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. 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

show all references

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

[1]

Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119

[2]

C. Connell McCluskey. Global stability for an SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2009, 6 (3) : 603-610. doi: 10.3934/mbe.2009.6.603

[3]

Aili Wang, Yanni Xiao, Robert A. Cheke. Global dynamics of a piece-wise epidemic model with switching vaccination strategy. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2915-2940. doi: 10.3934/dcdsb.2014.19.2915

[4]

Bin Fang, Xue-Zhi Li, Maia Martcheva, Li-Ming Cai. Global stability for a heroin model with two distributed delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 715-733. doi: 10.3934/dcdsb.2014.19.715

[5]

Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang. Backward bifurcation and global stability in an epidemic model with treatment and vaccination. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999

[6]

Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643

[7]

Kai Wang, Zhidong Teng, Xueliang Zhang. Dynamical behaviors of an Echinococcosis epidemic model with distributed delays. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1425-1445. doi: 10.3934/mbe.2017074

[8]

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

[9]

Shujing Gao, Dehui Xie, Lansun Chen. Pulse vaccination strategy in a delayed sir epidemic model with vertical transmission. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 77-86. doi: 10.3934/dcdsb.2007.7.77

[10]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1083-1106. doi: 10.3934/mbe.2015.12.1083

[11]

Gergely Röst, Jianhong Wu. SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2008, 5 (2) : 389-402. doi: 10.3934/mbe.2008.5.389

[12]

Shengqiang Liu, Lin Wang. Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy. Mathematical Biosciences & Engineering, 2010, 7 (3) : 675-685. doi: 10.3934/mbe.2010.7.675

[13]

Zhen Jin, Zhien Ma. The stability of an SIR epidemic model with time delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 101-109. doi: 10.3934/mbe.2006.3.101

[14]

Ábel Garab, Veronika Kovács, Tibor Krisztin. Global stability of a price model with multiple delays. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6855-6871. doi: 10.3934/dcds.2016098

[15]

Yali Yang, Sanyi Tang, Xiaohong Ren, Huiwen Zhao, Chenping Guo. Global stability and optimal control for a tuberculosis model with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 1009-1022. doi: 10.3934/dcdsb.2016.21.1009

[16]

Wei Feng, Xin Lu. Global stability in a class of reaction-diffusion systems with time-varying delays. Conference Publications, 1998, 1998 (Special) : 253-261. doi: 10.3934/proc.1998.1998.253

[17]

Emad Attia, Marek Bodnar, Urszula Foryś. Angiogenesis model with Erlang distributed delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 1-15. doi: 10.3934/mbe.2017001

[18]

Yasuhisa Saito. A global stability result for an N-species Lotka-Volterra food chain system with distributed time delays. Conference Publications, 2003, 2003 (Special) : 771-777. doi: 10.3934/proc.2003.2003.771

[19]

Islam A. Moneim, David Greenhalgh. Use Of A Periodic Vaccination Strategy To Control The Spread Of Epidemics With Seasonally Varying Contact Rate. Mathematical Biosciences & Engineering, 2005, 2 (3) : 591-611. doi: 10.3934/mbe.2005.2.591

[20]

Zhaohui Yuan, Xingfu Zou. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. Mathematical Biosciences & Engineering, 2013, 10 (2) : 483-498. doi: 10.3934/mbe.2013.10.483

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

[Back to Top]