys. B, 4 (2004), 843-852.
doi: 10.3934/dcdsb.2004.4.841. |
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