American Institute of Mathematical Sciences

Advanced
August  2004, 4(3): 595-605. doi: 10.3934/dcdsb.2004.4.595

Impulsive vaccination of sir epidemic models with nonlinear incidence rates

Jing Hui 1, and  Lansun Chen 2,

1. 

Department of Information and Computation Sciences, Guangxi Institute of Technology, Liuzhou 545006, China
2. 

Institute of Mathematics, Academy of Mathematics and System Sciences, Academia Sinia, Beijing 100080, China

Received  December 2002 Revised  March 2004 Published  May 2004

The impulsive vaccination strategies of the epidemic SIR models with nonlinear incidence rates $\beta I^{p}S^{q}$ are considered in this paper. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact periodic infection-free solution of the impulsive epidemic system and prove that the periodic infection-free solution is globally asymptotically stable. In order to apply vaccination pulses frequently enough so as to eradicate the disease, the threshold for the period of pulsing, i.e. $\tau _{max}$ is shown, further, by bifurcation theory, we obtain a supercritical bifurcation at this threshold, i.e. when $\tau>\tau_{max}$ and is closing to $\tau_{max}$, there is a stable positive periodic solution. Throughout the paper, we find impulsive epidemiological models with nonlinear incidence rates $\beta I^{p}S^{q}$ show a much wider range of dynamical behaviors than do those with bilinear incidence rate $\beta SI$ and our paper extends the previous results, at the same time, theoretical results show that pulse vaccination strategy is distinguished from the conventional strategies in leading to disease eradication at relatively low values of vaccination, therefore impulsive vaccination strategy provides a more natural, more effective vaccination strategy.
Keywords: stability, nonlinear incidence rates, bifurcation., Impulsive vaccination, threshold.
Mathematics Subject Classification: Primary: 92B05, 34C2.
Citation: Jing Hui, Lansun Chen. Impulsive vaccination of sir epidemic models with nonlinear incidence rates. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 595-605. doi: 10.3934/dcdsb.2004.4.595
[1]

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
[2]

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
[3]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with generalized nonlinear incidence and vaccination age. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 977-996. doi: 10.3934/dcdsb.2016.21.977
[4]

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
[5]

Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya. Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 61-74. doi: 10.3934/dcdsb.2011.15.61
[6]

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
[7]

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
[8]

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
[9]

Jinhuo Luo, Jin Wang, Hao Wang. Seasonal forcing and exponential threshold incidence in cholera dynamics. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2261-2290. doi: 10.3934/dcdsb.2017095
[10]

Jinling Zhou, Yu Yang, Cheng-Hsiung Hsu. Traveling waves for a nonlocal dispersal vaccination model with general incidence. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1469-1495. doi: 10.3934/dcdsb.2019236
[11]

Yu Yang, Jinling Zhou, Cheng-Hsiung Hsu. Critical traveling wave solutions for a vaccination model with general incidence. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1209-1225. doi: 10.3934/dcdsb.2021087
[12]

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
[13]

Yu Ji, Lan Liu. Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 133-149. doi: 10.3934/dcdsb.2016.21.133
[14]

Attila Dénes, Gergely Röst. Global stability for SIR and SIRS models with nonlinear incidence and removal terms via Dulac functions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1101-1117. doi: 10.3934/dcdsb.2016.21.1101
[15]

Tomás Caraballo, Mohamed El Fatini, Idriss Sekkak, Regragui Taki, Aziz Laaribi. A stochastic threshold for an epidemic model with isolation and a non linear incidence. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2513-2531. doi: 10.3934/cpaa.2020110
[16]

Meng Zhang, Kaiyuan Liu, Lansun Chen, Zeyu Li. State feedback impulsive control of computer worm and virus with saturated incidence. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1465-1478. doi: 10.3934/mbe.2018067
[17]

Yaxin Han, Zhenguo Bai. Threshold dynamics of a West Nile virus model with impulsive culling and incubation period. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021239
[18]

Pierre Gabriel. Global stability for the prion equation with general incidence. Mathematical Biosciences & Engineering, 2015, 12 (4) : 789-801. doi: 10.3934/mbe.2015.12.789
[19]

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
[20]

Wisdom S. Avusuglo, Kenzu Abdella, Wenying Feng. Stability analysis on an economic epidemiological model with vaccination. Mathematical Biosciences & Engineering, 2017, 14 (4) : 975-999. doi: 10.3934/mbe.2017051

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (226)
  • HTML views (0)
  • Cited by (53)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy