## Abstract and Applied Analysis

### An SIR Epidemic Model with Time Delay and General Nonlinear Incidence Rate

#### Abstract

An SIR epidemic model with nonlinear incidence rate and time delay is investigated. The disease transmission function and the rate that infected individuals recovered from the infected compartment are assumed to be governed by general functions $F(S,I)$ and $G(I)$, respectively. By constructing Lyapunov functionals and using the Lyapunov-LaSalle invariance principle, the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium is obtained. It is shown that the global properties of the system depend on both the properties of these general functions and the basic reproductive number ${R}_{0}$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 131257, 7 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412607565

Digital Object Identifier
doi:10.1155/2014/131257

Mathematical Reviews number (MathSciNet)
MR3173270

#### Citation

Li, Mingming; Liu, Xianning. An SIR Epidemic Model with Time Delay and General Nonlinear Incidence Rate. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 131257, 7 pages. doi:10.1155/2014/131257. https://projecteuclid.org/euclid.aaa/1412607565

#### References

• M. Gabriela, M. Gomes, L. J. White, and G. F. Medley, “The reinfection threshold,” Journal of Theoretical Biology, vol. 236, no. 1, pp. 111–113, 2005.
• Y. Zhou and H. Liu, “Stability of periodic solutions for an SIS model with pulse vaccination,” Mathematical and Computer Modelling, vol. 38, no. 3-4, pp. 299–308, 2003.
• A. Gray, D. Greenhalgh, L. Hu, X. Mao, and J. Pan, “A stochastic differential equation SIS epidemic model,” SIAM Journal on Applied Mathematics, vol. 71, no. 3, pp. 876–902, 2011.
• M. Song, W. Ma, and Y. Takeuchi, “Permanence of a delayed SIR epidemic model with density dependent birth rate,” Journal of Computational and Applied Mathematics, vol. 201, no. 2, pp. 389–394, 2007.
• W. Ma, M. Song, and Y. Takeuchi, “Global stability of an SIR epidemic model with time delay,” Applied Mathematics Letters, vol. 17, no. 10, pp. 1141–1145, 2004.
• F. Zhang, Z. Li, and F. Zhang, “Global stability of an SIR epidemic model with constant infectious period,” Applied Mathematics and Computation, vol. 199, no. 1, pp. 285–291, 2008.
• X. Liu, Y. Takeuchi, and S. Iwami, “SVIR epidemic models with vaccination strategies,” Journal of Theoretical Biology, vol. 253, no. 1, pp. 1–11, 2008.
• J. Wang, J. Zhang, and Z. Jin, “Analysis of an SIR model with bilinear incidence rate,” Nonlinear Analysis, vol. 11, no. 4, pp. 2390–2402, 2010.
• A. Korobeinikov and G. C. Wake, “Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models,” Applied Mathematics Letters, vol. 15, no. 8, pp. 955–960, 2002.
• S. M. O'Regan, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan, and A. V. Pokrovskii, “Lyapunov functions for SIR and SIRS epidemic models,” Applied Mathematics Letters, vol. 23, no. 4, pp. 446–448, 2010.
• L. Liu, X. Q. Zhao, and Y. Zhou, “A tuberculosis model with seasonality,” Bulletin of Mathematical Biology, vol. 72, no. 4, pp. 931–952, 2010.
• N. Yoshida and T. Hara, “Global stability of a delayed SIR epidemic model with density dependent birth and death rates,” Journal of Computational and Applied Mathematics, vol. 201, no. 2, pp. 339–347, 2007.
• R. M. Anderson and R. M. May, “Regulation and stability of host-parasite population interactions. I. Regulatory processes,” Journal of Animal Ecology, vol. 47, no. 1, pp. 219–267, 1978.
• C. Wei and L. Chen, “A delayed epidemic model with pulse vaccination,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 746951, 12 pages, 2008.
• J. Zhang, Z. Jin, Q. Liu, and Z. Zhang, “Analysis of a delayed SIR model with nonlinear incidence rate,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 636153, 16 pages, 2008.
• Z. Jiang and J. Wei, “Stability and bifurcation analysis in a delayed SIR model,” Chaos, Solitons and Fractals, vol. 35, no. 3, pp. 609–619, 2008.
• R. Xu and Z. Ma, “Stability of a delayed SIRS epidemic model with a nonlinear incidence rate,” Chaos, Solitons and Fractals, vol. 41, no. 5, pp. 2319–2325, 2009.
• R. Xu, Z. Ma, and Z. Wang, “Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity,” Computers & Mathematics with Applications, vol. 59, no. 9, pp. 3211–3221, 2010.
• R. Xu and Z. Ma, “Global stability of a SIR epidemic model with nonlinear incidence rate and time delay,” Nonlinear Analysis, vol. 10, no. 5, pp. 3175–3189, 2009.
• X. Zhang and X. Liu, “Backward bifurcation of an epidemic model with saturated treatment function,” Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 433–443, 2008.
• C. C. McCluskey, “Global stability for an SIR epidemic model with delay and nonlinear incidence,” Nonlinear Analysis, vol. 11, no. 4, pp. 3106–3109, 2010.
• A. Kaddar, “On the dynamics of a delayed SIR epidemic model with a modified saturated incidence rate,” Electronic Journal of Differential Equations, vol. 2009, no. 133, pp. 1–7, 2009.
• A. Kaddar, A. Abta, and H. T. Alaoui, “A comparison of delayed SIR and SEIR epidemic models,” Nonlinear Analysis, vol. 16, no. 2, pp. 181–190, 2011.
• A. Abta, A. Kaddar, and H. T. Alaoui, “Global stability for delay SIR and SEIR epidemic models with saturated incidence rates,” Electronic Journal of Differential Equations, vol. 2012, no. 23, pp. 1–13, 2012.
• Z. Liu, “Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates,” Nonlinear Analysis, vol. 14, no. 3, pp. 1286–1299, 2013.
• A. Korobeinikov and K. Philip Maini, “Nonliear incidence and stability of infectious diseasemodels,” Mathematical Medicine and Biology, vol. 22, pp. 113–128, 2005.
• A. Korobeinikov, “Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,” Bulletin of Mathematical Biology, vol. 68, no. 3, pp. 615–626, 2006.
• A. Korobeinikov, “Global properties of infectious disease models with nonlinear incidence,” Bulletin of Mathematical Biology, vol. 69, no. 6, pp. 1871–1886, 2007.
• A. Korobeinikov, “Stability of ecosystem: global properties of a general predator-prey model,” Mathematical Medicine and Biology, vol. 26, no. 4, pp. 309–321, 2009.
• G. Huang, Y. Takeuchi, W. Ma, and D. Wei, “Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate,” Bulletin of Mathematical Biology, vol. 72, no. 5, pp. 1192–1207, 2010.
• J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, NY, USA, 1993.
• S. F. Ellermeyer, “Competition in the chemostat: global asymptotic behavior of a model with delayed response in growth,” SIAM Journal on Applied Mathematics, vol. 54, no. 2, pp. 456–465, 1994.
• S. F. Ellermeyer, J. Hendrix, and N. Ghoochan, “A theoretical and empirical investigation of delayed growth response in the continuous culture of bacteria,” Journal of Theoretical Biology, vol. 222, no. 4, pp. 485–494, 2003.
• H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, NY, USA, 2011. \endinput