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2012, 9(3): 685-695. doi: 10.3934/mbe.2012.9.685

## Global properties of a delayed SIR epidemic model with multiple parallel infectious stages

 1 Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street Harbin, 150080 and College of Mathematics and Information Science, Xinyang Normal University, Xinyang, 464000, China 2 Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin, 150080

Received  October 2011 Revised  February 2012 Published  July 2012

In this paper, we study the global properties of an SIR epidemic model with distributed delays, where there are several parallel infective stages, and some of the infected cells are detected and treated, which others remain undetected and untreated. The model is analyzed by determining a basic reproduction number $R_0$, and by using Lyapunov functionals, we prove that the infection-free equilibrium $E^0$ of system (3) is globally asymptotically attractive when $R_0\leq 1$, and that the unique infected equilibrium $E^*$ of system (3) exists and it is globally asymptotically attractive when $R_0>1$.
Citation: Xia Wang, Shengqiang Liu. Global properties of a delayed SIR epidemic model with multiple parallel infectious stages. Mathematical Biosciences & Engineering, 2012, 9 (3) : 685-695. doi: 10.3934/mbe.2012.9.685
##### References:
 [1] K. L. Cooke, Stability analysis for a vector disease model, Rocky Mount. J. Math., 9 (1979), 31-42. [2] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4$^+$ T cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7. [3] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [4] S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biol. Dynam., 2 (2008), 140-153. [5] H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2. [6] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [7] V. Herz, S. Bonhoeffer, R. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus delay, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251. doi: 10.1073/pnas.93.14.7247. [8] J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. [9] A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71 (2009), 75-83. doi: 10.1007/s11538-008-9352-z. [10] A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60. [11] A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and nonlinear incidence rate, Math. Med. Biol., 26 (2009), 225-239. [12] 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: 10.1007/s11538-005-9037-9. [13] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. [14] M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x. [15] S. Liu, S. Wang and L. Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse, Nonlinear Anal. Real World Appl., 12 (2011), 119-127. doi: 10.1016/j.nonrwa.2010.06.001. [16] 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), 2434-2448. doi: 10.1137/090779322. [17] C. C. McCluskey, Global stability of an epidemic model with delay and general nonlinear incidence, Math. Biosci. and Eng., 7 (2010), 837-850. [18] P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122. [19] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. and Eng., 6 (2009), 603-610. [20] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014. [21] C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. Real World Appl., 11 (2010), 3106-3109. doi: 10.1016/j.nonrwa.2009.11.005. [22] K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109. doi: 10.1016/j.mbs.2011.11.002. [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), 931-947. doi: 10.1016/S0362-546X(99)00138-8. [24] J. Wang, G. Huang, Y. Takeuchi and S. Liu, Sveir epidemiological model with varying infectivity and distributed delays, Math. Biosci. and Eng., 8 (2011), 875-888. [25] X. Wang, Y. D. Tao and X. Y. Song, A delayed HIV-1 infection model with Beddington-DeAngelis functional response, Nonlinear Dyn., 62 (2010), 67-72. doi: 10.1007/s11071-010-9699-1.

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##### References:
 [1] K. L. Cooke, Stability analysis for a vector disease model, Rocky Mount. J. Math., 9 (1979), 31-42. [2] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4$^+$ T cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7. [3] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [4] S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biol. Dynam., 2 (2008), 140-153. [5] H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2. [6] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [7] V. Herz, S. Bonhoeffer, R. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus delay, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251. doi: 10.1073/pnas.93.14.7247. [8] J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. [9] A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71 (2009), 75-83. doi: 10.1007/s11538-008-9352-z. [10] A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60. [11] A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and nonlinear incidence rate, Math. Med. Biol., 26 (2009), 225-239. [12] 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: 10.1007/s11538-005-9037-9. [13] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. [14] M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x. [15] S. Liu, S. Wang and L. Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse, Nonlinear Anal. Real World Appl., 12 (2011), 119-127. doi: 10.1016/j.nonrwa.2010.06.001. [16] 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), 2434-2448. doi: 10.1137/090779322. [17] C. C. McCluskey, Global stability of an epidemic model with delay and general nonlinear incidence, Math. Biosci. and Eng., 7 (2010), 837-850. [18] P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122. [19] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. and Eng., 6 (2009), 603-610. [20] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014. [21] C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. Real World Appl., 11 (2010), 3106-3109. doi: 10.1016/j.nonrwa.2009.11.005. [22] K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109. doi: 10.1016/j.mbs.2011.11.002. [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), 931-947. doi: 10.1016/S0362-546X(99)00138-8. [24] J. Wang, G. Huang, Y. Takeuchi and S. Liu, Sveir epidemiological model with varying infectivity and distributed delays, Math. Biosci. and Eng., 8 (2011), 875-888. [25] X. Wang, Y. D. Tao and X. Y. Song, A delayed HIV-1 infection model with Beddington-DeAngelis functional response, Nonlinear Dyn., 62 (2010), 67-72. doi: 10.1007/s11071-010-9699-1.
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