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Mathematical Biosciences and Engineering (MBE)
 

An SIR epidemic model with vaccination in a patchy environment

Pages: 1141 - 1157, Volume 14, Issue 5/6, October/December 2017      doi:10.3934/mbe.2017059

 
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Qianqian Cui - School of Science, Nanjing University of Science and Technology, Nanjing 210094, China (email)
Zhipeng Qiu - School of Science, Nanjing University of Science and Technology, Nanjing 210094, China (email)
Ling Ding - School of Science, Nanjing University of Science and Technology, Nanjing 210094, China (email)

Abstract: In this paper, an SIR patch model with vaccination is formulated to investigate the effect of vaccination coverage and the impact of human mobility on the spread of diseases among patches. The control reproduction number $\mathfrak{R}_v$ is derived. It shows that the disease-free equilibrium is unique and is globally asymptotically stable if $\mathfrak{R}_v<1$, and unstable if $\mathfrak{R}_v>1$. The sufficient condition for the local stability of boundary equilibria and the existence of equilibria are obtained for the case $n=2$. Numerical simulations indicate that vaccines can control and prevent the outbreak of infectious in all patches while migration may magnify the outbreak size in one patch and shrink the outbreak size in other patch.

Keywords:  SIR model, vaccination, patchy environment, transmission dynamics, reproduction number.
Mathematics Subject Classification:  Primary: 34D23, 37N25; Secondary: 92B05.

Received: July 2016;      Accepted: November 2016;      Available Online: May 2017.

 References