Sveir epidemiological model with varying infectivity and distributed delays

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

Sveir epidemiological model with varying infectivity and distributed delays

Jinliang Wang ^1, , Gang Huang ^2, , Yasuhiro Takeuchi ^3, and Shengqiang Liu ^4,

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

Abstract
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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.

Keywords: global stability., vaccination strategy, distributed delays, varying infectivity, Epidemic model.

Mathematics Subject Classification: Primary: 34K20, 92D3.

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11. Google Scholar
[6]	Y. Hino, S. Murakami and T. Naito, "Functional-Differential Equations with Infinite Delay,", Lecture Notes in Mathematics, 1473 (1991). Google Scholar
[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. Google Scholar
[8]	J. R. Haddock, T. Krisztin and J. Terjéki, Invariance principles for autonomous functional-differential equations,, J. Integral equations, 10 (1985), 123. 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: 10.1007/s11538-009-9487-6. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, 191 (1993). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 5 (2008), 389. Google Scholar
[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. Google Scholar

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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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11. Google Scholar
[6]	Y. Hino, S. Murakami and T. Naito, "Functional-Differential Equations with Infinite Delay,", Lecture Notes in Mathematics, 1473 (1991). Google Scholar
[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. Google Scholar
[8]	J. R. Haddock, T. Krisztin and J. Terjéki, Invariance principles for autonomous functional-differential equations,, J. Integral equations, 10 (1985), 123. 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: 10.1007/s11538-009-9487-6. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, 191 (1993). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 5 (2008), 389. Google Scholar
[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. Google Scholar

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