American Institute of Mathematical Sciences

Advanced
2012, 9(2): 297-312. doi: 10.3934/mbe.2012.9.297

Global stability for epidemic model with constant latency and infectious periods

Gang Huang 1, , Edoardo Beretta 2, and  Yasuhiro Takeuchi 3,

1. 

School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, China
2. 

CIMAB, University of Milano, via C. Saldini 50, I20133 Milano
3. 

Graduate School of Science and Technology, Shizuoka University, Hamamatsu, 4328561, Japan

Received  August 2011 Revised  December 2011 Published  March 2012

In recent years many delay epidemiological models have been proposed to study at which stage of the epidemics the delays can destabilize the disease free equilibrium, or the endemic equilibrium, giving rise to stability switches. One of these models is the SEIR model with constant latency time and infectious periods [2], for which the authors have proved that the two delays are harmless in inducing stability switches. However, it is left open the problem of the global asymptotic stability of the endemic equilibrium whenever it exists. Even the Lyapunov functions approach, recently proposed by Huang and Takeuchi to study many delay epidemiological models, fails to work on this model. In this paper, an age-infection model is presented for the delay SEIR epidemic model, such that the properties of global asymptotic stability of the equilibria of the age-infection model imply the same properties for the original delay-differential epidemic model. By introducing suitable Lyapunov functions to study the global stability of the disease free equilibrium (when $\mathcal{R}_0\leq 1$) and of the endemic equilibria (whenever $ \mathcal{R}_0>1$) of the age-infection model, we can infer the corresponding global properties for the equilibria of the delay SEIR model in [2], thus proving that the endemic equilibrium in [2] is globally asymptotically stable whenever it exists.
    Furthermore, we also present a review of the SIR, SEIR epidemic models, with and without delays, appeared in literature, that can be seen as particular cases of the approach presented in the paper.
Keywords: epidemic model, infectious period, Global stability, age-structure..
Mathematics Subject Classification: Primary: 92D30, 34A34; Secondary: 34D20, 34D2.
Citation: 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
References:
[1]

J. Arino and P. van den Driessche, Time delays in epidemic models: Modeling and numerical considerations,, in, (2006), 539.   Google Scholar

[2]

E. Beretta and D. Breda, An SEIR epidemic model with constant latency time and infectious period,, Math. Biosci. Eng., 8 (2011), 931.  doi: 10.3934/mbe.2011.8.931.  Google Scholar

[3]

E. Beretta and Y. Takeuchi, Global stability of an SIR model with time delays,, J. Math. Biol., 33 (1995), 250.   Google Scholar

[4]

E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay,, Nonlinear Anal., 47 (2001), 4107.  doi: 10.1016/S0362-546X(01)00528-4.  Google Scholar

[5]

J. M. Cushing, "An Introduction to Structured Population Dynamics,", CBMS-NSF Regional Conference Series in Applied Mathematics, 71 (1998).   Google Scholar

[6]

Y. Enatsu, Y. Nakata and Y. Muroya, Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays,, Discrete and Continuous Dynamical Systems Ser. B, 15 (2011), 61.  doi: 10.3934/dcdsb.2011.15.61.  Google Scholar

[7]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate,, Bull. Math. Biol., 72 (2010), 1192.  doi: 10.1007/s11538-009-9487-6.  Google Scholar

[8]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125.  doi: 10.1007/s00285-010-0368-2.  Google Scholar

[9]

G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model,, SIAM J. Appl. Math., 72 (2012), 25.  doi: 10.1137/110826588.  Google Scholar

[10]

A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence,, Math. Biosci. Eng., 1 (2004), 57.  doi: 10.3934/mbe.2004.1.57.  Google Scholar

[11]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,, Bull. Math. Biol., 68 (2006), 615.  doi: 10.1007/s11538-005-9037-9.  Google Scholar

[12]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871.  doi: 10.1007/s11538-007-9196-y.  Google Scholar

[13]

M. Iannelli, "Mathematical Theory of Age-Structured Population Dynamics,", Applied Mathematical Monographs, 7 (1995).   Google Scholar

[14]

A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility,, (2011), (2011).   Google Scholar

[15]

P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Appl. Anal., 89 (2010), 1109.  doi: 10.1080/00036810903208122.  Google Scholar

[16]

C. C. McCluskey, Delay versus age-of-infection-global stability,, Appl. Math. Comput., 217 (2010), 3046.   Google Scholar

[17]

C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence,, Nonlinear Anal. RWA, 11 (2010), 3106.  doi: 10.1016/j.nonrwa.2009.11.005.  Google Scholar

[18]

C. C. McCluskey, Global stability of an SIR epidemic model with delay and general nonlinear incidence,, Math. Biosci. Eng., 7 (2010), 837.  doi: 10.3934/mbe.2010.7.837.  Google Scholar

[19]

P. van den Driessche, Some epidemiological models with delays,, in, (1994), 507.   Google Scholar

[20]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,", Marcel Dekker, (1985).   Google Scholar

[21]

R. Xu and Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay,, Nonlinear Anal. RWA, 10 (2009), 3175.  doi: 10.1016/j.nonrwa.2008.10.013.  Google Scholar

[22]

R. Xu and Y. Du, A delayed SIR epidemic model with saturation incidence and a constant infectious period,, J. Appl. Math. Comput., 35 (2011), 229.  doi: 10.1007/s12190-009-0353-3.  Google Scholar

show all references

References:
[1]

J. Arino and P. van den Driessche, Time delays in epidemic models: Modeling and numerical considerations,, in, (2006), 539.   Google Scholar

[2]

E. Beretta and D. Breda, An SEIR epidemic model with constant latency time and infectious period,, Math. Biosci. Eng., 8 (2011), 931.  doi: 10.3934/mbe.2011.8.931.  Google Scholar

[3]

E. Beretta and Y. Takeuchi, Global stability of an SIR model with time delays,, J. Math. Biol., 33 (1995), 250.   Google Scholar

[4]

E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay,, Nonlinear Anal., 47 (2001), 4107.  doi: 10.1016/S0362-546X(01)00528-4.  Google Scholar

[5]

J. M. Cushing, "An Introduction to Structured Population Dynamics,", CBMS-NSF Regional Conference Series in Applied Mathematics, 71 (1998).   Google Scholar

[6]

Y. Enatsu, Y. Nakata and Y. Muroya, Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays,, Discrete and Continuous Dynamical Systems Ser. B, 15 (2011), 61.  doi: 10.3934/dcdsb.2011.15.61.  Google Scholar

[7]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate,, Bull. Math. Biol., 72 (2010), 1192.  doi: 10.1007/s11538-009-9487-6.  Google Scholar

[8]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125.  doi: 10.1007/s00285-010-0368-2.  Google Scholar

[9]

G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model,, SIAM J. Appl. Math., 72 (2012), 25.  doi: 10.1137/110826588.  Google Scholar

[10]

A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence,, Math. Biosci. Eng., 1 (2004), 57.  doi: 10.3934/mbe.2004.1.57.  Google Scholar

[11]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,, Bull. Math. Biol., 68 (2006), 615.  doi: 10.1007/s11538-005-9037-9.  Google Scholar

[12]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871.  doi: 10.1007/s11538-007-9196-y.  Google Scholar

[13]

M. Iannelli, "Mathematical Theory of Age-Structured Population Dynamics,", Applied Mathematical Monographs, 7 (1995).   Google Scholar

[14]

A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility,, (2011), (2011).   Google Scholar

[15]

P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Appl. Anal., 89 (2010), 1109.  doi: 10.1080/00036810903208122.  Google Scholar

[16]

C. C. McCluskey, Delay versus age-of-infection-global stability,, Appl. Math. Comput., 217 (2010), 3046.   Google Scholar

[17]

C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence,, Nonlinear Anal. RWA, 11 (2010), 3106.  doi: 10.1016/j.nonrwa.2009.11.005.  Google Scholar

[18]

C. C. McCluskey, Global stability of an SIR epidemic model with delay and general nonlinear incidence,, Math. Biosci. Eng., 7 (2010), 837.  doi: 10.3934/mbe.2010.7.837.  Google Scholar

[19]

P. van den Driessche, Some epidemiological models with delays,, in, (1994), 507.   Google Scholar

[20]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,", Marcel Dekker, (1985).   Google Scholar

[21]

R. Xu and Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay,, Nonlinear Anal. RWA, 10 (2009), 3175.  doi: 10.1016/j.nonrwa.2008.10.013.  Google Scholar

[22]

R. Xu and Y. Du, A delayed SIR epidemic model with saturation incidence and a constant infectious period,, J. Appl. Math. Comput., 35 (2011), 229.  doi: 10.1007/s12190-009-0353-3.  Google Scholar

[1]

C. Connell McCluskey. Global stability for an $SEI$ model of infectious disease with age structure and immigration of infecteds. Mathematical Biosciences & Engineering, 2016, 13 (2) : 381-400. doi: 10.3934/mbe.2015008
[2]

Yoshiaki Muroya, Yoichi Enatsu, Huaixing Li. A note on the global stability of an SEIR epidemic model with constant latency time and infectious period. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 173-183. doi: 10.3934/dcdsb.2013.18.173
[3]

C. Connell McCluskey. Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes. Mathematical Biosciences & Engineering, 2012, 9 (4) : 819-841. doi: 10.3934/mbe.2012.9.819
[4]

Edoardo Beretta, Dimitri Breda. An SEIR epidemic model with constant latency time and infectious period. Mathematical Biosciences & Engineering, 2011, 8 (4) : 931-952. doi: 10.3934/mbe.2011.8.931
[5]

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
[6]

Xia Wang, Shengqiang Liu. Global properties of a delayed SIR epidemic model with multiple parallel infectious stages. Mathematical Biosciences & Engineering, 2012, 9 (3) : 685-695. doi: 10.3934/mbe.2012.9.685
[7]

Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109
[8]

Bin-Guo Wang, Wan-Tong Li, Liang Zhang. An almost periodic epidemic model with age structure in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 291-311. doi: 10.3934/dcdsb.2016.21.291
[9]

Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641
[10]

Yuming Chen, Junyuan Yang, Fengqin Zhang. The global stability of an SIRS model with infection age. Mathematical Biosciences & Engineering, 2014, 11 (3) : 449-469. doi: 10.3934/mbe.2014.11.449
[11]

Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva. Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1159-1186. doi: 10.3934/mbe.2017060
[12]

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
[13]

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
[14]

Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929
[15]

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
[16]

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
[17]

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
[18]

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
[19]

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
[20]

Hongbin Guo, Michael Yi Li. Global dynamics of a staged progression model for infectious diseases. Mathematical Biosciences & Engineering, 2006, 3 (3) : 513-525. doi: 10.3934/mbe.2006.3.513

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences