American Institute of Mathematical Sciences

Advanced
2010, 7(2): 347-361. doi: 10.3934/mbe.2010.7.347

Global stability for a class of discrete SIR epidemic models

Yoichi Enatsu 1, , Yukihiko Nakata 1, and  Yoshiaki Muroya 2,

1. 

Department of Pure and Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555, Japan, Japan
2. 

Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555, Japan

Received  July 2009 Revised  February 2010 Published  April 2010

In this paper, we propose a class of discrete SIR epidemic models which are derived from SIR epidemic models with distributed delays by using a variation of the backward Euler method. Applying a Lyapunov functional technique, it is shown that the global dynamics of each discrete SIR epidemic model are fully determined by a single threshold parameter and the effect of discrete time delays are harmless for the global stability of the endemic equilibrium of the model.
Keywords: globally asymptotic stability, distributed delays, Backward Euler method, Lyapunov functional., discrete SIR epidemic model.
Mathematics Subject Classification: Primary: 34K20, 34K25; Secondary: 92D3.
Citation: 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
[1]

Zhen Jin, Zhien Ma. The stability of an SIR epidemic model with time delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 101-109. doi: 10.3934/mbe.2006.3.101
[2]

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
[3]

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
[4]

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
[5]

Kai Wang, Zhidong Teng, Xueliang Zhang. Dynamical behaviors of an Echinococcosis epidemic model with distributed delays. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1425-1445. doi: 10.3934/mbe.2017074
[6]

Hermann Brunner, Chunhua Ou. On the asymptotic stability of Volterra functional equations with vanishing delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 397-406. doi: 10.3934/cpaa.2015.14.397
[7]

Fuke Wu, Xuerong Mao, Peter E. Kloeden. Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 885-903. doi: 10.3934/dcds.2013.33.885
[8]

Bin Fang, Xue-Zhi Li, Maia Martcheva, Li-Ming Cai. Global stability for a heroin model with two distributed delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 715-733. doi: 10.3934/dcdsb.2014.19.715
[9]

Miljana Jovanović, Vuk Vujović. Stability of stochastic heroin model with two distributed delays. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2407-2432. doi: 10.3934/dcdsb.2020016
[10]

Masaki Sekiguchi, Emiko Ishiwata, Yukihiko Nakata. Dynamics of an ultra-discrete SIR epidemic model with time delay. Mathematical Biosciences & Engineering, 2018, 15 (3) : 653-666. doi: 10.3934/mbe.2018029
[11]

Wanbiao Ma, Yasuhiro Takeuchi. Asymptotic properties of a delayed SIR epidemic model with density dependent birth rate. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 671-678. doi: 10.3934/dcdsb.2004.4.671
[12]

Chufen Wu, Peixuan Weng. Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 867-892. doi: 10.3934/dcdsb.2011.15.867
[13]

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
[14]

Xingwang Yu, Sanling Yuan. Asymptotic properties of a stochastic chemostat model with two distributed delays and nonlinear perturbation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2373-2390. doi: 10.3934/dcdsb.2020014
[15]

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
[16]

Hiroshi Ito. Input-to-state stability and Lyapunov functions with explicit domains for SIR model of infectious diseases. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5171-5196. doi: 10.3934/dcdsb.2020338
[17]

Emad Attia, Marek Bodnar, Urszula Foryś. Angiogenesis model with Erlang distributed delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 1-15. doi: 10.3934/mbe.2017001
[18]

Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159
[19]

Qianqian Cui, Zhipeng Qiu, Ling Ding. An SIR epidemic model with vaccination in a patchy environment. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1141-1157. doi: 10.3934/mbe.2017059
[20]

Yan Li, Wan-Tong Li, Guo Lin. Traveling waves of a delayed diffusive SIR epidemic model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1001-1022. doi: 10.3934/cpaa.2015.14.1001

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (150)
  • HTML views (0)
  • Cited by (40)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences