Bifurcation analysis of a discrete SIS model with
      bilinear incidence depending on new infection

Hui Cao; Yicang Zhou; Zhien Ma

doi:10.3934/mbe.2013.10.1399

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2013, 10(5&6): 1399-1417. doi: 10.3934/mbe.2013.10.1399

Bifurcation analysis of a discrete SIS model with bilinear incidence depending on new infection

Hui Cao ^1, , Yicang Zhou ^2, and Zhien Ma ^3,

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

Received August 2012 Revised March 2013 Published August 2013

Abstract
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Under a Creative Commons license

A discrete SIS epidemic model with the bilinear incidence depending on the new infection is formulated and studied. The condition for the global stability of the disease free equilibrium is obtained. The existence of the endemic equilibrium and its stability are investigated. More attention is paid to the existence of the saddle-node bifurcation, the flip bifurcation, and the Hopf bifurcation. Sufficient conditions for those bifurcations have been obtained. Numerical simulations are conducted to demonstrate our theoretical results and the complexity of the model.

Keywords: Hopf bifurcation., bilinear incidence, flip bifurcation, saddle-Node bifurcation, Discrete SIS model.

Mathematics Subject Classification: Primary: 37N25, 37G10; Secondary: 92B0.

Citation: Hui Cao, Yicang Zhou, Zhien Ma. Bifurcation analysis of a discrete SIS model with bilinear incidence depending on new infection. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1399-1417. doi: 10.3934/mbe.2013.10.1399

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	J. M. Grandmonet, Nonlinear difference equations, bifurcations and chaos: An introduction, Research in Economics, 62 (2008), 120-177. Google Scholar
[17]	J. Guckenheimer and P. Holmes, "Nonlinear Oscilations, Dynamical Systems, and Bifurcations of Vector Fields," Springer, New York, 1983. Google Scholar
[18]	M. P. Hassell, Density dependence in single-species populations, J. Anim. Ecol., 44 (1975), 283-289. doi: 10.2307/3863. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	R. M. May, Deterministic models with chaotic dynamics, Nature, 256 (1975), 165-166. Google Scholar
[25]	R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467. Google Scholar
[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. Google Scholar
[27]	X. Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, Comm. Appl. Nonl. Anal., 3 (1996), 43-66. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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}., (). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	J. M. Grandmonet, Nonlinear difference equations, bifurcations and chaos: An introduction, Research in Economics, 62 (2008), 120-177. Google Scholar
[17]	J. Guckenheimer and P. Holmes, "Nonlinear Oscilations, Dynamical Systems, and Bifurcations of Vector Fields," Springer, New York, 1983. Google Scholar
[18]	M. P. Hassell, Density dependence in single-species populations, J. Anim. Ecol., 44 (1975), 283-289. doi: 10.2307/3863. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	R. M. May, Deterministic models with chaotic dynamics, Nature, 256 (1975), 165-166. Google Scholar
[25]	R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467. Google Scholar
[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. Google Scholar
[27]	X. Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, Comm. Appl. Nonl. Anal., 3 (1996), 43-66. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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Bifurcation analysis of a discrete SIS model with bilinear incidence depending on new infection

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