# American Institute of Mathematical Sciences

2011, 8(3): 875-888. doi: 10.3934/mbe.2011.8.875

## Sveir epidemiological model with varying infectivity and distributed delays

 1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China 2 Graduate School of Science and Technology, Shizuoka University, Hamamatsu 4328561, Japan 3 Graduate School of Science and Technology, Faculty of Engineering, Shizuoka University, Hamamatsu 432-8561, Japan 4 Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin, 150080

Received  June 2010 Revised  December 2010 Published  June 2011

In this paper, based on an SEIR epidemiological model with distributed delays to account for varying infectivity, we introduce a vaccination compartment, leading to an SVEIR model. By employing direct Lyapunov method and LaSalle's invariance principle, we construct appropriate functionals that integrate over past states to establish global asymptotic stability conditions, which are completely determined by the basic reproduction number $\mathcal{R}_0^V$. More precisely, it is shown that, if $\mathcal{R}_0^V\leq 1$, then the disease free equilibrium is globally asymptotically stable; if $\mathcal{R}_0^V > 1$, then there exists a unique endemic equilibrium which is globally asymptotically stable. Mathematical results suggest that vaccination is helpful for disease control by decreasing the basic reproduction number. However, there is a necessary condition for successful elimination of disease. If the time for the vaccinees to obtain immunity or the possibility for them to be infected before acquiring immunity can be neglected, this condition would be satisfied and the disease can always be eradicated by some suitable vaccination strategies. This may lead to over-evaluating the effect of vaccination.
Citation: Jinliang Wang, Gang Huang, Yasuhiro Takeuchi, Shengqiang Liu. Sveir epidemiological model with varying infectivity and distributed delays. Mathematical Biosciences & Engineering, 2011, 8 (3) : 875-888. doi: 10.3934/mbe.2011.8.875
##### References:
 [1] F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations,, Funkcial. Ekvac., 31 (1988), 331. [2] E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay,, Nonlinear Anal., 47 (2001), 4107. doi: 10.1016/S0362-546X(01)00528-4. [3] A. B. Gumel, C. C. MuCluskey and J. Watmough, An SVEIR model for assessing potential impact of an imperfect anti-SARS vaccine,, Math. Biosci. Eng., 3 (2006), 485. [4] A. Gabbuti, L. Romano, P. Blanc, et al., Long-term immunogenicity of hepatitis B vaccination in a cohort of Italian healthy adolescents,, Vaccine, 25 (2007), 3129. doi: 10.1016/j.vaccine.2007.01.045. [5] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11. [6] Y. Hino, S. Murakami and T. Naito, "Functional-Differential Equations with Infinite Delay,", Lecture Notes in Mathematics, 1473 (1991). [7] J. R. Haddock and J. Terjéki, Liapunov-Razumikhin functions and an invariance principle for functional-differential equations,, J. Differential Equations, 48 (1983), 95. doi: 10.1016/0022-0396(83)90061-X. [8] J. R. Haddock, T. Krisztin and J. Terjéki, Invariance principles for autonomous functional-differential equations,, J. Integral equations, 10 (1985), 123. [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: 10.1007/s11538-009-9487-6. [10] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections,, SIAM J. Appl. Math., 70 (2010), 2693. doi: 10.1137/090780821. [11] 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. [12] A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871. doi: 10.1007/s11538-007-9196-y. [13] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, 191 (1993). [14] G. Li and Z. Jin, Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period,, Chaos Solitons Fractals, 25 (2005), 1177. doi: 10.1016/j.chaos.2004.11.062. [15] M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size,, Math. Biosci., 160 (1999), 191. doi: 10.1016/S0025-5564(99)00030-9. [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. doi: 10.1137/090779322. [17] M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology,, Math. Biosci., 125 (1995), 155. doi: 10.1016/0025-5564(95)92756-5. [18] X. Liu, Y. Takeuchi and S. Iwami, SVIR epidemic models with vaccination strategies,, J. Theo. Biol., 253 (2008), 1. doi: 10.1016/j.jtbi.2007.10.014. [19] S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy,, Math. Biosci. Eng., 7 (2010), 675. doi: 10.3934/mbe.2010.7.675. [20] P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Applicable Analysis, 89 (2010), 1109. doi: 10.1080/00036810903208122. [21] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603. doi: 10.3934/mbe.2009.6.603. [22] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete,, Nonlinear Anal. RWA., 11 (2010), 55. doi: 10.1016/j.nonrwa.2008.10.014. [23] G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 5 (2008), 389. [24] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6.

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##### References:
 [1] F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations,, Funkcial. Ekvac., 31 (1988), 331. [2] E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay,, Nonlinear Anal., 47 (2001), 4107. doi: 10.1016/S0362-546X(01)00528-4. [3] A. B. Gumel, C. C. MuCluskey and J. Watmough, An SVEIR model for assessing potential impact of an imperfect anti-SARS vaccine,, Math. Biosci. Eng., 3 (2006), 485. [4] A. Gabbuti, L. Romano, P. Blanc, et al., Long-term immunogenicity of hepatitis B vaccination in a cohort of Italian healthy adolescents,, Vaccine, 25 (2007), 3129. doi: 10.1016/j.vaccine.2007.01.045. [5] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11. [6] Y. Hino, S. Murakami and T. Naito, "Functional-Differential Equations with Infinite Delay,", Lecture Notes in Mathematics, 1473 (1991). [7] J. R. Haddock and J. Terjéki, Liapunov-Razumikhin functions and an invariance principle for functional-differential equations,, J. Differential Equations, 48 (1983), 95. doi: 10.1016/0022-0396(83)90061-X. [8] J. R. Haddock, T. Krisztin and J. Terjéki, Invariance principles for autonomous functional-differential equations,, J. Integral equations, 10 (1985), 123. [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: 10.1007/s11538-009-9487-6. [10] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections,, SIAM J. Appl. Math., 70 (2010), 2693. doi: 10.1137/090780821. [11] 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. [12] A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871. doi: 10.1007/s11538-007-9196-y. [13] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, 191 (1993). [14] G. Li and Z. Jin, Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period,, Chaos Solitons Fractals, 25 (2005), 1177. doi: 10.1016/j.chaos.2004.11.062. [15] M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size,, Math. Biosci., 160 (1999), 191. doi: 10.1016/S0025-5564(99)00030-9. [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. doi: 10.1137/090779322. [17] M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology,, Math. Biosci., 125 (1995), 155. doi: 10.1016/0025-5564(95)92756-5. [18] X. Liu, Y. Takeuchi and S. Iwami, SVIR epidemic models with vaccination strategies,, J. Theo. Biol., 253 (2008), 1. doi: 10.1016/j.jtbi.2007.10.014. [19] S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy,, Math. Biosci. Eng., 7 (2010), 675. doi: 10.3934/mbe.2010.7.675. [20] P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Applicable Analysis, 89 (2010), 1109. doi: 10.1080/00036810903208122. [21] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603. doi: 10.3934/mbe.2009.6.603. [22] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete,, Nonlinear Anal. RWA., 11 (2010), 55. doi: 10.1016/j.nonrwa.2008.10.014. [23] G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 5 (2008), 389. [24] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6.
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