    January  2011, 15(1): 93-112. doi: 10.3934/dcdsb.2011.15.93

## Bifurcations of an SIRS epidemic model with nonlinear incidence rate

 1 Department of Applied Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, China 2 Department of Mathematics, East China Normal University, Shanghai 200062, China 3 Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083 4 Department of Mathematics, The University of Miami, P.O. Box 249085, Coral Gables, Florida 33124

Received  December 2009 Revised  July 2010 Published  October 2010

The main purpose of this paper is to explore the dynamics of an epidemic model with a general nonlinear incidence $\beta SI^p/(1+\alpha I^q)$. The existence and stability of multiple endemic equilibria of the epidemic model are analyzed. Local bifurcation theory is applied to explore the rich dynamical behavior of the model. Normal forms of the model are derived for different types of bifurcations, including Hopf and Bogdanov-Takens bifurcations. Concretely speaking, the first Lyapunov coefficient is computed to determine various types of Hopf bifurcations. Next, with the help of the Bogdanov-Takens normal form, a family of homoclinic orbits is arising when a Hopf and a saddle-node bifurcation merge. Finally, some numerical results and simulations are presented to illustrate these theoretical results.
Citation: Zhixing Hu, Ping Bi, Wanbiao Ma, Shigui Ruan. Bifurcations of an SIRS epidemic model with nonlinear incidence rate. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 93-112. doi: 10.3934/dcdsb.2011.15.93
##### References:
  M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96. doi: doi:10.1016/j.mbs.2004.01.003.  Google Scholar  M. E. Alexander and S. M. Moghadas, Bifurcation analysis of an SIRS epidemic model with generalized incidence, SIAM J. Appl. Math., 65 (2005), 1794-1816. Google Scholar  V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61. doi: doi:10.1016/0025-5564(78)90006-8.  Google Scholar  W. R. Derrick and P. van den Driessche, Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population, Discrete Contin. Dyn. Syst. Ser. B, 2 (2003), 299-309. doi: doi:10.3934/dcdsb.2003.3.299.  Google Scholar  Z. Feng and H. R. Thieme, Recurrent outbreaks of childhood disease revisited: The impact of isolation, Math. Biosci., 128 (1995), 93-130. doi: doi:10.1016/0025-5564(94)00069-C.  Google Scholar  P. Glendinning, "Stability, Instability and Chaos," Cambridge University Press, Cambridge, 1994. Google Scholar  B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation," Lecture Notes Series, vol. 41, Cambridge University Press, Cambridge, 1981. Google Scholar  H. W. Hethcote, The mathematics of infectious disease, SIAM Rev., 42 (2000), 599-653. doi: doi:10.1137/S0036144500371907.  Google Scholar  H. W. Hethcote and S. A. Levin, Periodicity in epidemiological models, in "Applied Mathematical Ecology" (Trieste, 1986), Biomathematics 18, Springer-Verlag, Berlin, 1989, pp. 193-211. Google Scholar  H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287. doi: doi:10.1007/BF00160539.  Google Scholar  Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," 3rd edition, Appl. Math. Sci. 112, Springer-Verlag, New York, 2004. Google Scholar  G. Li and W. Wang, Bifurcation analysis of an epidemic model with nonlinear incidence, Appl. Math. Comput., 214 (2009), 411-423. doi: doi:10.1016/j.amc.2009.04.012.  Google Scholar  W. M. Liu, H. W. Hetchote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: doi:10.1007/BF00277162.  Google Scholar  W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204. doi: doi:10.1007/BF00276956.  Google Scholar  M. Lizana and J. Rivero, Multiparametric bifurcations for a model in epidemiology, J. Math. Biol., 35 (1996), 21-36. doi: doi:10.1007/s002850050040.  Google Scholar  S. M. Moghadas, Analysis of an epidemic model with bistable equilibria using the Poincaré index, Appl. Math. Comput., 149 (2004), 689-702. doi: doi:10.1016/S0096-3003(03)00171-1.  Google Scholar  S. M. Moghadas and M. E. Alexander, Bifurcations of an epidemic model with non-linear incidence and infection-dependent removal rate, Math. Med. Biol., 23 (2006), 231-254. doi: doi:10.1093/imammb/dql011. Google Scholar  S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163. doi: doi:10.1016/S0022-0396(02)00089-X.  Google Scholar  Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639. doi: doi:10.1137/070700966.  Google Scholar  W. Wang, Epidemic models with nonlinear infection forces, Math. Biosci. Eng., 3 (2006), 267-279. Google Scholar  S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," 2nd edition, Springer-Verlag, New York, 2004. Google Scholar  D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429. doi: doi:10.1016/j.mbs.2006.09.025.  Google Scholar

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##### References:
  M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96. doi: doi:10.1016/j.mbs.2004.01.003.  Google Scholar  M. E. Alexander and S. M. Moghadas, Bifurcation analysis of an SIRS epidemic model with generalized incidence, SIAM J. Appl. Math., 65 (2005), 1794-1816. Google Scholar  V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61. doi: doi:10.1016/0025-5564(78)90006-8.  Google Scholar  W. R. Derrick and P. van den Driessche, Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population, Discrete Contin. Dyn. Syst. Ser. B, 2 (2003), 299-309. doi: doi:10.3934/dcdsb.2003.3.299.  Google Scholar  Z. Feng and H. R. Thieme, Recurrent outbreaks of childhood disease revisited: The impact of isolation, Math. Biosci., 128 (1995), 93-130. doi: doi:10.1016/0025-5564(94)00069-C.  Google Scholar  P. Glendinning, "Stability, Instability and Chaos," Cambridge University Press, Cambridge, 1994. Google Scholar  B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation," Lecture Notes Series, vol. 41, Cambridge University Press, Cambridge, 1981. Google Scholar  H. W. Hethcote, The mathematics of infectious disease, SIAM Rev., 42 (2000), 599-653. doi: doi:10.1137/S0036144500371907.  Google Scholar  H. W. Hethcote and S. A. Levin, Periodicity in epidemiological models, in "Applied Mathematical Ecology" (Trieste, 1986), Biomathematics 18, Springer-Verlag, Berlin, 1989, pp. 193-211. Google Scholar  H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287. doi: doi:10.1007/BF00160539.  Google Scholar  Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," 3rd edition, Appl. Math. Sci. 112, Springer-Verlag, New York, 2004. Google Scholar  G. Li and W. Wang, Bifurcation analysis of an epidemic model with nonlinear incidence, Appl. Math. Comput., 214 (2009), 411-423. doi: doi:10.1016/j.amc.2009.04.012.  Google Scholar  W. M. Liu, H. W. Hetchote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: doi:10.1007/BF00277162.  Google Scholar  W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204. doi: doi:10.1007/BF00276956.  Google Scholar  M. Lizana and J. Rivero, Multiparametric bifurcations for a model in epidemiology, J. Math. Biol., 35 (1996), 21-36. doi: doi:10.1007/s002850050040.  Google Scholar  S. M. Moghadas, Analysis of an epidemic model with bistable equilibria using the Poincaré index, Appl. Math. Comput., 149 (2004), 689-702. doi: doi:10.1016/S0096-3003(03)00171-1.  Google Scholar  S. M. Moghadas and M. E. Alexander, Bifurcations of an epidemic model with non-linear incidence and infection-dependent removal rate, Math. Med. Biol., 23 (2006), 231-254. doi: doi:10.1093/imammb/dql011. Google Scholar  S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163. doi: doi:10.1016/S0022-0396(02)00089-X.  Google Scholar  Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639. doi: doi:10.1137/070700966.  Google Scholar  W. Wang, Epidemic models with nonlinear infection forces, Math. Biosci. Eng., 3 (2006), 267-279. Google Scholar  S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," 2nd edition, Springer-Verlag, New York, 2004. Google Scholar  D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429. doi: doi:10.1016/j.mbs.2006.09.025.  Google Scholar
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