An SIR epidemic model with vaccination in a patchy environment

  • 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.

    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

    Related Papers:

  • 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.


    加载中
    [1] [ M. E. Alexander,C. S. Bowman,S. M. Moghadas,R. Summers,A. B. Gumel,B. M. Sahai, A vaccination model for transmission dynamics of influenza, SIAM J. Appl. Dyn. Syst., 3 (2004): 503-524.
    [2] [ J. Arino,C. C. Mccluskey,P. van den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003): 260-276.
    [3] [ J. Arino,R. Jordan,P. van den Driessche, Quarantine in a multi-species epidemics model with spatial dynamics, Math. Biosci., 206 (2007): 46-60.
    [4] [ P. Auger,E. Kouokam,G. Sallet,M. Tchuente,B. Tsanou, The Ross-Macdonald model in a patchy environment, Math. Biosci., 216 (2008): 123-131.
    [5] [ A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Philadelphia, 1994.
    [6] [ World Health Organization, Ebola response roadmap -Situation report update 3 December 2014, website: http://apps.who.int/iris/bitstream/10665/144806/1/roadmapsitrep_3Dec2014_eng.pdf
    [7] [ F. Brauer,P. van den Driessche,L. Wang, Oscillations in a patchy environment disease model, Math. Biosci., 215 (2008): 1-10.
    [8] [ C. Castillo-Chavez and H. Thieme, Asymptotically autonomous epidemic models, in O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, Springer, Berlin, 1995, 33–35.
    [9] [ O. Diekmann,J. A. P. Heesterbeek,M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2010): 873-885.
    [10] [ P. van den Driessche,J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002): 29-48.
    [11] [ M. C. Eisenberg,Z. Shuai,J. H. Tien,P. van den Driessche, A cholera model in a patchy environment with water and human movement, Math. Biosci., 246 (2013): 105-112.
    [12] [ How Many Ebola Patients Have Been Treated Outside of Africa? website: http://ritholtz.com/2014/10/how-many-ebola-patients-have-been-treated-outside-africa/.
    [13] [ D. Gao,S. Ruan, A multipathc malaria model with Logistic growth populations, SIAM J. Appl. Math., 72 (2012): 819-841.
    [14] [ D. Gao and S. Ruan, Malaria Models with Spatial Effects, Analyzing and Modeling Spatial and Temporal Dynamics of Infectious Diseases, Wiley, 2015.
    [15] [ D. Gao,S. Ruan, An SIS patch model with variable transmission coefficients, Math. Biosci., 232 (2011): 110-115.
    [16] [ H. Guo,M. Li,Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14 (2006): 259-284.
    [17] [ Vaccination, website: https://en.wikipedia.org/wiki/Vaccination.
    [18] [ Immunisation Advisory Centre, A Brief History of Vaccination, website: http://www.immune.org.nz/brief-history-vaccination.
    [19] [ K. E. Jones,N. G. Patel,M. A. Levy,A. Storeygard,D. Balk,J. L. Gittleman,P. Daszak, Global trends in emerging infectious diseases, Nature, 451 (2008): 990-993.
    [20] [ J. P. LaSalle, The Stability of Dynamical systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.
    [21] [ S. Ruan,W. Wang,S. A. Levin, The effect of global travel on the spread of SARS, Math. Biosci. Eng., 3 (2006): 205-218.
    [22] [ H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University, 1995.
    [23] [ C. Sun,W. Yang,J. Arino,K. Khan, Effect of media-induced social distancing on disease transmission in a two patch setting, Math. Biosci., 230 (2011): 87-95.
    [24] [ W. Wang,X. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004): 97-112.
    [25] [ W. Wang,X. Zhao, An epidemic model with population dispersal and infection period, SIAM J. Appl. Math., 66 (2006): 1454-1472.
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3003) PDF downloads(631) Cited by(5)

Article outline

Figures and Tables

Figures(6)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog