Research ArticleNo Access

Study on a susceptible–infected–vaccinated model with delay and proportional vaccination

College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, P. R. China

and

College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, P. R. China

Key Laboratory of Operations Research and Control of Universities in Fujian, Fuzhou University, Fuzhou 350116, P. R. China

E-mail Address: weifengying@fzu.edu.cn

Corresponding author.

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https://doi.org/10.1142/S1793524518501024Cited by:4 (Source: Crossref)

Abstract

A susceptible–infected–vaccinated epidemic model with proportional vaccination and generalized nonlinear rate is formulated and investigated in the paper. We show that the stochastic epidemic model admits a unique and global positive solution with probability one when constructing a proper $C^{2}$ -function therewith. Then a sufficient condition that guarantees the disappearances of diseases is derived when the indicator $R_{0} < 1$ . Further, if $R_{0} > 1$ , then we obtain that the solution is weakly permanent with probability one. We also derived the sufficient conditions of the persistence in the mean for the susceptible and infected when another indicator ${\tilde{R}}_{0} > 1$ .

Keywords:

AMSC: 60H10, 34F05, 93E03

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References

1. G. O. Agaba, Y. N. Kyrychko and K. B. Blyuss, Dynamics of vaccination in a time-delayed epidemic model with awareness, Math. Biosci. 294 (2017) 92–99. Web of Science, Google Scholar
2. B. Cao, M. Shan, Q. Zhang et al., A stochastic SIS epidemic model with vaccination, Physica A 486 (2017) 127–143. Web of Science, Google Scholar
3. L. Chen and F. Wei, Persistence and distribution of a stochastic susceptible–infected–recovered epidemic model with varying population size, Physica A 483 (2017) 386–397. Web of Science, Google Scholar
4. K. Fan, Y. Zhang, S. Gao et al., A class of stochastic delayed SIR epidemic models with generalized nonlinear incidence rate and temporary immunity, Physica A 481 (2017) 198–208. Web of Science, Google Scholar
5. A. Gray, D. Greenhalgh, L. Hu et al., A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71 (2011) 876–902. Web of Science, Google Scholar
6. D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43(3) (2001) 525–546. Web of Science, Google Scholar
7. W. O. Kermack and A. G. McKendric, A contribution to the mathematical theory of epidemics. 1, Proc. R. Soc. London A 115 (1927) 700–721. Google Scholar
8. Q. Liu, Q. Chen and D. Jiang, The threshold of a stochastic delayed SIR epidemic model with temporary immunity, Physica A 450 (2016) 115–125. Web of Science, Google Scholar
9. Q. Liu, D. Jiang, N. Shi et al., Asymptotic behaviors of a stochastic delayed SIR epidemic model with nonlinear incidence, Commun. Nonlinear Sci. Numer. Simul. 40 (2016) 89–99. Web of Science, Google Scholar
10. L. M. Stolerman, D. Coombs and S. Boatto, SIR-network model and its application to Dengue fever, SIAM J. Appl. Math. 75 (2015) 2581–2609. Web of Science, Google Scholar
11. Z. Teng and L. Wang, Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate, Physica A 451 (2016) 507–518. Web of Science, Google Scholar
12. F. Xu and R. Cressman, Voluntary vaccination strategy and the spread of sexually transmitted diseases, Math. Biosci. 274 (2016) 94–107. Web of Science, Google Scholar
13. Y. Zhang, Y. Li, Q. Zhang et al., Behavior of a stochastic SIR epidemic model with saturated incidence and vaccination rules, Physica A 501 (2018) 178–187. Web of Science, Google Scholar
14. D. Zhao, T. Zhang and S. Yuan, The threshold of a stochastic SIVS epidemic model with nonlinear saturated incidence, Physica A 443 (2016) 372–379. Web of Science, Google Scholar