American Institute of Mathematical Sciences

Advanced
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

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.  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.  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.  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.  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.   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.  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.  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.   Google Scholar

[10]

H. Cao, Y. Zhou and B. Song, Complex dynamics of discrete SEIS models with simple demography,, Discrete Dyn. Nat. Soc., (2011).  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.  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, (2002), 153.   Google Scholar

[13]

C. Celik and O. Duman, Allee effect in a discrete-time predator-prey system,, Chaos Soliton. Fract., 40 (2009), 1956.  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.  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.   Google Scholar

[17]

J. Guckenheimer and P. Holmes, "Nonlinear Oscilations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer, (1983).   Google Scholar

[18]

M. P. Hassell, Density dependence in single-species populations,, J. Anim. Ecol., 44 (1975), 283.  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.  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.  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.  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.  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.  doi: 10.1016/0022-5193(75)90078-8.  Google Scholar

[24]

R. M. May, Deterministic models with chaotic dynamics,, Nature, 256 (1975), 165.   Google Scholar

[25]

R. M. May, Simple mathematical models with very complicated dynamics,, Nature, 261 (1976), 459.   Google Scholar

[26]

H. R. Thieme, Covergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755.  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.   Google Scholar

[28]

Y. Zhou and H. Cao, Discrete tuberculosis transmission models and their application,, in, 57 (2010), 83.   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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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.  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.   Google Scholar

[10]

H. Cao, Y. Zhou and B. Song, Complex dynamics of discrete SEIS models with simple demography,, Discrete Dyn. Nat. Soc., (2011).  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.  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, (2002), 153.   Google Scholar

[13]

C. Celik and O. Duman, Allee effect in a discrete-time predator-prey system,, Chaos Soliton. Fract., 40 (2009), 1956.  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.  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.   Google Scholar

[17]

J. Guckenheimer and P. Holmes, "Nonlinear Oscilations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer, (1983).   Google Scholar

[18]

M. P. Hassell, Density dependence in single-species populations,, J. Anim. Ecol., 44 (1975), 283.  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.  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.  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.  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.  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.  doi: 10.1016/0022-5193(75)90078-8.  Google Scholar

[24]

R. M. May, Deterministic models with chaotic dynamics,, Nature, 256 (1975), 165.   Google Scholar

[25]

R. M. May, Simple mathematical models with very complicated dynamics,, Nature, 261 (1976), 459.   Google Scholar

[26]

H. R. Thieme, Covergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755.  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.   Google Scholar

[28]

Y. Zhou and H. Cao, Discrete tuberculosis transmission models and their application,, in, 57 (2010), 83.   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.  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.  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.  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.  doi: 10.3934/dcdsb.2004.4.841.  Google Scholar

[1]

Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419
[2]

Ping Liu, Junping Shi, Yuwen Wang. A double saddle-node bifurcation theorem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2923-2933. doi: 10.3934/cpaa.2013.12.2923
[3]

Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203
[4]

Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233
[5]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
[6]

Zhiqin Qiao, Deming Zhu, Qiuying Lu. Bifurcation of a heterodimensional cycle with weak inclination flip. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1009-1025. doi: 10.3934/dcdsb.2012.17.1009
[7]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[8]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[9]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[10]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152
[11]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020208
[12]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
[13]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325
[14]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735
[15]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[16]

Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891
[17]

Jisun Lim, Seongwon Lee, Yangjin Kim. Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3575-3602. doi: 10.3934/dcdsb.2016111
[18]

Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657
[19]

Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046
[20]

Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences