# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems - B, 2011, 15 (1) : 93-112. doi: 10.3934/dcdsb.2011.15.93
##### References:
 [1] 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. [2] 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. [3] 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. [4] 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. [5] 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. [6] P. Glendinning, "Stability, Instability and Chaos," Cambridge University Press, Cambridge, 1994. [7] 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. [8] H. W. Hethcote, The mathematics of infectious disease, SIAM Rev., 42 (2000), 599-653. doi: doi:10.1137/S0036144500371907. [9] 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. [10] 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. [11] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," 3rd edition, Appl. Math. Sci. 112, Springer-Verlag, New York, 2004. [12] 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. [13] 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. [14] 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. [15] M. Lizana and J. Rivero, Multiparametric bifurcations for a model in epidemiology, J. Math. Biol., 35 (1996), 21-36. doi: doi:10.1007/s002850050040. [16] 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. [17] 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. [18] 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. [19] 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. [20] W. Wang, Epidemic models with nonlinear infection forces, Math. Biosci. Eng., 3 (2006), 267-279. [21] S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," 2nd edition, Springer-Verlag, New York, 2004. [22] 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.

show all references

##### References:
 [1] 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. [2] 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. [3] 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. [4] 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. [5] 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. [6] P. Glendinning, "Stability, Instability and Chaos," Cambridge University Press, Cambridge, 1994. [7] 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. [8] H. W. Hethcote, The mathematics of infectious disease, SIAM Rev., 42 (2000), 599-653. doi: doi:10.1137/S0036144500371907. [9] 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. [10] 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. [11] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," 3rd edition, Appl. Math. Sci. 112, Springer-Verlag, New York, 2004. [12] 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. [13] 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. [14] 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. [15] M. Lizana and J. Rivero, Multiparametric bifurcations for a model in epidemiology, J. Math. Biol., 35 (1996), 21-36. doi: doi:10.1007/s002850050040. [16] 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. [17] 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. [18] 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. [19] 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. [20] W. Wang, Epidemic models with nonlinear infection forces, Math. Biosci. Eng., 3 (2006), 267-279. [21] S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," 2nd edition, Springer-Verlag, New York, 2004. [22] 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.
 [1] Min Lu, Chuang Xiang, Jicai Huang. Bogdanov-Takens bifurcation in a SIRS epidemic model with a generalized nonmonotone incidence rate. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3125-3138. doi: 10.3934/dcdss.2020115 [2] Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3253-3269. doi: 10.3934/dcdss.2020130 [3] Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041 [4] Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 [5] Xun Cao, Xianyong Chen, Weihua Jiang. Bogdanov-Takens bifurcation with $Z_2$ symmetry and spatiotemporal dynamics in diffusive Rosenzweig-MacArthur model involving nonlocal prey competition. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3747-3785. doi: 10.3934/dcds.2022031 [6] Yoichi Enatsu, Yukihiko Nakata. Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate. Mathematical Biosciences & Engineering, 2014, 11 (4) : 785-805. doi: 10.3934/mbe.2014.11.785 [7] Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057 [8] Bing Zeng, Pei Yu. A hierarchical parametric analysis on Hopf bifurcation of an epidemic model. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022069 [9] Qin Pan, Jicai Huang, Qihua Huang. Global dynamics and bifurcations in a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3533-3561. doi: 10.3934/dcdsb.2021195 [10] Shouying Huang, Jifa Jiang. Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate. Mathematical Biosciences & Engineering, 2016, 13 (4) : 723-739. doi: 10.3934/mbe.2016016 [11] Hebai Chen, Xingwu Chen, Jianhua Xie. Global phase portrait of a degenerate Bogdanov-Takens system with symmetry. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1273-1293. doi: 10.3934/dcdsb.2017062 [12] Hebai Chen, Xingwu Chen. Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ). Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4141-4170. doi: 10.3934/dcdsb.2018130 [13] Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6131-6154. doi: 10.3934/dcdsb.2021010 [14] 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 [15] Yu Yang, Dongmei Xiao. Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 195-211. doi: 10.3934/dcdsb.2010.13.195 [16] C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837 [17] Yu Yang, Lan Zou, Tonghua Zhang, Yancong Xu. Dynamical analysis of a diffusive SIRS model with general incidence rate. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2433-2451. doi: 10.3934/dcdsb.2020017 [18] Yanan Zhao, Yuguo Lin, Daqing Jiang, Xuerong Mao, Yong Li. Stationary distribution of stochastic SIRS epidemic model with standard incidence. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2363-2378. doi: 10.3934/dcdsb.2016051 [19] Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264 [20] Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang. Backward bifurcation and global stability in an epidemic model with treatment and vaccination. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999

2021 Impact Factor: 1.497