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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), 331-347.  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), 4107-4115. 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), 43-61. 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), 31-42. 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), 365-382. 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), 803-833.  Google Scholar

[7]

B.-S. Goh, Stability of some multispecies population models, in "Modeling and Differential Equations in Biology," Dekker, New York, 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), 11-41.  Google Scholar

[10]

H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: doi:10.1016/0025-5564(76)90132-2.  Google Scholar

[11]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. 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), 1192-1207. 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), 1871-1886. 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), 113-128. doi: doi:10.1093/imammb/dqi001.  Google Scholar

[16]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, vol 191, Academic Press, Cambridge, 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), 2434-2448. 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), 581-591. 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), 1109-1140. 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), 603-610. 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), 55-59. 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), 3106-3109. 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), 931-947. 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), 3175-3189. 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), 331-347.  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), 4107-4115. 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), 43-61. 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), 31-42. 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), 365-382. 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), 803-833.  Google Scholar

[7]

B.-S. Goh, Stability of some multispecies population models, in "Modeling and Differential Equations in Biology," Dekker, New York, 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), 11-41.  Google Scholar

[10]

H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: doi:10.1016/0025-5564(76)90132-2.  Google Scholar

[11]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. 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), 1192-1207. 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), 1871-1886. 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), 113-128. doi: doi:10.1093/imammb/dqi001.  Google Scholar

[16]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, vol 191, Academic Press, Cambridge, 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), 2434-2448. 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), 581-591. 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), 1109-1140. 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), 603-610. 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), 55-59. 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), 3106-3109. 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), 931-947. 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), 3175-3189. doi: doi:10.1016/j.nonrwa.2008.10.013.  Google Scholar

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