August  2004, 4(3): 635-642. doi: 10.3934/dcdsb.2004.4.635

Stability analysis for SIS epidemic models with vaccination and constant population size

1. 

Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an, 710049, China

2. 

Department of Applied Mathematics, College of Science, Xian Jiaotong University, Xian 710049, China

Received  August 2002 Revised  November 2003 Published  May 2004

This paper investigates two types of SIS epidemic model with vaccination and constant population size to determine to the thresholds, equilibria, and stabilities. One of SIS models is a delay differential equations, in which the period of immunity due to vaccination is a constant. Another is an ordinary differential equations, in which the loss of immunity due to vaccination is in the exponent form. We find all of their thresholds respectively, and compare them. The disease-free equilibrium is globally asymptotically stable if the threshold is not greater than one; the endemic equilibrium is globally asymptotically stable if the threshold is greater than one.
Citation: Jianquan Li, Zhien Ma. Stability analysis for SIS epidemic models with vaccination and constant population size. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 635-642. doi: 10.3934/dcdsb.2004.4.635
[1]

Qingming Gou, Wendi Wang. Global stability of two epidemic models. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 333-345. doi: 10.3934/dcdsb.2007.8.333

[2]

Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang. Backward bifurcation and global stability in an epidemic model with treatment and vaccination. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999

[3]

Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643

[4]

Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya. Global stability for a class of discrete SIR epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (2) : 347-361. doi: 10.3934/mbe.2010.7.347

[5]

Eunha Shim. A note on epidemic models with infective immigrants and vaccination. Mathematical Biosciences & Engineering, 2006, 3 (3) : 557-566. doi: 10.3934/mbe.2006.3.557

[6]

Jing Hui, Lansun Chen. Impulsive vaccination of sir epidemic models with nonlinear incidence rates. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 595-605. doi: 10.3934/dcdsb.2004.4.595

[7]

Aili Wang, Yanni Xiao, Robert A. Cheke. Global dynamics of a piece-wise epidemic model with switching vaccination strategy. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2915-2940. doi: 10.3934/dcdsb.2014.19.2915

[8]

Jinliang Wang, Xianning Liu, Toshikazu Kuniya, Jingmei Pang. Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2795-2812. doi: 10.3934/dcdsb.2017151

[9]

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

[10]

Yali Yang, Sanyi Tang, Xiaohong Ren, Huiwen Zhao, Chenping Guo. Global stability and optimal control for a tuberculosis model with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 1009-1022. doi: 10.3934/dcdsb.2016.21.1009

[11]

Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119

[12]

Jianquan Li, Zhien Ma, Fred Brauer. Global analysis of discrete-time SI and SIS epidemic models. Mathematical Biosciences & Engineering, 2007, 4 (4) : 699-710. doi: 10.3934/mbe.2007.4.699

[13]

Dong-Uk Hwang, S. Boccaletti, Y. Moreno, R. López-Ruiz. Thresholds for Epidemic Outbreaks in Finite Scale-Free Networks. Mathematical Biosciences & Engineering, 2005, 2 (2) : 317-327. doi: 10.3934/mbe.2005.2.317

[14]

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

[15]

Gang Huang, Edoardo Beretta, Yasuhiro Takeuchi. Global stability for epidemic model with constant latency and infectious periods. Mathematical Biosciences & Engineering, 2012, 9 (2) : 297-312. doi: 10.3934/mbe.2012.9.297

[16]

Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971

[17]

Toshikazu Kuniya, Yoshiaki Muroya. Global stability of a multi-group SIS epidemic model for population migration. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1105-1118. doi: 10.3934/dcdsb.2014.19.1105

[18]

Yongli Cai, Yun Kang, Weiming Wang. Global stability of the steady states of an epidemic model incorporating intervention strategies. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1071-1089. doi: 10.3934/mbe.2017056

[19]

Jing-Jing Xiang, Juan Wang, Li-Ming Cai. Global stability of the dengue disease transmission models. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2217-2232. doi: 10.3934/dcdsb.2015.20.2217

[20]

Paul Georgescu, Hong Zhang, Daniel Maxin. The global stability of coexisting equilibria for three models of mutualism. Mathematical Biosciences & Engineering, 2016, 13 (1) : 101-118. doi: 10.3934/mbe.2016.13.101

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]