American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Erratum to: Investigating the steady state of multicellular sheroids by revisiting the two-fluid model
  • MBE Home
  • This Issue
  • Next Article
    A minimal mathematical model for the initial molecular interactions of death receptor signalling
2012, 9(3): 685-695. doi: 10.3934/mbe.2012.9.685

Global properties of a delayed SIR epidemic model with multiple parallel infectious stages

Xia Wang 1, and  Shengqiang Liu 2,

1. 

Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street Harbin, 150080 and College of Mathematics and Information Science, Xinyang Normal University, Xinyang, 464000, China
2. 

Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin, 150080

Received  October 2011 Revised  February 2012 Published  July 2012

In this paper, we study the global properties of an SIR epidemic model with distributed delays, where there are several parallel infective stages, and some of the infected cells are detected and treated, which others remain undetected and untreated. The model is analyzed by determining a basic reproduction number $R_0$, and by using Lyapunov functionals, we prove that the infection-free equilibrium $E^0$ of system (3) is globally asymptotically attractive when $R_0\leq 1$, and that the unique infected equilibrium $E^*$ of system (3) exists and it is globally asymptotically attractive when $R_0>1$.
Keywords: distributed delay, global attractivity, Lyapounov functional., SIR epidemic model.
Mathematics Subject Classification: 34C05, 92D2.
Citation: 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
References:
[1]

K. L. Cooke, Stability analysis for a vector disease model,, Rocky Mount. J. Math., 9 (1979), 31.   Google Scholar

[2]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4$^+$ T cells,, Math. Biosci., 165 (2000), 27.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[3]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365.  doi: 10.1007/BF00178324.  Google Scholar

[4]

S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection,, J. Biol. Dynam., 2 (2008), 140.   Google Scholar

[5]

H. W. Hethcote, Qualitative analyses of communicable disease models,, Math. Biosci., 28 (1976), 335.  doi: 10.1016/0025-5564(76)90132-2.  Google Scholar

[6]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev., 42 (2000), 599.  doi: 10.1137/S0036144500371907.  Google Scholar

[7]

V. Herz, S. Bonhoeffer, R. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus delay,, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247.  doi: 10.1073/pnas.93.14.7247.  Google Scholar

[8]

J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations,", Applied Mathematical Sciences, 99 (1993).   Google Scholar

[9]

A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages,, Bull. Math. Biol., 71 (2009), 75.  doi: 10.1007/s11538-008-9352-z.  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.   Google Scholar

[11]

A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and nonlinear incidence rate,, Math. Med. Biol., 26 (2009), 225.   Google Scholar

[12]

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

[13]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, 191 (1993).   Google Scholar

[14]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492.  doi: 10.1007/s11538-010-9503-x.  Google Scholar

[15]

S. Liu, S. Wang and L. Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse,, Nonlinear Anal. Real World Appl., 12 (2011), 119.  doi: 10.1016/j.nonrwa.2010.06.001.  Google Scholar

[16]

M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434.  doi: 10.1137/090779322.  Google Scholar

[17]

C. C. McCluskey, Global stability of an epidemic model with delay and general nonlinear incidence,, Math. Biosci. and Eng., 7 (2010), 837.   Google Scholar

[18]

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

[19]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. and Eng., 6 (2009), 603.   Google Scholar

[20]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete,, Nonlinear Anal. Real World Appl., 11 (2010), 55.  doi: 10.1016/j.nonrwa.2008.10.014.  Google Scholar

[21]

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

[22]

K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data,, Math. Biosci., 235 (2012), 98.  doi: 10.1016/j.mbs.2011.11.002.  Google Scholar

[23]

Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times,, Nonlinear Anal., 42 (2000), 931.  doi: 10.1016/S0362-546X(99)00138-8.  Google Scholar

[24]

J. Wang, G. Huang, Y. Takeuchi and S. Liu, Sveir epidemiological model with varying infectivity and distributed delays,, Math. Biosci. and Eng., 8 (2011), 875.   Google Scholar

[25]

X. Wang, Y. D. Tao and X. Y. Song, A delayed HIV-1 infection model with Beddington-DeAngelis functional response,, Nonlinear Dyn., 62 (2010), 67.  doi: 10.1007/s11071-010-9699-1.  Google Scholar

show all references

References:
[1]

K. L. Cooke, Stability analysis for a vector disease model,, Rocky Mount. J. Math., 9 (1979), 31.   Google Scholar

[2]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4$^+$ T cells,, Math. Biosci., 165 (2000), 27.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[3]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365.  doi: 10.1007/BF00178324.  Google Scholar

[4]

S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection,, J. Biol. Dynam., 2 (2008), 140.   Google Scholar

[5]

H. W. Hethcote, Qualitative analyses of communicable disease models,, Math. Biosci., 28 (1976), 335.  doi: 10.1016/0025-5564(76)90132-2.  Google Scholar

[6]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev., 42 (2000), 599.  doi: 10.1137/S0036144500371907.  Google Scholar

[7]

V. Herz, S. Bonhoeffer, R. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus delay,, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247.  doi: 10.1073/pnas.93.14.7247.  Google Scholar

[8]

J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations,", Applied Mathematical Sciences, 99 (1993).   Google Scholar

[9]

A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages,, Bull. Math. Biol., 71 (2009), 75.  doi: 10.1007/s11538-008-9352-z.  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.   Google Scholar

[11]

A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and nonlinear incidence rate,, Math. Med. Biol., 26 (2009), 225.   Google Scholar

[12]

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

[13]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, 191 (1993).   Google Scholar

[14]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492.  doi: 10.1007/s11538-010-9503-x.  Google Scholar

[15]

S. Liu, S. Wang and L. Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse,, Nonlinear Anal. Real World Appl., 12 (2011), 119.  doi: 10.1016/j.nonrwa.2010.06.001.  Google Scholar

[16]

M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434.  doi: 10.1137/090779322.  Google Scholar

[17]

C. C. McCluskey, Global stability of an epidemic model with delay and general nonlinear incidence,, Math. Biosci. and Eng., 7 (2010), 837.   Google Scholar

[18]

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

[19]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. and Eng., 6 (2009), 603.   Google Scholar

[20]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete,, Nonlinear Anal. Real World Appl., 11 (2010), 55.  doi: 10.1016/j.nonrwa.2008.10.014.  Google Scholar

[21]

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

[22]

K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data,, Math. Biosci., 235 (2012), 98.  doi: 10.1016/j.mbs.2011.11.002.  Google Scholar

[23]

Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times,, Nonlinear Anal., 42 (2000), 931.  doi: 10.1016/S0362-546X(99)00138-8.  Google Scholar

[24]

J. Wang, G. Huang, Y. Takeuchi and S. Liu, Sveir epidemiological model with varying infectivity and distributed delays,, Math. Biosci. and Eng., 8 (2011), 875.   Google Scholar

[25]

X. Wang, Y. D. Tao and X. Y. Song, A delayed HIV-1 infection model with Beddington-DeAngelis functional response,, Nonlinear Dyn., 62 (2010), 67.  doi: 10.1007/s11071-010-9699-1.  Google Scholar

[1]

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
[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]

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
[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]

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
[6]

Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080
[7]

Y. Chen, L. Wang. Global attractivity of a circadian pacemaker model in a periodic environment. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 277-288. doi: 10.3934/dcdsb.2005.5.277
[8]

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
[9]

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
[10]

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
[11]

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
[12]

Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793
[13]

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
[14]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1083-1106. doi: 10.3934/mbe.2015.12.1083
[15]

Amitava Mukhopadhyay, Andrea De Gaetano, Ovide Arino. Modeling the intra-venous glucose tolerance test: A global study for a single-distributed-delay model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 407-417. doi: 10.3934/dcdsb.2004.4.407
[16]

Qun Liu, Daqing Jiang, Ningzhong Shi, Tasawar Hayat, Ahmed Alsaedi. Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2479-2500. doi: 10.3934/dcdsb.2017127
[17]

G. A. Enciso, E. D. Sontag. Global attractivity, I/O monotone small-gain theorems, and biological delay systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 549-578. doi: 10.3934/dcds.2006.14.549
[18]

Shujing Gao, Dehui Xie, Lansun Chen. Pulse vaccination strategy in a delayed sir epidemic model with vertical transmission. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 77-86. doi: 10.3934/dcdsb.2007.7.77
[19]

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
[20]

Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039

2018 Impact Factor: 1.313

Tools

Metrics

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

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences