Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 34K20, 34K25; Secondary: 92D30.

 Citation:

•  [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. [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. [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. [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. [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. [6] J. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, 1993. [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. [8] H. W. Hethcote, The Mathematics of Infectious Diseases, SIAM Rev., 42 (2000), 599-653. [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. [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. [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. [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. [13] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, San Diego, 1993. [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. [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. [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. [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. [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. [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. [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. [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. [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.