American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Time variations in the generation time of an infectious disease: Implications for sampling to appropriately quantify transmission potential
  • MBE Home
  • This Issue
  • Next Article
    A stoichiometrically derived algal growth model and its global analysis
2010, 7(4): 837-850. doi: 10.3934/mbe.2010.7.837

Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence

C. Connell McCluskey 1,

1. 

Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada

Received  March 2010 Revised  July 2010 Published  October 2010

An SIR model with distributed delay and a general incidence function is studied. Conditions are given under which the system exhibits threshold behaviour: the disease-free equilibrium is globally asymptotically stable if R0<1 and globally attracting if R0=1; if R0>1, then the unique endemic equilibrium is globally asymptotically stable. The global stability proofs use a Lyapunov functional and do not require uniform persistence to be shown a priori. It is shown that the given conditions are satisfied by several common forms of the incidence function.
Keywords: Global Stability, Lyapunov Functional, Nonlinear Incidence., Delay.
Mathematics Subject Classification: Primary: 34K20, 92D3.
Citation: 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
References:
[1]

F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations,, Funkcial. Ekvac., 31 (1988), 331.   Google Scholar

[2]

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

[3]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model,, Math. Biosci., 42 (1978), 43.  doi: doi:10.1016/0025-5564(78)90006-8.  Google Scholar

[4]

K. L. Cooke, Stability analysis for a vector disease model,, Rocky Mount. J. Math., 9 (1979), 31.  doi: doi:10.1216/RMJ-1979-9-1-31.  Google Scholar

[5]

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

[6]

Z. Feng and H. Thieme, Endemic models for the spread of infectious diseases with arbitrarily distributed disease stages I: General theory,, SIAM J. Appl. Math., 61 (2000), 803.   Google Scholar

[7]

B.-S. Goh, Stability of some multispecies population models,, in, (1980).   Google Scholar

[8]

J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations,", Springer-Verlag, (1993).   Google Scholar

[9]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.   Google Scholar

[10]

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

[11]

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

[12]

Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Springer-Verlag, (1993).   Google Scholar

[13]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stabilty for delay SIR and SEIR epidemic models with nonlinear incidence rate,, Bull. Math. Biol., 72 (2009), 1192.  doi: doi:10.1007/s11538-009-9487-6.  Google Scholar

[14]

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

[15]

A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models,, Math. Med. and Biol., 22 (2005), 113.  doi: doi:10.1093/imammb/dqi001.  Google Scholar

[16]

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

[17]

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: doi:10.1137/090779322.  Google Scholar

[18]

W. Ma, Y. Takeuchi, T. Hara and E. Beretta, Permanence of an SIR epidemic model with distributed time delays,, Tohoku Math. J., 54 (2002), 581.  doi: doi:10.2748/tmj/1113247650.  Google Scholar

[19]

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

[20]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. and Eng., 6 (2009), 603.  doi: doi:10.3934/mbe.2009.6.603.  Google Scholar

[21]

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

[22]

C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence,, Nonlinear Anal. RWA, 11 (2010), 3106.  doi: doi:10.1016/j.nonrwa.2009.11.005.  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: doi:10.1016/S0362-546X(99)00138-8.  Google Scholar

[24]

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: doi:10.1016/j.nonrwa.2008.10.013.  Google Scholar

show all references

References:
[1]

F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations,, Funkcial. Ekvac., 31 (1988), 331.   Google Scholar

[2]

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

[3]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model,, Math. Biosci., 42 (1978), 43.  doi: doi:10.1016/0025-5564(78)90006-8.  Google Scholar

[4]

K. L. Cooke, Stability analysis for a vector disease model,, Rocky Mount. J. Math., 9 (1979), 31.  doi: doi:10.1216/RMJ-1979-9-1-31.  Google Scholar

[5]

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

[6]

Z. Feng and H. Thieme, Endemic models for the spread of infectious diseases with arbitrarily distributed disease stages I: General theory,, SIAM J. Appl. Math., 61 (2000), 803.   Google Scholar

[7]

B.-S. Goh, Stability of some multispecies population models,, in, (1980).   Google Scholar

[8]

J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations,", Springer-Verlag, (1993).   Google Scholar

[9]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.   Google Scholar

[10]

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

[11]

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

[12]

Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Springer-Verlag, (1993).   Google Scholar

[13]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stabilty for delay SIR and SEIR epidemic models with nonlinear incidence rate,, Bull. Math. Biol., 72 (2009), 1192.  doi: doi:10.1007/s11538-009-9487-6.  Google Scholar

[14]

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

[15]

A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models,, Math. Med. and Biol., 22 (2005), 113.  doi: doi:10.1093/imammb/dqi001.  Google Scholar

[16]

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

[17]

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: doi:10.1137/090779322.  Google Scholar

[18]

W. Ma, Y. Takeuchi, T. Hara and E. Beretta, Permanence of an SIR epidemic model with distributed time delays,, Tohoku Math. J., 54 (2002), 581.  doi: doi:10.2748/tmj/1113247650.  Google Scholar

[19]

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

[20]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. and Eng., 6 (2009), 603.  doi: doi:10.3934/mbe.2009.6.603.  Google Scholar

[21]

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

[22]

C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence,, Nonlinear Anal. RWA, 11 (2010), 3106.  doi: doi:10.1016/j.nonrwa.2009.11.005.  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: doi:10.1016/S0362-546X(99)00138-8.  Google Scholar

[24]

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: doi:10.1016/j.nonrwa.2008.10.013.  Google Scholar

[1]

Andrei Korobeinikov, Philip K. Maini. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Mathematical Biosciences & Engineering, 2004, 1 (1) : 57-60. doi: 10.3934/mbe.2004.1.57
[2]

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
[3]

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 & Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057
[4]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with generalized nonlinear incidence and vaccination age. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 977-996. doi: 10.3934/dcdsb.2016.21.977
[5]

Attila Dénes, Gergely Röst. Global stability for SIR and SIRS models with nonlinear incidence and removal terms via Dulac functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1101-1117. doi: 10.3934/dcdsb.2016.21.1101
[6]

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
[7]

Yu Ji, Lan Liu. Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 133-149. doi: 10.3934/dcdsb.2016.21.133
[8]

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
[9]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727
[10]

Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493
[11]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169
[12]

Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082
[13]

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
[14]

Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214
[15]

Yu Ji. Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 525-536. doi: 10.3934/mbe.2015.12.525
[16]

Ting Guo, Haihong Liu, Chenglin Xu, Fang Yan. Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4223-4242. doi: 10.3934/dcdsb.2018134
[17]

Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115
[18]

Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038
[19]

C. Connell McCluskey. Global stability for an SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2009, 6 (3) : 603-610. doi: 10.3934/mbe.2009.6.603
[20]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (32)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences