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

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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

Yoichi Enatsu ^1, , Yukihiko Nakata ^2, and Yoshiaki Muroya ^3,

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

Abstract
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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.

Keywords: Lyapunov functional., distributed delays, permanence, SIR epidemic models, nonlinear incidence rate, global asymptotic stability.

Mathematics Subject Classification: Primary: 34K20, 34K25; Secondary: 92D3.

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. 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. 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. 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. 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. 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), 28 (1976), 335. 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. 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. 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. 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. 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. doi: doi:10.1007/s11538-007-9196-y. Google Scholar
[13]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press, (1993). Google Scholar
[14]	J. LaSalle and S. Lefschetz, "Stability by Liapunov's Direct Method, with Applications,", Mathematics in Science and Engineering, 4 (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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), 28 (1976), 335. 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. 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. 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. 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. 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. doi: doi:10.1007/s11538-007-9196-y. Google Scholar
[13]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press, (1993). Google Scholar
[14]	J. LaSalle and S. Lefschetz, "Stability by Liapunov's Direct Method, with Applications,", Mathematics in Science and Engineering, 4 (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. 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. 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. 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. 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. 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. 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. 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. doi: doi:10.1016/j.nonrwa.2008.10.013. Google Scholar

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