# American Institute of Mathematical Sciences

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

 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

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.
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. [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. 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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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##### 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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