# American Institute of Mathematical Sciences

January  2011, 15(1): 61-74. doi: 10.3934/dcdsb.2011.15.61

## Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays

 1 Department of Pure and Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555 2 Basque Center for Applied Mathematics, Bizkaia Technology Park, Building 500 E-48160 Derio, Spain 3 Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555

Received  October 2009 Revised  April 2010 Published  October 2010

In this paper, we establish the global asymptotic stability of equilibria for an SIR model of infectious diseases with distributed time delays governed by a wide class of nonlinear incidence rates. We obtain the global properties of the model by proving the permanence and constructing a suitable Lyapunov functional. Under some suitable assumptions on the nonlinear term in the incidence rate, the global dynamics of the model is completely determined by the basic reproduction number $R_0$ and the distributed delays do not influence the global dynamics of the model.
Citation: 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
##### References:
 [1] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I, Nature, 280 (1979), 361-367. doi: doi:10.1038/280361a0.  Google Scholar [2] E. Beretta and Y. Takeuchi, Convergence results in SIR epidemic models with varying population size, Nonlinear Anal., 28 (1997), 1909-1921. doi: doi:10.1016/S0362-546X(96)00035-1.  Google Scholar [3] G. C. Brown and R. Hasibuan, Conidial discharge and transmission efficiency of "Neozygites floridana," an entomopathogenic fungus infeccting two-spotted spider mites under laboratory conditions, J. Invertebr. Pathol., 65 (1995), 10-16. doi: doi:10.1006/jipa.1995.1002.  Google Scholar [4] 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 [5] K. L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42. doi: doi:10.1216/RMJ-1979-9-1-31.  Google Scholar [6] J. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, 1993.  Google Scholar [7] 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 [8] H. W. Hethcote, The Mathematics of Infectious Diseases, SIAM Rev., 42 (2000), 599-653.  Google Scholar [9] 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), 1192-1207. doi: doi:10.1007/s11538-009-9487-6.  Google Scholar [10] A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128. doi: doi:10.1093/imammb/dqi001.  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), 615-626. doi: 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), 1871-1886. doi: doi:10.1007/s11538-007-9196-y.  Google Scholar [13] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, San Diego, 1993.  Google Scholar [14] J. LaSalle and S. Lefschetz, "Stability by Liapunov's Direct Method, with Applications," Mathematics in Science and Engineering, 4, Academic Press, New York-London, 1961.  Google Scholar [15] W. Ma, Y. Takeuchi, T. Hara and E. Beretta, Permanence of an SIR epidemic model with distributed delays, Tohoku. Math. J., 54 (2002), 581-591. doi: doi:10.2748/tmj/1113247650.  Google Scholar [16] W. Ma, M. Song and Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett., 17 (2004), 1141-1145. doi: doi:10.1016/j.aml.2003.11.005.  Google Scholar [17] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-Distributed or discrete, Nonl. Anal. RWA., 11 (2010), 55-59. doi: doi:10.1016/j.nonrwa.2008.10.014.  Google Scholar [18] C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonl. Anal. RWA., 11 (2010), 3106-3109. doi: doi:10.1016/j.nonrwa.2009.11.005.  Google Scholar [19] M. Song, W. Ma and Y. Takeuchi, Permanence of a delayed SIR epidemic model with density dependent birth rate, J. Comp. Appl. Math., 201 (2007), 389-394. doi: doi:10.1016/j.cam.2005.12.039.  Google Scholar [20] 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 [21] W. Wang, Global behavior of an SEIRS epidemic model with time delays, Appl. Math. Lett., 15 (2002), 423-428. doi: doi:10.1016/S0893-9659(01)00153-7.  Google Scholar [22] R. Xu and Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonl. Anal. RWA., 10 (2009), 3175-3189. doi: doi:10.1016/j.nonrwa.2008.10.013.  Google Scholar

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##### References:
 [1] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I, Nature, 280 (1979), 361-367. doi: doi:10.1038/280361a0.  Google Scholar [2] E. Beretta and Y. Takeuchi, Convergence results in SIR epidemic models with varying population size, Nonlinear Anal., 28 (1997), 1909-1921. doi: doi:10.1016/S0362-546X(96)00035-1.  Google Scholar [3] G. C. Brown and R. Hasibuan, Conidial discharge and transmission efficiency of "Neozygites floridana," an entomopathogenic fungus infeccting two-spotted spider mites under laboratory conditions, J. Invertebr. Pathol., 65 (1995), 10-16. doi: doi:10.1006/jipa.1995.1002.  Google Scholar [4] 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 [5] K. L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42. doi: doi:10.1216/RMJ-1979-9-1-31.  Google Scholar [6] J. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, 1993.  Google Scholar [7] 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 [8] H. W. Hethcote, The Mathematics of Infectious Diseases, SIAM Rev., 42 (2000), 599-653.  Google Scholar [9] 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), 1192-1207. doi: doi:10.1007/s11538-009-9487-6.  Google Scholar [10] A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128. doi: doi:10.1093/imammb/dqi001.  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), 615-626. doi: 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), 1871-1886. doi: doi:10.1007/s11538-007-9196-y.  Google Scholar [13] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, San Diego, 1993.  Google Scholar [14] J. LaSalle and S. Lefschetz, "Stability by Liapunov's Direct Method, with Applications," Mathematics in Science and Engineering, 4, Academic Press, New York-London, 1961.  Google Scholar [15] W. Ma, Y. Takeuchi, T. Hara and E. Beretta, Permanence of an SIR epidemic model with distributed delays, Tohoku. Math. J., 54 (2002), 581-591. doi: doi:10.2748/tmj/1113247650.  Google Scholar [16] W. Ma, M. Song and Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett., 17 (2004), 1141-1145. doi: doi:10.1016/j.aml.2003.11.005.  Google Scholar [17] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-Distributed or discrete, Nonl. Anal. RWA., 11 (2010), 55-59. doi: doi:10.1016/j.nonrwa.2008.10.014.  Google Scholar [18] C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonl. Anal. RWA., 11 (2010), 3106-3109. doi: doi:10.1016/j.nonrwa.2009.11.005.  Google Scholar [19] M. Song, W. Ma and Y. Takeuchi, Permanence of a delayed SIR epidemic model with density dependent birth rate, J. Comp. Appl. Math., 201 (2007), 389-394. doi: doi:10.1016/j.cam.2005.12.039.  Google Scholar [20] 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 [21] W. Wang, Global behavior of an SEIRS epidemic model with time delays, Appl. Math. Lett., 15 (2002), 423-428. doi: doi:10.1016/S0893-9659(01)00153-7.  Google Scholar [22] R. Xu and Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonl. Anal. RWA., 10 (2009), 3175-3189. doi: doi:10.1016/j.nonrwa.2008.10.013.  Google Scholar
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