An SIR epidemic model with vaccination in a patchy environment

Qianqian Cui; Zhipeng Qiu; Ling Ding; Qianqian Cui; Zhipeng Qiu; Ling Ding

doi:10.3934/mbe.2017059

Mathematical Biosciences and Engineering

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

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An SIR epidemic model with vaccination in a patchy environment

School of Science, Nanjing University of Science and Technology, Nanjing 210094, China

Received: 01 July 2016 Accepted: 01 November 2016 Published: 01 October 2017
MSC : Primary: 34D23, 37N25; Secondary: 92B05

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 $\mathfrak{R}_v \lt 1$ , and unstable if $\mathfrak{R}_v \gt 1$ . 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.
- SIR model,
- vaccination,
- patchy environment,
- transmission dynamics,
- reproduction number
Citation: Qianqian Cui, Zhipeng Qiu, Ling Ding. An SIR epidemic model with vaccination in a patchy environment[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1141-1157. doi: 10.3934/mbe.2017059

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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 $\mathfrak{R}_v \lt 1$ , and unstable if $\mathfrak{R}_v \gt 1$ . 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.

References

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