# 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
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