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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

 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.
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. [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. [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. [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. [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. [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. [7] B.-S. Goh, Stability of some multispecies population models,, in, (1980). [8] J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations,", Springer-Verlag, (1993). [9] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11. [10] H. W. Hethcote, Qualitative analyses of communicable disease models,, Math. Biosci., 28 (1976), 335. doi: doi:10.1016/0025-5564(76)90132-2. [11] H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: doi:10.1137/S0036144500371907. [12] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Springer-Verlag, (1993). [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. [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. [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. [16] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, 191 (1993). [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. [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. [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. [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. [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. [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. [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. [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.

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##### References:
 [1] F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations,, Funkcial. Ekvac., 31 (1988), 331. [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. [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. [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. [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. [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. [7] B.-S. Goh, Stability of some multispecies population models,, in, (1980). [8] J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations,", Springer-Verlag, (1993). [9] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11. [10] H. W. Hethcote, Qualitative analyses of communicable disease models,, Math. Biosci., 28 (1976), 335. doi: doi:10.1016/0025-5564(76)90132-2. [11] H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: doi:10.1137/S0036144500371907. [12] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Springer-Verlag, (1993). [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. [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. [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. [16] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, 191 (1993). [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. [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. [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. [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. [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. [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. [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. [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.
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