
Mathematical Biosciences and Engineering, 2017, 14(5&6): 11411157. doi: 10.3934/mbe.2017059
Export file:
Format
 RIS(for EndNote,Reference Manager,ProCite)
 BibTex
 Text
Content
 Citation Only
 Citation and Abstract
An SIR epidemic model with vaccination in a patchy environment
. School of Science, Nanjing University of Science and Technology, Nanjing 210094, China
Received: , Accepted: , Published:
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 diseasefree 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.
References
[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): 503524.
[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): 260276.
[3] J. Arino,R. Jordan,P. van den Driessche, Quarantine in a multispecies epidemics model with spatial dynamics, Math. Biosci., 206 (2007): 4660.
[4] P. Auger,E. Kouokam,G. Sallet,M. Tchuente,B. Tsanou, The RossMacdonald model in a patchy environment, Math. Biosci., 216 (2008): 123131.
[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): 110.
[8] C. CastilloChavez 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 nextgeneration matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2010): 873885.
[10] P. van den Driessche,J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002): 2948.
[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): 105112.
[12] How Many Ebola Patients Have Been Treated Outside of Africa? website: http://ritholtz.com/2014/10/howmanyebolapatientshavebeentreatedoutsideafrica/.
[13] D. Gao,S. Ruan, A multipathc malaria model with Logistic growth populations, SIAM J. Appl. Math., 72 (2012): 819841.
[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): 110115.
[16] H. Guo,M. Li,Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14 (2006): 259284.
[17] Vaccination, website: https://en.wikipedia.org/wiki/Vaccination.
[18] Immunisation Advisory Centre, A Brief History of Vaccination, website: http://www.immune.org.nz/briefhistoryvaccination.
[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): 990993.
[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): 205218.
[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 mediainduced social distancing on disease transmission in a two patch setting, Math. Biosci., 230 (2011): 8795.
[24] W. Wang,X. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004): 97112.
[25] W. Wang,X. Zhao, An epidemic model with population dispersal and infection period, SIAM J. Appl. Math., 66 (2006): 14541472.
Copyright Info: © 2017, Zhipeng Qiu, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)